Classification, Delineation 

and Measurement of 

Nonparallel Folds

GEOLOGICAL SURVEY PROFESSIONAL PAPER 314-E

Classification, Delineation 

and Measurement of 

Nonparallel Folds

By JOHN B. MERTIE, Jr.

CONTRIBUTIONS TO GENERAL GEOLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 314-E

The thesis of this paper is that the stratigraphic 

traces of most nonpar allel folds ^ in sections selected 

to show the maximum or minimum curvature^ may 

be represented approximately by one or more families 

of curves that are analytically related

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1959
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

The U. S. Geological Survey Library has cataloged this publication as follows:

Mertie, John Beaver, Jr., 1888-

Classification, delineation and measurement of nonparallel 

folds. Washington, U. S. Govt Print. Off., 1959.

iii, 91-124 p. diagrs. 29 cm. (U. S. Geological Survey. Professional paper 314-E. Contributions to general geology)

"Literature cited": p. 121.

1. Folds (Geology) 2. Geology, Stratigraphic. i. Title. (Series: U. S. Geological Survey. Professional paper 314-E. Series: 

U. S. Geological Survey. Shorter contributions to general geology)

551.87

For sale by the Superintendent of Documents, U. S. Government Printing Office 

Washington 25, D. C. - Price 30 cents
CONTENTS

Page

Abstract_ ________________________________________ 91

Introduction. _-_____--___--_-______._-______-____-_ 91

Plane curves______---_-----_____-_--______-___-__ 92

Classification of folds________________________________ 95

Cylindrical folds.-___-___-_-__-__________-______-__- 97

Parallel folds_-_---__-_----_____--__._-____--_ 97

Similar folds.__________________________________ 99

Cognate folds._________________________________ 103

Composite folds. _-----_--__-_-____-____________ 105

Summary. _____________________________________ 106

Page

Noncylindrical folds_______________________________ 107

Parallel domed surfaces________________________ 109

Similar domes surfaces._________________________ 110

Cognate domed surfaces.________________________ 111

Composite domed surfaces_______________________ 111

Construction and application of ellipses_____________ 112

Stratigraphic thickness--_____---_-_-__-------------_ 116

Analysis of folds____________________________________ 118

Literature cited___________________________________ 121

Index.._______________________________________ 123

ILLUSTRATIONS

FIGURE 16. Central semiellipses, congruent hybrid elliptic 

arcs, and prolate and oblate similar semiellipses exemplifying certain types of 

structural features _____________________

17. Prolate and oblate similar elliptic arcs and 

similar semiellipses exemplifying certain 

types of structural features____________

18. Prolate similar semiellipses and elliptic arcs 

and prolate cognate and oblate cognate 

semiellipses exemplifying certain types of 

structural features _____________________

19. Prolate cognate and oblate cognate semiellipses and prolate cognate elliptic arcs 

exemplifying certain types of structural 

features ______________________________

20. Prolate cognate semiellipses and elliptic arcs 

and oblate cognate elliptic arcs exemplifying certain types of structural features. _

Page

94

98

100

102

104

Page

21. Oblate similar quarter ellipses, oblate similar 

and prolate cognate elliptic arcs, sections 

of a composite elongate dome, and application of trammel of Archimedes in construction of ellipses_______________________ 108

22. Nineteen prolate semiellipses with eccentricities ranging from fc=0.312 to fc = 0.999_ 113

23. Nineteen oblate semiellipses with eccentricities ranging from k=0.312 to k=0.999_ 114

24. Transverse geologic profile of a cognate syncline reconstructed by the use of elliptic 

arcs_______________________________ 115

25. Diagram showing measurement of the area

between two noncoaxial elliptic arcs____ 117

26. Diagram showing measurement of the length

of an elliptic arc_______________________ 118

27. Diagram illustrating angle of contingence 

(x), length of arc (s), and radius of curvature (#)--____---_--_----___________ 120

in

SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS

By JOHN B. MERTIE, JR.

ABSTRACT

Simple folds are divided primarily into two classes, cylindrical and noncylindrical. The true nature of the stratigraphic 

surfaces of all folds is unknown, but it is the thesis of this 

paper that they may be described approximately as cylinders 

and quadrics. The traces of the stratigraphic surfaces are 

likewise unknown, but they are represented empirically in 

selected profiles by algebraic curves. Cylindrical and noncylindrical folds are divided into four genera that depend upon 

the character of their stratigraphic traces. These are designated as parallel, similar, cognate, and composite folds.

All cylindrical folds have axial lines. They are subdivided 

into species according to whether the fold has an axial plane 

or a curved axial surface. Cylindrical parallel folds are unique 

in that they necessarily have axial planes. Cylindrical similar 

folds are most simply defined as those whose traces, in a plane 

normal to the axial line, are curves that are reproducible from 

one another by nondistortional enlargement or reduction. Similar traces are illustrated by elliptic arcs of the same eccentricity, at the same or different scales. Cylindrical cognate 

folds are defined as those whose traces, in the selected profile, 

are nonparallel nonsimilar curves that are analytically related 

but differ in the values of their assigned parameters. Cognate 

traces are represented by ellipses with different eccentricities. 

Cylindrical composite folds are of three kinds. One has related but mixed stratigraphic traces; a second consists of hybrid 

traces, illustrated by quarter ellipses, either similarly placed or 

rotated 90° to one another, that are joined where their tangents 

are parallel; and a third has analytically unrelated traces.

Noncylindrical folds are defined to include quaquaversal folds, 

elongate or canoe-shaped domes, and the plunging ends of cylindrical folds. All such structures are characterized by an absence of axial lines, though they have analogous indices that 

are described as apical lines. Canoe-shaped folds have principal sections that are comparable to the axial planes and 

surfaces of cylindrical folds and to the cross sections normal 

thereto. The apical lines are either plane or space curves; the 

principal sections may be either plane or curved surfaces. 

Plunging folds have some of the characteristics of canoe-shaped 

folds. The stratigraphic surfaces of noncylindrical folds are 

illustrated by spheroids, ellipsoids, and modifications thereof. 

The traces of the stratigraphic surfaces in the principal sections 

are analogous to those in the selected profiles of cylindrical 

folds.

Parallel folds, both cylindrical and noncylindrical, are represented in cross section by higher plane curves that may be 

precisely generated by means of evolutes and involutes. All 

nonparallel folds, however, must be shown approximately in 

cross section by empirical curves. Ellipses were chosen for

this purpose, first, because they are simple curves of the second 

degree; second, because their eccentricities are variable; third, 

because the arcs of an ellipse at the ends of its major and 

minor axes are parallel, and this permits the delineation of 

dips of 90° at the base of a fold; and finally, because this 

parallelism facilitates the construction of hybridized traces. 

The empirical curves thus generated by the use of ellipses are 

extensively modified by variations in the eccentricity, by change 

of scale, and by several types of linear translation, to simulate 

the various kinds of nonparallel folds that are required to fit 

the geological data.

Many ellipses of different eccentricity and scale are needed 

in the empirical representation of nonparallel folds. These 

are generated as glissettes, by application of the trammel of 

Archimedes, the construction and use of which are described. 

As a further aid, 38 prolate and oblate ellipses of different eccentricity are illustrated.

Stratigraphic thickness, in the selected profiles, is defined as 

the area between two elliptic tracer divided by the length of a 

medial elliptic arc. The area between the elliptic arcs is obtained either by graphical integration or by formula. The 

length of an elliptic arc requires an evaluation of Legendre's 

E function, as given in a table of elliptic functions. Graphic 

linear integration may also be used. As an aid in this work, 

the lengths of 99 semiellipses, with major semiaxes of 10, and 

minor semiaxes ranging from 9.9 to 0.1, have been computed 

and tabulated. This table, with a trivial amount of interpolation, gives the semiperimeters of all ellipses, regardless of the 

lengths of their semiaxes. An empirical formula for the length 

of an ellipse is also presented.

Methods for the mathematical analysis of the traces of cylindrical and noncylindrical folds are not treated at length in this 

paper, but the value of such work is stressed. Some elementary 

methods are outlined, and attention is called to the applicability 

of intrinsic equations for the simplification of such work.

INTRODUCTION

The surfaces of folds are not simple geometrical 

figures, but they do correspond approximately, within 

certain limits, to recognizable geometric patterns, such 

that they may be studied, described, and measured in 

simplified geometrical terms, with the understood proviso that all results are approximations. Most field 

geologists appreciate these limitations. The terms 

"parallel" and "similar" folds and folding are those 

most commonly used in geologic descriptions; but these 

terms are so loosely applied that it behooves a writer

91
92 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

on such topics to define accurately, or as accurately as 

the facts permit, the meaning of his nomenclature. 

Folds whose stratigraphic surfaces approach parallelism are the most amenable to exact delineation and 

measurement. Nonparallel folds, of which similar 

folds will be shown to be merely one genus, are much 

more difficult to analyze and delineate.

Parallel folds have been so systemized and conventionalized by the method of evolute and involutes that 

any group of geologists, applying this method to the 

same structural data, will reconstruct identical geologic sections. Likewise, different geologists will obtain the same values for the stratigraphic thickness, 

depth or distance to a stratum, and other stratigraphic 

dimensions. It would be desirable if such concordant 

results could be obtained for nonparallel folds; but 

this is impossible because, unlike the involutes that 

represent the traces of parallel stratigraphic surfaces, 

the geometrical character of nonparallel traces is unknown. Such traces, however, may be conventionalized and represented approximately in accordance with 

the structural data in possession of the geologist. For 

reasons later to be stated, this empirical representation 

is attempted by the use of elliptic arcs that are so selected as to match approximately the dips in a profile 

and the geologist's interpretation of such dips. Nonparallel folds that are Reconstructed in this manner 

should be easier to draw and should be more comparable with one another; and as the traces are drawn as 

definite curves, it should be easier to compute stratigraphic thickness and related measurements. More 

precise analytical methods could also be used advantageously in the classification of folds and in studies pertaining to the mechanics of folding, but such work has 

not yet been attempted. A few elementary methods, 

however, are suggested on pages 118-121 of this paper.

The following definitions and inferences are meant 

to apply to simple arches, cylindrical or noncylindrical, 

that may be upright, tilted, recumbent, or reversed, 

with or without axial lines or axial planes. A simple 

arch is defined as a fold whose stratigraphic surfaces 

attain either a maximum or a minimum curvature along 

a line or at a point but have no points or zones at 

which discontinuities, singularities, or inflections occur. The stratigraphic traces of a simple arch, in a 

section selected to show maximum or minimum curvature, are continuous curves without discontinuities, 

singularities, or points of the first or higher orders of 

inflection. Stated in this manner, these definitions include tilted and recumbent folds without reference to 

any derived curves and without recourse to any transformation of coordinates. A generalization of the definition of a simple arch, to include surfaces with zones

of inflection, would permit the inclusion of monoclinal 

folds. By combining numbers of such simple arches, 

more complex folds may be simulated.

PLANE CURVES

The thesis of this paper is that the stratigraphic 

traces of most nonparallel folds, in sections selected 

to show the maximum or minimum curvature, may be 

represented approximately by one or more families of 

curves that are analytically related. The selection of 

a suitable family of curves for a particular class or 

genus of folds is largely a matter of convenience, as 

the true curvature of folds has not been investigated. 

Attention naturally centers on the conic sections, as 

these are simple curves that are easy to construct.

The use of circular arcs is both unrealistic and impracticable. The parabola is unsuitable for two reasons. Its eccentricity is constant, so that it is impossible to show two or more conic traces with different 

eccentricities; and second, its limbs approach parallelism only at infinity, so that it is impossible to show 

dips of 90° on both flanks of an upright symmetrical 

structure. The hyperbola meets the first objection but 

is vulnerable to the second, as its limbs nowhere approach parallelism. The ellipse, however, meets the 

necessary specifications, as its eccentricity is variable 

and its arcs are parallel at the ends of the major and 

minor axes. These properties lend variability to the 

delineation of stratigraphic traces, permit the desired 

charting of dips of 90° on both flanks of certain folds, 

and also facilitate the hybridization of the traces. The 

ellipse has therefore been selected to illustrate theoretically the types of folds that may exist and to reconstruct empirically the actual folds that occur in 

nature.

The terms "parallel," "similar," "cognate," and 

"composite," applied in this paper to the traces and 

surfaces of folds, must be defined. With the exception of concentric circles, no two conies of the same 

species may be drawn parallel to one another. For 

example, it is impossible to construct two parallel ellipses, though a series of curves of another family may 

be drawn parallel to an ellipse and to one another, 

according to Leibnitz's definition of parallelism. This 

construction is shown in figure 16J., where 3 curves are 

drawn parallel to a central ellipse with an eccentricity 

of k=0.800. These 3 parallel curves have an algebraic equation of the eighth degree with 4 constants, 

whereas the central ellipse is a curve of the second degree with only 2 essential constants. It is generally 

true that parallel curves, excepting concentric circles, 

have rather involved equations, so that parallel traces 

and parallel surfaces are best treated geometrically.
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 98

For practical purposes, however, certain ellipses may 

be constructed that simulate parallelism, as shown in 

figure 19(7. Strange though it may seem, parallel 

curves may intersect one another, or may actually be 

self-intersecting. Both these conditions are illustrated 

in figure 16Z?, where one curve is drawn parallel to and 

outside an ellipse with an eccentricity of k=0.923; and 

two other intersecting and self-intersecting curves are 

drawn parallel to and inside the same ellipse. Necessarily, such intersecting curves are excluded in the consideration of parallel stratigraphic traces; and only 

involutes are used, which conform with the intuitive 

concept of parallelism.

Similar curves may be denned either analytically or 

geometrically. In general, similar curves are produced 

if the original curve is subjected to a transformation of 

similitude, as if the plane containing the curve were an 

elastic membrane that was stretched uniformly in all 

directions. Referring to the general equation of the

conic 

=Q

two conies are similar if 

&-4AC

and they are both similar and similarly placed ifA__B__C_ A'~B'~C'

(i)

(2)

(3)

Moreover, the expressions B2 4AC and A + C are invariant under a rotation of axes, or under a change in 

origin. Therefore, similarity depends only upon the 

constants of the terms of the second degree. In general, families of ellipses and families of hyperbolas are 

not similar, but as B2 4AC vanishes for every parabola, equation 2 likewise vanishes, and it follows that 

all parabolas are similar. The central equations for 

the circle and the equilateral hyperbola, respectively 

a?2±2/2 =#2 , have one constant that obviously affects 

both variables equally and thus meet the condition of 

similarity. Ellipses and hyperbolas, with central

equations, respectively of ^±|j=l, may be similar

only if their major and minor semiaxes are identical, 

or if their ratio a: b is constant.

The geometric definition of similarity is simpler. 

Two curves are similar and similarly placed if radii 

vectors drawn to the first from some selected point are 

in constant ratio to parallel radii vectors drawn to the 

second from some other point. If two such points exist, an unlimited number of others can be found. More 

simply stated, similar curves may be produced from 

one another by uniform enlargement or reduction; and

congruent or identical curves are a special type of similar curve, where the enlargement or reduction is unity. 

Therefore the transformation of one curve to a similar 

curve alters the sizes of the figures but preserves their 

shapes.

Circles, parabolas, and equilateral hyperbolas are 

similar conies because they can be transformed by uniform enlargement or reduction without change of 

shape. Two ellipses or two hyperbolas, however, are 

similar only if they are congruent or if they have the 

same eccentricity. Similar conies are similarly placed 

if they have the same or parallel axes. In general, all 

parallel plane sections of a quadric surface are similar 

conies.

A term is needed to describe two or more dissimilar 

curves that are nonparallel, noncongruent, and nonsimilar, but are analytically related by having the same 

general equation. A set of ellipses with different eccentricities or a family of parabolic catenaries will illustrate the type. For curves of this kind the writer 

proposes and in this paper utilizes the designation cognate curves, with special application to ellipses. The 

adjective cognate, thus applied, has no recognized 

mathematical acceptance but is used merely for descriptive convenience.

Composite curves are also utilized. These are produced by joining arcs of ellipses with the same or different eccentricities at points where their tangents are 

parallel. The principal application is in the construction of the so-called hybrid folds and their traces. 

Thus, two quarter ellipses, with the same or different 

eccentricities, may be joined at the ends of their major 

or minor axes or at other points where their tangents 

are parallel. The same operation may also be performed after one quarter ellipse has been revolved 90° 

with regard to another. Two such quarter eljlipses, 

after they are joined, constitute a composite curtfe that 

is no longer of the second degree but instead is a $igher 

plane curve of unknown character. These hybrid 

curves cannot be represented exactly by any analytical 

expression but are known as piecewise functions that 

may be characterized within 1 range by 1 equation and 

within another range by a different equation. Two 

joined quarter ellipses are thus represented piece Wise by 

two elliptic equations having different and possibly reversed constant parameters. They might be described, 

however, as closely as desired, by some empirical equation. Hybrid curves are illustrated in this paper by 

arcs of ellipses, but they may also be constructed from 

other curves. Cassinian ovals, for example, could be 

similarly utilized.

The parallel, similar, cognate, and composite curves 

discussed in this paper are similarly placed, though
94 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 95

some are afterwards rotated; and initially they are constructed with collinear major and minor axes, in vertical or horizontal positions. Initially they are also 

equally spaced, representing the traces of strata of the 

same thickness, though in actual usage they will be 

unevenly spaced to represent beds of different thickness. Linear translation from their original positions, 

parallel to either or both axes, results in the production of curves that are partly or wholly noncollinear. 

Such translations are used to delineate strata that 

thicken, thin, or disappear entirely, either at the apices 

or along the flanks of folds, and also to illustrate folds 

with inclined axial planes or curved axial surfaces. 

The pinching out, or complete disappearance, of strata, 

either at the apex or along the flanks of a fold, implies 

the tangency of adjacent stratigraphic traces in some 

part or parts of the fold. The types of folds in which 

this structural feature is possible are later described.

All the folds illustrated in the following pages are 

not claimed to exist in nature. Some freakish profiles 

are shown merely to demonstrate the versatility of the 

method of empirical charting by the use of elliptic 

arcs.

CLASSIFICATION OF FOLDS

Simple folds may be divided into two principal 

classes, cylindrical and noncylindrical. The existence 

of flexures that are essentially cylindrical in character 

is so commonplace in regions of geosynclinal orogeny 

as to require no special proof. Noncylindrical folds 

that are essentially cupolas or domes with approximately a circular plan are likewise generally recognized under the designation of quaquaversal folds. 

These two classes, however, do not exhaust the field but 

instead may be regarded as the end members of a much 

larger group of structures, to which no suitable name 

can be applied. Quaquaversal folds, for example, 

grade into domes that have a pronounced elongation, 

designated in this paper as canoe-shaped folds or elongate domes. Cylindrical folds, on the other hand, become noncylindrical as they plunge, provided the 

plunge has not been caused by tilting subsequent to the 

folding. Such plunging folds are intermediate in 

character between cylindrical and canoe-shaped folds 

and have some of the properties of both.

The axial line is the most fundamental of the various 

stratigraphic dimensions, because most of the others

depend directly or indirectly upon it; and any classification of folds will depend upon the presence or absence of axial lines. An unequivocal definition is therefore needed. On every stratigraphic surface of a cylindrical fold, there exists a right line along which the 

curvature of the surface attains either a maximum or 

a minimum value, depending, respectively, on whether 

the cross section of the fold is a prolate or an oblate 

figure. Many stratigraphic surfaces, however, exist in 

a cylindrical fold, and therefore many axial lines are 

present; but these are parallel and may be regarded 

vectorially as the single axial line of the fold. Thus 

the axial line has the dimensional properties of a lineation, with a direction and possibly a plunge but without spatial position. It should be emphasized that the 

axial line is defined in terms of a cylindrical fold, and 

the inference follows that axial lines are absent in noncylindrical folds. Curved lines that lie along the crests 

of noncylindrical stratigraphic surfaces are analogous 

to axial lines and are here designated as apical lines. 

These may be plane or space curves and may or may 

not be parallel, though ordinarily they are nonparallel.

The axial plane is a property of cylindrical folds 

that must be defined in terms of the axial line. If a 

set of axial lines is parallel and coplanar, they lie in 

and define an axial plane. If they are parallel and noncoplanar, they lie in and define a cylindrically curved 

surface, which is here designated as the "axial surface." 

Figures ISA, B, G, F; 20#, C; 21/>; and 24 show the 

traces of axial surfaces of this kind.

Noncylindrical folds include quaquaversal folds, 

canoe-shaped or elongate folds, and the plunging ends 

of cylindrical folds. A quaquaversal fold is approximately equidimensional in horizontal outline, with an 

unlimited number of apical lines; hence no unique 

apical line exists. Similarly an unlimited number of 

planes or surfaces normal to the stratigraphic surfaces 

may be constructed to pass through the apex of thedome. Therefore the terms "axial plane" and "axial 

surface" are also inapplicable.

An elongate dome, however, has one set of apical 

lines, either plane or space curves, which follow the 

crests of the successive stratigraphic surfaces in the 

same direction as the elongation of the fold. These 

apical lines are the loci of the minimum or elongate 

curvature of the strata; and the totality of such lines

fi. Central semlelllpses, congruent hybrid elliptic arcs, and prolate and oblate similar semielllpses exemplifying certain types of structural features. 

A, central semlellipse, bounded by three parallel curves of the eighth degree. Represents the profile of one type of parallel fold. 

B, central semiellipse, with 1 outer and 2 intersecting inner parallel curves of the eighth degree. Illustrates parallel curves with discontinuities that are unsuitable for the

reconstruction of folds.

C, congruent hybrid elliptic arcs illustrating the profile of a congruent similar fold. Diagram comparable to Van Hlse's original drawing of a similar fold. 

D, prolate similar semlellipses Represents the profile of a similar fold, with dips of 90° everywhere along its base. Strata thin proportionately from apex to base of fold. 

E, oblate similar semlelllpses. Represents the profile of a similar fold, with dips of 90° everywhere along its base. Strata thicken proportionately from apex to base of fold. 

F, oblate similar elliptic arcs, with noncoaxial horizontal axes. Represents the profile of a similar fold, in which dips of 90° are absent.

476376 59 2
96 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

defines a surface that is called the major or longitudinal section of the fold. A surface that passes through 

the vertices of all the stratigraphic surfaces and intersects orthogonally all the elongate apical lines, constitutes the minor or transverse section of the fold. 

Collectively these two upright surfaces are called the 

principal sections of a canoe-shaped or elongated dome, 

and obviously either or both may be plane or curved 

surfaces. A third section, not generally a principal 

section, is indicated by the horizontal plan of the fold. 

In conformity with these definitions, a canoe-shaped 

fold is one whose elongate apical lines, lying in the 

major section, are continuously curved and grade nowhere into linear segments. If the principal section 

of a canoe-shaped fold is a curved surface, a plane 

surface that passes through the apex of some selected 

stratigraphic surface and is normal to an elongate 

apical line at that point, will show the general character of the fold. Principal plane sections drawn 

parallel and normal to coplanar apical lines are illustrated in figure 2IE.

The plunging end of a cylindrical fold is generally 

noncylindrical, but the cylindrical and noncylindrical 

sectors are continuous and are threfore difficult to treat 

separately. The length of a cylindrical fold, however, 

is ordinarily much greater than its plunging end. The 

composite structure may best be regarded and described 

as a cylindrical fold over some stated longitudinal interval, and a noncylindrical fold over some other specified range. The plunging end of a cylindrical fold has 

the general characteristics of an elongate dome, but 

owing to structural modifications in the ends of such 

folds, the apical lines are commonly neither parallel 

nor coplanar. Yet it is possible for nonparallel but 

coplanar apical lines to lie within and define a plane 

that is an extension of the axial plane of the cylindrical 

fold. With such a structural relationship, the term 

"axial plane" is warranted. Generally, however, the 

terms "axial plane" and "axial surface" are not strictly 

applicable to the plunging ends of cylindrical folds.

Profiles drawn across a plunging fold, where it has 

lost its cylindrical character, do not have the properties of analogous profiles along the principal sections 

of a canoe-shaped fold. Both are normal to the stratigraphic surfaces, and generally both are curved surfaces ; but the transverse section of a canoe-shaped fold 

passes through the apices of all the stratigraphic surfaces, whereas the analogous section across the plunging part of a cylindrical fold passes through some arbitrarily selected point on an arbitrarily selected apical 

line, and is normal to the other apical lines at random 

points. A plane profile, however, may be drawn normal

to some apical line and tangent to the curved profile, or 

better still, as a secant plane intersecting the curved 

surface. A profile of this sort will closely approximate 

the true cross section of the plunging fold, and will be 

analogous to the cross section of the cylindrical part 

of the fold.

Cylindrical folds have heretofore been divided into 

"parallel" and "similar" folds, but these two terms are 

commonly undefined and are used differently than in 

this paper. The term "concentric" fold is used by 

some authors as a synonym for "parallel" folds, but 

only circles and spheres are concentric; and it is easy 

to show from a sequence of three or more dips in a 

profile that stratigraphic traces are generally noncircular. A fourfold division of cylindrical folds is 

here recognized, based upon the geometry of the stratigraphic surfaces and their traces. These four genera 

are designated as parallel, similar, cognate, and composite folds, all of which have axial lines. A cylindrical parallel fold necessarily has parallel stratigraphic 

surfaces and an axial plane. Cylindrical similar, cognate, and composite folds have, respectively, similar, 

cognate, and composite stratigraphic surfaces but may 

have either axial planes or axial surfaces. Cylindrical 

similar and cognate folds are subdivided into species 

according to whether they have axial planes or axial 

surfaces. Cylindrical composite folds, however, are 

subdivided first into subgenera on the basis of their 

stratigraphic traces and thereafter into species as stated 

above. Noncylindrical folds are likewise divided into 

four genera, but further subdivisions into specific types 

appears to be too difficult and involved to be useful.

A classification of folds that accords with the definitions already stated is presented herewith:

I. Cylindrical folds 

A. Parallel folds 

B. Similar folds

1. With axial plane

2. With curved axial surface 

C. Cognate folds

1. With axial plane

2. With curved axial surface 

D. Composite folds

1. Mixed but related stratigraphic traces 

a. With axial plane 

b. With curved axial surface

2. Hybrid stratigraphic traces

a. Unrotated

aa. With axial plane

bb. With curved axial surface

b. Rotated 90°

aa. With axial plane

bb. With curved axial surface

3. Unrelated stratigraphic traces 

a. With axial plane 

b. With curved axial surface
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 97

II. Noncylindrieal folds

A. Parallel domed surfaces 

B. Similar domed surfaces 

C. Cognate domed surfaces 

D. Composite domed surfaces

CYLINDRICAL FOLDS

Cylindrical folds have cylindrical stratigraphic surfaces over some stated or implied longitudinal limit. 

No assumption is made regarding the cross section of 

a cylindrical fold, and therefore no assumption is made 

regarding the character of its stratigraphic traces in a 

section normal to the axial line. These curved traces 

may be any of the myriads of continuous curves, here 

classified as parallel, similar, cognate, and composite.

The term "symmetry," as applied to a cylindrical 

fold, should mean that an axial plane exists which is 

approximately a plane of symmetry, such that the 

same stratigraphic curvature, sequence, and thickness 

of beds appear on both sides of it. Otherwise stated, 

the half of the fold on one side of the axial plane must 

be identical with the half on the other side. Symmetry 

vanishes if a curved axial surface is present. Symmetrical similar, cognate, and composite cylindrical 

folds should therefore be less common than symmetrical parallel folds.

The major and minor axes of the ellipses used in 

this paper to illustrate and simulate cylindrical folds 

are drawn initially as vertical and horizontal lines. 

The traces of cylindrical folds constructed in this manner represent upright anticlinal arches that have vertical axial planes. But these figures may be rotated 

so as to illustrate inclined, recumbent, or synclinal 

folds, as required by the structural data. All dips will 

be modified numerically by rotation, but generically 

and intrinsically they will remain invariant. Thus 

the comments regarding dips of 90° apply only to upright folds; and such dips can appear on both flanks 

of an upright fold only where complete semiellipses 

are utilized.

Five variables may be used in the reconstruction of 

nonparallel folds by the application of elliptic arcs. 

These are, first, a choice in the eccentricity of the elliptic arcs; second, change of scale, defined geometrically 

as uniform or nondistortional enlargement or reduction; third, linear translation of the stratigraphic 

traces parallel either to the vertical or horizontal axes 

of the ellipses; fourth, a choice between proportional 

and differential linear translation of each trace; and 

fifth, rotation of the elliptic axes and stratigraphic 

traces.

The first two of these variables require no further 

explanation. Linear translation results when one or 

both semiaxes depart from coaxiality without rotation.

Vertical and horizontal translations produce markedly 

different effects. Proportional linear translation 

means a displacement parallel to either or both axes, 

such that the stratigraphic intervals on one or both 

axes are increased or decreased proportionately. Differential linear translation results in corresponding 

displacements that are nonproportional to the initial 

stratigraphic intervals. These are called differential 

displacements. A rotatory displacement of 90° of the 

elliptic axes is also permissible, but any other rotation 

would cause the stratigraphic traces to intersect.

The major types of linear translation are shown 

below.

I. Translation parallel to the vertical axes, thus displacing

the horizontal axes.

A. Translation proportional to the thickness of beds. 

B. Translation nonproportional, or differential.

II. Translation parallel to the horizontal axes, thus displacing

the vertical axes.

A. Translation proportional to the thickness of beds. 

B. Translation nonproportional, or differential. 

III. Translation parallel to and displacing both axes.

A. Translation proportional to the thickness of beds. 

B. Translation nonproportional, or differential. 

C. Translation both proportional and differential.

1. Proportional vertical and horizontal displacements.

2. Proportional vertical and differential horizontal 

displacements.

3. Differential vertical and proportional horizontal 

displacements.

4. Differential vertical and horizontal displacements.

PARALLEL FOLDS

Parallel folds are defined as folds whose stratigraphic 

surfaces are essentially parallel. The traces of sueh 

surfaces in sections normal to the axial line are involutes, which constitute one class of parallel lines. The 

strata of all folds must be stretched in relation to their 

original lengths; but this is an overall effect that may 

be accompanied locally by stratigraphic thickening as 

well as thinning. Parallel folds are conceived to be 

those in which the strata have not been thickened but 

instead have been uniformly stretched and thinned. 

It is doubtful if all the strata of any fold are uniformly 

stretched, so that truly parallel folds may not exist in 

nature. Yet in actual folding, particularly in open 

folding, many competent beds show so little differential thickening and thinning that for practical purposes their thickness may be regarded as constant. The 

terms "parallel folds" and "folding" therefore appear 

to be warranted, and descriptions and measurements 

based upon this interpretation are permissible and 

useful.
98 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY
CLASSIFICATION, DELINEATION, AND MEASUREMENT QP NONPARALLEL FOLDS 99

Cylindrical parallel folds may also be defined as 

those whose stratigraphic surfaces, over some limited 

longitudinal range, may be generated by two or more 

parallel coplanar right lines that move parallel to 

themselves, and in such a manner that the plane in 

which they lie remains normal to the surfaces generated. This condition is best visualized in a plane section normal to the axial line, wherein the traces of the 

stratigraphic surfaces are parallel curves, the traces of 

the generating lines are collinear points, and a right 

line connecting these points is normal to each of the 

stratigraphic traces. No assumption is made regarding the cross section of a parallel fold, except that the 

stratigraphic traces must be parallel or approximately 

so. These traces may thus be any continuous curves, 

though it is improbable that they are ever concentric 

circular arcs. The delineation and measurement of 

parallel folds have been treated by the writer (1940, 

1944, 1947, 1948) in four earlier papers. It suffices 

here to emphasize that the traces of parallel folds, in 

sections normal to the axial line, are not represented 

empirically as elliptic or any other kind of preselected 

arcs. Instead they are accurately delineated as involutes derived from an evolute that is generated as the 

envelope of a set of lines drawn normal to the dips 

charted on a geologic profile.

SIMILAR FOLDS

Cylindrical nonparallel folds have axial lines over 

some stated or implied longitudinal limit, but the 

traces of the stratigraphic surfaces, in a plane normal 

to the axial line, are commonly either similar or cognate curves, though these traces may also be one of 

several types of composite curves.

The term "similar fold," meaning in reality a similar cylindrical fold, was introduced by Van Hise 

(1896) and has been used by many geologists to describe all sorts of nonparallel folds. Van Hise's drawing of the traces of a similar fold, in a plane normal 

to the axial line, consists of congruent or identical 

curves of unspecified character that are similarly placed 

but are separated from one another by a linear translation parallel to the trace of the axial plane. An equivalent type of folding is shown in figure 16(7, where the 

congruent curves are formed by joining arcs of an 

ellipse having an eccentricity of 0.943. These congruent curves are composite and hybrid and are therefore 

higher plane curves. Dips of 90° may be shown for 

such folds along the flanks of the outer stratigraphic 

surface but not elsewhere in the fold. Only moderate 

dips appear in figure 16(7, because complete semiellipses 

are not used. Instead only arcs of the semiellipse were 

chosen and translated.

Congruent similar folds, however, constitute only 

one type of similar folding. If the traces of the stratigraphic surfaces, in the specified section, are noncongruent, nonintersecting, similar curves that are similarly placed, the fold is likewise similar. In three 

dimensions, similar surfaces, cylindrical or noncylindrical, are defined as those having point-to-point correspondence, such that the distance between any two 

points of one surface is invariably the same multiple 

of the distance between two corresponding points on 

the other surface. It is doubtful whether many cylindrical folds have stratigraphic surfaces that meet 

exactly the requirements of similarity. It is permissible, however, to utilize the term for folds that meet 

approximately the required conditions, just as we use 

the term "parallel folds" for those whose traces are only 

approximately parallel.

Similar folds are represented in this discussion by 

elliptic arcs of the same eccentricity. Therefore an 

initial choice exists in the selection of some particular 

semiellipse, but no further choice is permitted. Three 

other variables, however, are commonly utilized in the 

construction of similar traces. These are, first, change 

of scale; second, linear translation of the stratigraphic 

traces parallel either to the vertical or horizontal elliptic axes, or to both; and third, a choice between proportional and .differential linear translation for each 

trace. It is also possible to have a similar fold represented by hybridized stratigraphic traces, if one such 

trace is repeated at different scales.

FIGTTEE 17. Prolate and oblate similar elliptic arcs and similar semlellipses exemplifying certain types of structural features.

A, prolate similar elliptic arcs, with noncoaxial horizontal axes and an innermost semiellipse. Represents the profile of a similar fold, with dips of 90° only at the inner 

base of the fold. Strata thicken proportionately from apex to base.

B, prolate similar elliptic arcs, with noncoaxial horizontal axes and a semiellipse next to the innermost curve. Represents the profile of a similar fold, in which the thickness of the strata changes differentially from the apex to the base.

C, prolate similar elliptic arcs, with noncoaxial horizontal axes and an outermost semiellipse. Represents a similar fold with dips of 90° only at the outer base of the fold. 

Strata thin proportionately from apex to base.

D, prolate similar elliptic arcs, with noncoaxial horizontal axes and outermost and innermost semlellipses. Represents the profile of a similar fold, In which the thickness 

of the strata changes differentially from the apex to the base.

E, oblate similar elliptic arcs, with noncoaxial horizontal axes, an Innermost semiellipse, and all curves tangent to one another at one end of their minor semiaxes. Represents the profile of a similar fold, in which the strata pinch out entirely at the apex. This is the limiting form of a similar supratenuous fold.

F, prolate similar semiellipses, with noncoaxial vertical axes. Represents the profile of a similar fold, with dips of 90° everywhere along its base, and an Inclined axial 

plane. Shows proportional thinning of thejstrata on the right side of the fold and proportional thickening on the left side. Configuration produced by proportional 

horizontal translation.
100 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 101

The eccentricity of the elliptic traces of a similar 

fold is first selected. Thereafter change of scale is 

necessarily used, in order to illustrate the number and 

thickness of the strata. In most of the accompanying 

diagrams, the successive strata are shown as equal in 

thickness, in order to facilitate comparisons between 

different constructions, though in the reconstruction of 

actual folds, the strata will have different thicknesses. 

If the dips along a profile cannot be made to correspond with those indicated by the chosen elliptic arcs, a 

new set of arcs with a different eccentricity must be 

tried. If translation is not used, the construction requires coaxial semiellipses, though the complete semiellipses may not be used; but for either construction 

the resulting fold will be symmetrical to the vertical 

axis.

A construction using only changes in scale is first 

attempted. If the major axes of the semiellipses are 

vertical, such a fold will show strata that are uniformly 

thicker along the trace of the axial plane than at the 

base of the fold; but if they are horizontal, the reverse 

will be true. These conditions are illustrated, respectively, in figures 16Z> and E. Under this construction 

the complete disappearance of the strata, either at the 

apex or along the outer flanks of the fold, cannot be 

shown; but dips of 90° may be shown entirely across 

the base of the fold. Dips of 90°, however, are not 

necessarily present, as only parts of the semiellipses 

may be utilized; and where this is done, the dips that 

do appear at the base of the fold will depend partly 

upon the eccentricity of the ellipses and partly upon 

the lengths of the elliptic arcs that are used. This is 

exemplified in figure 16F, where only parts of semiellipses with an eccentricity of 0.600 are shown. This 

feature is likewise illustrated in the congruent similar 

curves of figure 16(7.

Linear translation introduces the second variable in 

the representation of similar folds. Case I-A is first 

considered. The major axes of a set of similar noncongruent ellipses illustrated in figure 16Z> are used 

again in a vertical position in figure 17#; but each of

the three inner curves has been translated vertically 

downward, in proportion to the original stratigraphic 

intervals, leaving the outer curve untranslated. Under 

this construction the thickness of each stratum is proportionately increased, but dips of 90° appear only 

along the outer flanks of the fold at its base. If, however, the three outer curves of figure 16Z> are translated 

proportionately downward, leaving the inner curve untranslated, the thickness of each stratum along the vertical axis will be proportionately decreased, as shown 

in figure VIA ; or if desired, the strata may be made to 

disappear entirely at the apex of the fold, as shown in 

figure 17#. In figure 17'A and E, dips of 90° may be 

shown only along the inner base of the fold. All constructions that produce proportional thickening or thinning along the vertical axis produce a nonproportional 

thinning of strata along the base of the fold, from the 

center outward to the flanks. Also, under the assumption of type I-A, no cross section of a similar fold with 

elliptic traces may show a dip of 90° along its base at 

any point intermediate between the center of the fold 

and its outer flanks.

Figures VIA and C portray opposite effects, and it 

therefore follows that a proper choice of stratigraphic 

intervals along the vertical axis can be made to match 

approximately those along the horizontal axis; this 

produces the traces of a pseudoparallel fold. The approximation to stratigraphic parallelism can be made 

closer by utilizing a slightly nonproportional spacing 

on the vertical axis, so that each stratigraphic interval 

on the vertical axis matches exactly its corresponding 

interval on the horizontal axis; but such a construction would introduce a slight nonuniformity in the 

stratigraphic spacing on both axes. The approximation to parallel folding may be made still closer by the 

use of cognate curves, as shown in figure 19(7. Any of 

these three constructions would be difficult to discriminate from parallel folding, except by accurate measurement.

The third variable in the construction of similar 

folds is introduced by type I-B, where the linear transFIGUKE 18. Prolate similar semiellipses and elliptic arcs and prolate cognate and oblate cognate semlellipses exemplifying certain types of Btructural features. 

A, prolate similar semiellipses, with noncoaxial vertical axes. Represents the profile of a similar fold with dips of 90° everywhere along its base, and a curved axial surface

Shows differential thickening and thinning of strata at base of fold. Configuration produced by differential horizontal translation. 

B, prolate similar semiellipses with noncoaxlal vertical axes. Represents the profile of a similar fold with dips of 90° everywhere along its base, and a curved axial surface In

reverse of Illustration A. Shows differential thickening and thinning of strata at base of fold. Configuration produced by differential horizontal translation. 

C, prolate similar elliptic arcs with noncoaxial vertical and noncoaxial horizontal axes and an outermost semielllpse. Represents a curved axial surface and the profile of

a similar fold with dips of 90° only at the outer base of the fold. Shows differential thickening and thinning of strata at base of fold. Configuration produced by proportional vertical and differential horizontal translation. 

D, prolate cognate semlellipses. Represents the profile of a cognate fold, with dips of 90" everywhere along Its base. Shows proportional thinning of strata from apex to

base of fold. 

E, oblate cognate semiellipses. Represents the profile of a cognate fold, with dips of 90" everywhere along Its base. Shows proportional thinning of strata from apex to

base of fold. } 

F, a semlellipse next to the innermost curve and prolate similar elliptic arcs with noncoaxial vertical and noncoaxial horizontal axes. Represents a strongly curved axial

surface and the profile of a similar fold with limited basal dips of 90". Shows differential thickening and thinning of strata at base of fold. Configuration produced by

differential vertical and differential horizontal translation.
102 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

FIGURE 19. Prolate cognate and oblate cognate semiellipses and prolate cognate elliptic arcs exemplifying certain types of structural features. 

A, prolate cognate semiellipses. Represents the profile of a cognate fold, with dips of 00° everywhere along Its base. Shows proportional thickening of strata from apex to

base of fold. 

B, oblate cognate semiellipses that are tangent to one another at one end of their minor semiaxes. Represents the profile of a cognate fold, in which the strata pinch out

entirely at the apex. This is the limiting form of a cognate supratenuoiis fold. '

C, prolate cognate semiellipses with equal increments of length for the major and minor semiaxis. Represents the profile of a cognate fold that simulates a parallel fold. 

D, prolate cognate semiellipses that are tangent to one another at both ends of their minor semiaxes. Represents the profile of a cognate fold, in which the strata pinch out

entirely at the base on both flanks. 

B, oblate cognate semiellipses. Represents the profile of a cognate fold, with dips of 90° everywhere along its base. Shows proportional thickening of strata from apex to

base of fold. 

F, prolate cognate elliptic arcs with noncoaxial horizontal axes, and a semiellipse next to the innermost curve. Represents the profile of a cognate fold with limited basal

dips of 90°. Shows differential changes in thickness cf strata from apex to base of fold. Configuration produced by differential vertical translation.

lation is nonproportional. Vertical differential translation makes it possible to show dips of 90° along the 

base of a fold, either on the outer or inner stratigraphic 

surfaces, on both the outer and inner surfaces, on one 

or more of the intermediate surfaces, or on none of 

these surfaces. The only limitation is that dips of 90°

may not be shown entirely across the base of a fold. 

In these various delineations the thickness of the strata 

may increase along the vertical axis downward, upward, or in some erratic manner that may be required 

to simulate a stratigraphic sequence. Figure 17B illustrates a set of stratigraphic traces in which only the
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 103

surface next to the innermost one shows a dip of 90° at 

the base of the fold, and the thickness of the strata increase differentially downward. Figure 17Z? illustrates a fold in which only the outer and inner stratigraphic surfaces show dips of 90° at the base of the 

fold, and the stratigraphic intervals decrease nonproportionately downward.

Proportional horizontal translation, as in type II-A, 

results in additional effects; and these are further 

diversified in type II-B, where the horizontal translation is differential. Proportional horizontal translation is illustrated in figure 177^, which is derived 

from figure 16Z? merely by moving the four semiellipses laterally by amounts proportional to the original stratigraphic separations along the horizontal axis. 

This movement produces an inclined axial plane, whose 

trace appears in the figure; but this position of th« 

axial plane, with regard to the stratigraphic traces, 

is different from what would have been produced if 

figure 16Z? had merely been tilted. Stratigraphic 

asymmetry has been introduced; and dips of 90° are 

also preserved everywhere across the base of the fold. 

By still further horizontal translation, all the strata 

can be made to thin uniformly to zero, or in other 

words to disappear completely on one outer flank 

of the fold.

Nonproportional, or differential, horizontal translation of similar ellipses is shown in figure 18J. and B. 

These traces are produced from figure 16Z? by a differential lateral movement of the semiellipses, done in 

such a manner that the strata on the right flank are 

thinned respectively by negative and positive increments. Both figures show the traces of curved axial 

surfaces, but these traces are reversed in their general 

inclinations. The delineation of dips of 90° across 

the base of the fold remains unchanged. Figure 1651, 

where the minor 'axes of the semiellipses are vertically 

placed, could likewise be transformed to show either 

a tilted axial plane or a curved axial surface. Further variations could be produced by operating in the 

same way on figure 17J.-Z>.

Combinations of vertical 'and horizontal translations 

constitute type III; but as these displacements may be 

proportional, differential, or both, several major combinations may exist, with numerous minor variations 

depending upon the vertical or horizontal placement 

of the axes of the semiellipses, the preexisting thickening or thinning of stratigraphic intervals, the directions and amounts of the dips, and other factors. The 

principal combinations are tabulated as types III-C-1, 

III-C-2, III-C-3, and III-C-4. Only two illustrations are given. Figure 17<7, which shows proportional vertical translation, has been transformed by

476376 59 3

differential horizontal translation, to produce the fold 

shown in figure 18<7. Similarly figure 175, which 

shows one type of differential vertical translation, has 

been transformed by differential horizontal translation to produce the fold shown in figure ISF. Folds 

that are represented by figure 17<7, where only parts 

of the semiellipses are used, can likewise be transformed by vertical, horizontal, or both vertical and 

horizontal translations, either proportionately, differentially, or both proportionately and differentially. 

Thus it appears that with linear translations the number of similar folds that may be constructed is very 

large; and no general statements are therefore warranted regarding all the effects that may be produced.

COGNATE FOLDS

Cylindrical cognate folds comprise nonparallel 

stratigraphic surfaces whose traces, in a plane normal 

to the axial line, are cognate curves. Three variables 

may be used in the delineation of cognate folds by the 

use of ellipses; first, a changing eccentricity in each 

of the stratigraphic surfaces; second, linear translation of the traces parallel either to the vertical or the 

horizontal axis, or to both; and third, the choice between proportional and differential linear translation. 

Change of scale and congruency are inadmissible, as 

either of these conditions would produce similar traces 

and folds. Variable eccentricity, however, is so effective in producing a variety of folds that it eliminates in large measure the need for translation parallel 

to the vertical axis. Probably more cognate than similar cylindrical folds occur in nature. A simple test, 

regardless of the nature of the curvature, is to determine whether all the traces of the stratigraphic surfaces, in a specified section, can be made to superpose 

on one another, with or without change of scale, followed by linear translation. If this is possible, the 

traces are similar; if not, they are probably cognate, 

though the possibility remains that they may be 

composite.

Thickening, thinning, or the complete disappearance 

of strata, either at the apex or along the flanks of a 

symmetrical fold, may readily be shown in cross section 

by the use of cognate curves, whereas the complete disappearance of strata cannot be shown along both flanks 

of a symmetrical fold by the use of similar curves. 

Moreover, the degree of thickening or thinning can be 

made to vary between much wider limits than for similar folds. Dips of 90° may be shown concurrently with 

these effects, if desired, entirely across the base of a 

cognate fold. Finally, the eccentricities may be chosen 

to produce stratigraphic traces which resemble parallel 

curves so closely that the difference is determinable only
104 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

FIGTTEE 20. Prolate cognate semiellipses and elliptic arcs and oblate cognate elliptic arcs exemplifying certain types of structural features. 

A, oblate cognate elliptic arcs with noncoaxial horizontal axes, and an outermost semiellipse. Represents the profile of a cognate fold, with dips of 90° only at the outer

base of the fold. Shows differential changes in the thickness of strata from apex to base of fold. Configuration produced by differential vertical translation. 

B, prolate cognate elliptic arcs with noncoaxial vertical and noncoaxial horizontal axes and an innermost semiellipse. Represents the profile of a cognate fold with a strongly

curved axial surface and dips of 90° only at the inner base of the fold. Shows differential thickening and thinning of strata throughout the fold. Configuration produced

by differential vertical and differential horizontal translation. 

C, prolate cognate semiellipses with noncoaxial vertical axes. Represents the profile of a cognate fold with a curved axial surface and dips of 90° everywhere along its base.

Shows differential thickening and thinning of strata along base of fold. Configuration produced by differential horizontal translation. 

D, prolate cognate semiellipses with noncoaxial vertical axes. Represents the profile of a cognate fold with dips of 90° everywhere along its base, and an inclined axial plane.

Shows proportional thinning of strata on right side of fold and proportional thickening on left side. Configuration produced by proportional horizontal translation. 

E, four prolate elliptic arcs bounded by a curve of the eighth degree. The three inner curves are similar; the third and fourth are cognate; and the fourth and fifth are

parallel. Represents the profile of a composite fold of mixed but related stratigraphic surfaces. 

F, one set of prolate similar quarter ellipses joined to a set of prolate cognate elliptic arcs. Represents the profile of a hybrid composite fold, with dips of 90° along the base

of the left half of the fold.

by close measurement. Horizontal translation, or a 

combination of horizontal and vertical translations, 

however, are needed to represent folds with curved

axial surfaces. Only a few of the unlimited types of 

cognate folds will be illustrated. 

Thickening of strata at the apices of cognate folds is
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 105

illustrated in figure 18Z> and E, and thinning at the 

apices, in figure 19^4. and E. The arcs of figure 18Z> 

and 19Z? are identical, but one set is rotated 90° with 

respect to the other. The same is true of figures 18E 

and 19-4. Disappearance of strata at the apex and 

along the outer flanks of a fold are shown, respectively, 

in figures 19B and 19Z>. The simulation of parallel 

folding is accomplished in figure 190 by utilizing equal 

increments of the major and minor semiaxes of the ellipses. Thus the four curves shown in this figure are 

parallel to one another at the ends of the elliptic axes, 

but not elsewhere, so that the deviation from complete 

parallelism is scarcely noticeable. Figures 18Z>, E, and 

19A-D show dips of 90° across the entire base of the 

folds. This type of folding may be considered unrealistic; but if not applicable, dips of moderate degree 

may be shown at the base of the fold by utilizing only 

parts of the semiellipses, as shown in figure ITtf, or by 

using vertical linear translation, either proportional or 

differential.

It has been shown that strata may thin or disappear 

completely at the apices of both similar and cognate 

folds that they may be made to disappear along one 

outer flank of a similar fold by horizontal translation, 

and that they may be caused to disappear along both 

outer flanks of a cognate fold without recourse to 

translation. Where the thinning or disappearance occurs at the apex of a fold, the term "supratenuous fold" 

has been applied by JSTevin (1942). Hills (1953), however, has pointed out that supratenuous folds may originate from several causes, so that they cannot be regarded as a distinct genetic type. Such flexures may 

be simulated as similar folds, by the use of linear vertical translation; or as cognate folds, by variable eccentricity, without or with such translation. Supratenuous folds are therefore not a distinct geometric 

type but instead are merely a subspecies of similar or 

of cognate folds.

Linear translation, vertical, horizontal, or both, and 

either proportional or differential, lead to an unlimited 

number of cognate structures. Added to these variables 

is also the use of partial instead of entire semiellipses. 

Figure 197?, for example, is constructed from figure 

18Z>, by the use of differential vertical translation, accompanied by the utilization of 1 semiellipse and 3 

partial semiellipses. A dip of 90° is thus depicted only 

for the trace whose eccentricity is 0.821. Figure 2(Li, 

on the other hand, is constructed from figure \§E by the 

same method, but the strata in this figure thicken upward instead of downward, and a dip of 90° at the 

flanks is shown only for the outer trace, whose eccentricity is 0.843.

Figures 205, 0, and Z> are produced from figure 18Z>. 

Figure 20Z>, by proportional horizontal translation, 

shows proportional thinning on one flank of a fold, 

and proportional thickening on the other. Figure 20(7, 

produced by differential horizontal translation, shows 

differential thinning on one flank of a fold and differential thickening on the other. Figure 20Z> has an 

axial plane; figure 206^ shows a curved axial surface: 

and both are drawn to show dips of 90° entirely across 

the base of the fold. Figure 205 is produced by a combination of differential vertical and differential horizontal translation, and the use of partial semielliptic 

arcs, with the development of a sharply curved axial 

surface, and the delineation of a dip of 90° only on the 

inner stratigraphic trace. Innumerable other constructions may be made.

COMPOSITE FOLDS

Three principal types of composite cylindrical folds 

are recognized, each of which may be modified by the 

application of vertical, horizontal, or combined vertical 

and horizontal translations; and such translation may 

either be proportional or differential. Horizontal 

translation results in tilted axial planes or curved axial 

surfaces. Moreover, the first and third types may be 

modified extensively by the methods utilized to produce the second type. The subject is thus a large one 

that cannot be treated exhaustively; but the permissible and practical types will occur to field geologists 

in the investigation of particular folds.

The first type comprises folds in which the traces 

of the stratigraphic surfaces, in a plane normal to the 

axial line, are various combinations of parallel, similar, and cognate curves. The possible combinations 

may be parallel and similar, parallel and cognate, similar and cognate, or parallel, similar, and cognate. An 

example of the latter type is shown in figure 20Z?. The 

two lowest traces in this diagram are congruent; the 

second and third, moving upward in the figure, are 

noncongruent but similar; the third and fourth are 

cognate; and the fourth and fifth are parallel, the fifth 

being a curve of the eighth degree parallel to the ellipse whose eccentricity is 0.843. None of the first four 

curves is a complete semiellipse, and therefore no dips 

of 90° are shown at the base of the fold.

Hybrid stratigraphic traces may produce many types 

of composite folds, all of which will be unsymmetrical. 

Hybridization is produced by joining two quarter ellipses at the ends of one or the other of their semiaxis 

or at other points where their tangents are parallel. 

This construction may be achieved by either of two 

general methods, of which one does not require rota-
106 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

tion of the quarter ellipses, whereas the other requires 

the rotation of one set of quarter ellipses through an 

angle of 90° with regard to the other set. Numerous 

modifications of each of these two subspecies are possible.

Some of the cross sections of folds that may be produced without rotation are now mentioned. Consider 

the traces of any of the folds heretofore illustrated by 

semiellipses or parts thereof. Bifurcate these traces 

along the vertical or near-vertical axis of the ellipses, 

and by translation parallel to this axis move one set 

of quarter ellipses upward or downward with respect 

to the other, joining the stratigraphic traces as required. Hybrid curves will result that no longer have 

elliptical curvature but instead are higher plane curves. 

Hybrid curves generated in this way are such that the 

quarter-elliptical traces remain parallel, similar, cognate, or of whatever type they originally were, so that 

each half of the fold retains its original classification. 

The whole fold, however, belongs to none of these original types but is instead a hybridized variety of a composite fold.

Hybridized composite folds produced without rotation may be further generalized. Thus, a set of quarter ellipses of one kind may be joined to a similarly 

placed set of quarter ellipses of another kind, along 

their vertical or near-vertical common axis. Such a 

fold is illustrated in figure 20/?, where a set of similar 

quarter ellipses are joined to a set of cognate quarter 

ellipses. The resulting cross section resembles somewhat those shown in figures 177^ and 20Z>, but the axis 

is vertical instead of inclined. Figure 2QF however, 

may be further modified, if desired, by vertical translation, either proportional or differential. After the 

composite fold has been constructed, it may be still 

further modified by additional vertical translation, by 

horizontal translation, or by both, in the manner shown 

in figures 1*7F; ISA, B, G, F; and 20B-D.

Rotation of the quarter ellipses adds another variable that greatly diversifies the types of hybridized 

composite folds, but in general the effect is to produce 

folds of marked asymmetry. The traces that are 

transformed may be either identical or different, and 

their junction may be made either at the ends of the 

major or minor axes or at other points where their 

tangents are parallel. Consider the similar fold illustrated in figure 16Z>. If two sets of quarter ellipses are 

suitably revolved, translated, and connected, they can 

be so arranged as to produce the asymmetrical hybrid 

fold illustrated in figure 21A. Figure 21B shows two 

sets of similar quarter ellipses of different eccentricity 

that have been revolved, translated, and joined in the

same way. In this construction, however, the quarter 

ellipses have been translated proportionately, both 

vertically and horizontally, to produce a tilted axial 

plane.

The traces of cognate folds may also be hybridized 

by rotation, with or without translation, to produce 

composite folds. Thus in figure 21 (7,4 traces of eccentricities 0.484, 0.533, 0.600, and 0.700 have been combined with another set of revolved traces having 

eccentricities of 0.843, 0.835, 0.821, and 0.781. Proportional horizontal translation has also been used to produce the inclined axial plane whose trace is shown in 

the figure. Differential horizontal translation of the 

same curves results in a fold with a curved axial surface, as illustrated in figure 21Z>. No illustration, except figure 16 <7, is presented of cognate or similar traces 

that are joined at points other than the ends of the 

major and minor axes, though this procedure is altogether feasible.

A third type of composite folds has a different origin. 

Cognate folds have been required by definition to possess the same equation, with parametric variations. 

The different strata in a fold have been deformed by 

the same forces, and it is therefore probable that successive traces are analytically related. Yet it may be 

possible to have stratigraphic surfaces that are analytically unrelated. Thus the traces of the surfaces 

of a cylindrical fold, in a plane normal to the axial 

line, might conceivably be arcs of an ellipse, a parabolic 

catenary, a cassinian oval, and a sine curve. Such a 

possibility needs to be observed and mapped before 

folds of this type can be listed as proved species. If 

existent, it is obvious that related types can be produced by the different methods of translation.

SUMMARY

Four genera of cylindrical folds have been discussed, 

with species that depend primarily upon whether they 

have axial planes or curved axial surfaces. Parallel 

folds are determined by the test of constant, or sensibly 

constant, stratigraphic thickness of each bed; and until 

this test has been applied, they should be interpreted 

as similar or cognate. From 19(7 illustrates a pseudoparallel fold that is really cognate. Likewise, similar 

folds should be regarded as cognate until the test has 

been applied of reproducibility of the stratigraphic 

traces by nondistortional enlargement or reduction, including congruency. Composite folds have been classified into three subgenera, which indeed might be further expanded; and each of these may be divided into 

species, subspecies, and varieties. The term "composite folds," as defined above, is broad enough to include
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 107

all simple cylindrical folds in nature that may not be 

classified as parallel, similar, or cognate, or approximately so. Many geologists will object that adequate 

data are not ordinarily available for utilization of this 

classification. This is true; but the classification may 

stimulate the acquisition of the necessary data.

The traces of nonparallel cylindrical folds, in a 

plane normal to the axial line, are not claimed to have 

elliptical curvatures; but ellipses are the simplest 

figures that can be used to match approximately the 

stratigraphic curvatures and tangents thereto that 

exist in nature. One cannot ordinarily measure the 

curvature of stratigraphic surfaces or traces thereof; 

but strikes and dips can be translated into curvature 

and approximately into elliptic curvature. Stratigraphic thickness, or other stratigraphic measurements 

in folded beds are most commonly made indirectly, 

either numerically or graphically, but whatever 

method is used must be based upon some assumption 

regarding stratigraphic curvature. The theory of 

evolute and involutes provides a perfect assumption 

for parallel folds; but the reconstruction of nonparallel folds, unfortunately, cannot be based upon such a 

unique theorem. Elliptic arcs, however, will provide 

a satisfactory first approximation to the true curvature, and this permits a logical and at the same time 

conventional method for the reconstruction of folds 

and for making consistent stratigraphic measurements. 

Nonparallel folds have not yet been studied from an 

analytical or even a geometrical point of view, and 

such studies must precede a proper understanding of 

the genesis of folding. No genetic interpretations are 

therefore associated with the geometrical classification 

of folds presented in this paper. The mechanics of 

folding awaits an adequate analysis of folds.

NONCYLINDRICAL FOLDS

Noncylindrical folds have been defined to include 

quaquaversal folds, elongate domes or canoe-shaped 

folds, and the plunging ends of cylindrical folds. 

Canoe-shaped folds and plunging folds have been 

shown to have apical lines that are analogous to the 

axial lines of cylindrical folds, but these lines may be 

either plane or space curves. A canoe-shaped fold 

has major and minor sections, known as principal sections, that may be either plane or curved surfaces. 

The minor section of a canoe-shaped fold is analogous 

to the profile of a cylindrical fold, normal to the axial 

line. The terms "axial plane" and "axial surface" are 

not applied to canoe-shaped folds. The medial surface of a plunging fold, however, may possibly be 

designated as an axial plane or axial surface, if continuity exists with such surfaces in a cylindrical fold.

Quaquaversal and canoe-shaped folds comprise the 

flexures under present discussion. The maximum curvature of a canoe-shaped fold will generally appear in 

its minor section, and the complimentary minimum 

curvature in its major section. The structure of an 

elongate dome must therefore be determined in two 

principal sections normal to one another, whereas the 

structure of a cylindrical fold can be understood from 

numerous cross sections parallel to one another. Such 

sets of orthogonal profiles render necessary a threedimensional evaluation of canoe-shaoed folds.

A group of surfaces, analogous to cylinders, is 

needed to describe and illustrate empirically the stratigraphic surfaces of domed structures. Three of the 

5 fundamental quadric surfaces, together with 3 surfaces of revolution, are suitable for this purpose. 

These, together with their principal Cartesian sections, 

are given in the following tabulation:

Surfaces Principal sections

1. Ellipsoid of revolution (spheroid) Circle; unlimited congruent ellipses.

2. Paraboloid of revolution_ _ _ Circle; unlimited congruent parabolas.

3. Biparted hyperboloid of revolution. Circle; unlimited congruent hyperbolas.

4. Ellipsoid (triaxial). _ Three ellipses.

5. Elliptic paraboloid_____ _ Ellipse; two parabolas.

6. Biparted hyperboloid- Ellipse; two hyperbolas.

One principal section for each quadric surface will 

have to lie in a horizontal or nearly horizontal plane; 

and it therefore follows that surfaces 1, 2, and 3, if 

their circular sections were made horizontal, could be 

used advantageously to represent quaquaversal folds. 

It is equally obvious that surfaces 4, 5, and 6, if an 

elliptic section were placed in a horizontal position, 

could be used to represent elongate or canoe-shaped 

folds. Eeasons have been given, however, why elliptic 

sections are preferred in the delineation of the stratigraphic traces of cylindrical folds. If this limitation 

is extended to domes, surfaces 1 and 4 appear to be the 

most suitable, the first for quaquaversal folds and the 

fourth for canoe-shaped folds. In the delineation of 

domes it is commonly assumed that the applied force 

acted vertically upward; but if the applied force acted 

obliquely upward, or if its direction of application 

changed during the formation of the dome, or if the 

dome was tilted during or after its formation, the 

principal sections will be either inclined planes or 

curved surfaces.

Domed surfaces with a circular horizontal outline 

may be generated as surfaces of revolution, but canoeshaped surfaces may be produced only by more involved 

modes of graphical or mechanical assembly. All sur-
108 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

FIGURE 21. Oblate similar quarter ellipses, oblate similar and prolate cognate elliptic arcs, sections of a composite elongate dome, and application of trammel of Archimedes 

in construction of ellipses.

A, one set of oblate similar quarter ellipses joined to a set of prolate similar elliptic arcs of the same eccentricity. Represents the profile of a hybrid composite fold with dips 

of 90° along the base of the left half of the fold.

B, one set of oblate similar elliptic arcs joined to a set of prolate similar elliptic arcs. Represents the profile of a hybrid composite fold with an inclined axial plane. Configuration produced by proportional horizontal translation.

C, one set of prolate cognate elliptic arcs joined to another set of prolate cognate elliptic arcs. Represents the profile of a hybrid composite fold with an inclined axial plane. 

Configuration produced by proportional horizontal translation.

D, one set of prolate cognate elliptic arcs joined to another set of prolate cognate elliptic arcs. Represents the profile of a hybrid composite fold with a curved axial surface. 

Configuration produced by differential horizontal translation.

E, two principal sections of a composite elongate dome, drawn in perspective. The major section consists of oblate cognate elliptic arcs; the minor section consists of prolate 

similar elliptic arcs.

F, application of the trammel of Archimedes in projecting as a glissette an oblate ellipse with an eccentricity of fc=0.884. The semidiameters are a-30 and 6=14.
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 109

faces of revolution, regardless of whether the generating curves are parallel, similar, cognate, or composite, 

show an unlimited number of plane sections that intersect in the axis of revolution. Thus a dome with an 

equidimensional horizontal outline may be represented 

approximately as a set of prolate or oblate surfaces, of 

which at least one may be constructed as a spheroid; 

the shapes of the others will depend upon the nature 

of the stratigraphic traces. If the dome is elongate, 

the best choice for the first stratigraphic surface is an 

ellipsoid, bounded by other surfaces that may or may 

not be ellipsoidal.

The exact configuration of domes is not definitely 

known. If they were thought to have the shapes of 

quadric and related surfaces, noncylindrical folds could 

be divided primarily into subclasses according to the 

preceding tabulation. But quadrics are simpler surfaces than cylinders, as their sections yield curves of 

the second degree, whereas the sections of cylinders may 

show higher plane curves. True cylindrical folds are 

therefore more likely to occur in nature than are 

quadric folds; and in any classification the concept of 

cylindrical folding will withstand the acquisition of 

new knowledge, whereas the idea of quadric folding 

probably will not. Hence, it seems best, until more information is available, to subdivide noncylindrical folds 

primarily upon the nature of their stratigraphic surfaces in principal sections and to utilize the concept of 

quadric folding only secondarily for descriptive purposes.

The existence of two principal sections in domes introduces a complication not present in cylindrical folds. 

Horizontal translation, or combinations of vertical and 

horizontal translations, of the curves that represent 

stratigraphic traces in cylindrical folds have been 

shown to produce inclined axial planes and curved axial 

surfaces. But canoe-shaped folds have two principal or 

orthogonal sections, and quaquaversal folds have an unlimited number. Hence, vertical and horizontal translations of similar, cognate, and composite curves can 

be applied independently along two sections normal to 

each other. The translations, moreover, need not be in 

the same sense, or of the same magnitude; and they may 

thus generate not merely curved but warped or twisted 

sections lying between the principal sections. This effect can apply both to canoe-shaped and to quaquaversal folds, producing marked asymmetry in such 

structures.

Asymmetry may also be produced in another way. 

Transverse and longitudinal sections of a canoe-shaped 

fold, or two orthogonal sections in a quaquaversal fold, 

are not necessarily related in their general geometric 

configurations. Thus one section may show similar

stratigraphic traces, and the other may show cognate 

or composite traces. By using the three types of composite surfaces, it appears that at least 10 such general 

combinations are possible. Twisted as well as simply 

curved profiles may thus result in sections lying between the principal or orthogonal sections. Such 

flexures would really be composite domes, because 

mixed stratigraphic traces are present in orthogonal 

sections. A simple type of canoe-shaped fold with a 

vertical axis, showing similar curvature in one principal section and cognate curvature in the other, is illustrated in figure 21/T.

Domes may thus include many kinds of unsymmetrical domed surfaces; for if n types of curvature or translation are possible in one principal section and n types 

in another, it follows that n2 types of composite curvature may characterize such folds as a whole. No attempt is made in this paper to enumerate and describe 

these many species of domes. Instead, related types of 

curvature are utilized in both principal sections; and 

translation, if any, is conceived to be applied equally. 

It is desirable also to avoid complex figures, for which 

reason the principal sections of all domes are interpreted as plane profiles.

PARALLEL DOMED SURFACES

Parallel domed surfaces may be constructed by either 

of two general methods. Two orthogonal sections of 

a quaquaversal fold or the two principal sections of a 

canoe-shaped fold may be drawn as involutes, generated 

from evolutes derived from the dips along these sections. This is the most realistic method, if the geological data are available, as it involves no assumption regarding elliptic or any other curvature for the stratigraphic traces.

The second method requires that one stratigraphic 

trace be fitted to an elliptic arc, after which parallel 

traces are drawn. No two ellipses can be parallel, and 

it therefore is impossible to have a domed structure with 

parallel spheroidal or ellipsoidal surfaces. But just as 

it is possible to have an ellipse bordered by a set of 

curves that are parallel to the ellipse and to one 

another, so it is possible to have a spheroid or an ellipsoid bounded by surfaces that are parallel to the 

given surface and to one another. Surfaces parallel to 

the other conicoids could likewise be constructed; and 

in fact, stratigraphic traces may be drawn parallel to 

any continuous surface that itself meets the stratigraphic requirements.

Surfaces parallel to a semispheriod are easily generrated as surfaces of revolution. Thus if figure WA is 

rotated 180°, either about its vertical or its horizontal 

axis, parallel surfaces will be generated. With the first
110 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

construction, a prolate parallel quaquaversal fold will 

be simulated; but from oblate parallel curves, oblate 

quaquaversal folds may also be produced. Surfaces of 

revolution may likewise be generated parallel to a 

paraboloid of revolution or to a hyperboloid of revolution; and more generally any algebraic or transcendental surface of revolution could be used as the 

original stratigraphic surface.

Surfaces parallel to a semiellipsoid cannot be produced as surfaces of revolution but are conceived to be 

generated as follows. If normals to the ellipsoid are 

constructed at many closely adjacent points and are 

extended both inward and outward from the ellipsoidal 

surface, a set of nonparallel lines will result, resembling 

penetrating quills. If fixed distances are laid off 

equally on all these normals, both inside and outside 

the ellipsoid, the resulting points will define the loci of 

a set of surfaces parallel to the ellipsoid. The inner 

surfaces may or may not qualify as stratigraphic surfaces, as shown in figure 165; but all surfaces of this 

kind generated outside the ellipsoid, and some of those 

inside the ellipsoid, may be used as parallel stratigraphic surfaces to illustrate parallel canoe-shaped 

folds. By the same method, surfaces may be constructed parallel to an elliptic paraboloid, a biparted 

hyperboloid, or any other continuous algebraic or transcendental surface.

Parallel domed surfaces, either quaquaversal or canoe-shaped, necessarily have plane orthogonal or principal sections. Such profiles, however, may be inclined 

for either of two reasons. First, the direction of the 

force applied to produce the dome may have been inclined to the vertical; and second, the dome may have 

been tilted subsequent to its formation. Any variation 

in the direction of the applied force during its period 

of operation, or any differential in the direction of later 

tilting, would probably have destroyed the parallelism 

of the stratigraphic surfaces. Parallel domed surfaces 

therefore have at least a negative significance in the 

mechanics of folding.

SIMILAR DOMED SURFACES

Similar surfaces are defined as those which have a 

point-to-point correspondence such that the distance 

between any two points on one surface is invariably 

the same multiple of the distance between two corresponding points on the second surface. Also the areas 

of similar surfaces are to each other as the squares of

the corresponding distances between them. Similar 

spheroids and ellipsoids conform with these general 

requirements.

Three variables were shown to apply in the delineation of cylindrical similar folds; and these apply 

equally well in the construction of quaquaversal and 

canoe-shaped similar folds. The effects are thickening, thinning, or the disappearance of strata at the 

apex of a fold, and a thickening or thinning of strata 

along the outer flanks; differential thickening or thinning of strata along the axial plane or at the base of the 

fold; the production of asymmetry, accompanied either 

by proportional or differential thickening or thinning 

of strata in one-half of the fold; and the generation 

of tilted axial planes or curved axial surfaces.

Similar domed surfaces may be constructed as spheroidal surfaces of revolution or as assemblies of ellipsoids. Any of the sets of symmetrical similar curves 

shown in figures IQD-F and 11A-D could be revolved 

180° about their vertical axes to produce either prolate 

or oblate quaquaversal structures, depending upon the 

character of the original profiles. They may be rotated about their horizontal axes only if dips of 90° 

are shown everywhere across the base of the fold, for 

otherwise the resulting surfaces would show discontinuities. But such sets of surfaces would have one 

principal section showing circular stratigraphic traces, 

which are inadmissible. The same geometric difficulty 

applies also to parallel, cognate, and composite curves 

that might otherwise be revolved about their horizontal 

axes. Thus the similar curves shown in.figures 16Z>, 

16#, 17F, ISA, and 185, and the parallel curves of 

figure 16A9 should not be revolved about their horizontal axes to illustrate domal structures. Figures 17F, 

18JL, B, C, and F cannot be revolved about any vertical 

lines.

Similar quaquaversal folds may also comprise sets 

of similar paraboloids of revolution or similar hyperboloids of revolution. And in general, similar qiia-' 

quaversal folds may be produced from sets of any similar algebraic or transcendental surfaces that can be 

generated as surfaces of revolution. All the similar 

surfaces of revolution heretofore mentioned are symmetrical in 1 or 2 orthogonal sections; but completely 

asymmetrical folds may be produced by various combinations of vertical and horizontal translations of these 

surfaces.

Elongate similar domes are not constructed as surfaces of revolution. Similar ellipsoids, for example,
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 111

are ellipsoids whose principal sections are similar 

ellipses; that is, ellipses with the same or proportional semiaxes. If the semiaxes are identical, the 

ellipsoids are congruent. But an ellipsoid whose semiaxes are 15, 12, and 9 is similar to another ellipsoid 

with semiaxes of 10, 8, and 6. A dome consisting of 

an assembly of ellipsoidal surfaces conforming to this 

restriction illustrates an elongate similar fold; and 

such a fold may be constructed either as a prolate or 

an oblate fold. Commonly the longest and median 

axes of canoe-shaped folds lie approximately in the 

plane of the horizon; but this is not necessarily so, as 

the axes of greatest and least lengths may occupy this 

position.

Elongate similar folds may likewise be constructed 

from similar elliptic paraboloids, similar biparted hyperboloids, or in general from many sets of similar 

surfaces of degree higher than the second. The principal sections of all these similar canoe-shaped folds 

may be plane or curved, depending upon the deviation 

from coaxiality of the sets of surfaces that are utilized.

COGNATE DOMED SURFACES

Cognate curves have been shown to produce a greater 

variety of cylindrical folds than similar curves, owing 

principally to the permissible variation in the eccentricity of each stratigraphic trace. Cognate curves are 

also useful in permitting dips of 90° in many more 

folds than is possible with similar curves; and in addition it is possible to show the complete disappearance 

of strata along both outer flanks of a fold. All these 

effects can likewise be produced in cognate domed 

surfaces.

Cognate quaquaversal domes may be constructed as 

sets of spheroidal surfaces of revolution with cognate 

ellipses as the generating elements. Figures 18Z>, E\ 

19J., C", E, F\ and 20 J., for example, may be rotated 

about their vertical axes to simulate domed surfaces of 

revolution. But in order to avoid circular stratigraphic 

traces, figures 18Z>, E, and 19J., C", E should not be 

revolved about their horizontal axes. Figures 205, C, 

D illustrate unsymmetrical curves that cannot be revolved about any vertical lines.

Cognate quaquaversal surfaces of revolution may 

also comprise cognate hyperboloids of revolution, but 

not paraboloids of revolution, as the generating elements of the latter are parabolas, which are similar 

curves. Cognate quaquaversal or canoe-shaped folds 

may also comprise sets of algebraic or transcendental 

surfaces, represented by equations that differ only in

the values of their constant parameters. A family of 

catenoids will illustrate this possibility in the generation of a quaquaversal fold. Parenthetically, the catenoids are the only minimal surfaces of revolution, for 

which reason they may have some structural significance if they are found to exist in quaquaversal folds.

Elongate cognate domes may be constructed by assembling similarly placed cognate ellipsoids, which may 

or may not be coaxial, depending on whether or not 

linear translation is utilized. The simplest construction 

is the assembly of coaxial cognate ellipsoids. Four such 

ellipsoids, for example, might have semiaxes 16,12, and 

8; 13,10, and 7; 10,8, and 6; and 7,6, and 5. The eccentricities of the elliptic arcs in all the principal sections of 

such surfaces would be different; and in fact, the same 

would be true of their horizontal sections. The halves 

of these ellipsoids, oriented with the major axes 

either in a vertical or in a horizontal position, will 

constitute two sets of cognate stratigraphic surfaces, 

illustrating respectively a prolate and an oblate canoeshaped cognate fold.

Elongate cognate domes could likewise be represented by an assembly of cognate biparted hyperboloids; and these might also be constructed from 

numerous sets of related surfaces of higher degree. 

All cognate domes produced with or without rotation, 

in the manner outlined above, could be greatly modified 

by different combinations of vertical and horizontal 

translation, to produce many types of unsymmetrical 

folds. It is clear that the number of quaquaversal 

and canoe-shaped cognate folds that are possible is 

very large.

COMPOSITE DOMED SURFACES

Composite domes, like composite cylindrical folds, 

have at least three types of surfaces, which are classified according to the character of the stratigraphic 

traces. Some of the curves that simulate these traces 

are amenable to rotation, so that the resulting figures 

may be used to illustrate specialized types of domes. 

An example is figure 20#, which can be revolved about 

its vertical axis to produce a prolate composite quaquaversal dome, though it cannot be revolved about the 

horizontal line at its base. A difference in the original 

profile could have resulted in an oblate composite quaquaversal dome. Figures 20^ and 2L4-Z>, on the other 

hand, cannot be rotated about any horizontal or vertical line without producing discontinuities.

Composite curves may also be assembled without rotation to produce a large variety of unsymmetrical
112 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

quaquaversal and elongate domes, with principal sections that are either plane or curved surfaces. These 

sections are not necessarily related in their geometrical 

configurations, so that composite domes will be much 

more numerous and complex than similar or cognate 

cylindrical folds. For all practical purposes, one is 

constrained to study mainly the principal sections, 

which are represented conventionally and approximately as plane profiles.

Quaquaversal and canoe-shaped composite domes are 

thus constructed by all the methods heretofore described in the production of composite cylindrical 

folds. The number of possible domal structures is too 

large to attempt their enumeration. Figure 2L£r, 

drawn in perspective, illustrates theoretically a simple 

elongate composite dome with two kinds of plane principal sections. The major section, parallel to the 

elongation of the fold, comprises cognate ellipses, having eccentricities ranging from 0.775 to 0.947; the 

minor or cross section consists of a set of similar ellipses at different scales, each with an eccentricity of 

0.745. It should be noted that neither principal section in this figure comprises composite stratigraphic 

traces; but the fold as a whole is composite because 

these two sections illustrate different species of profiles.

CONSTRUCTION AND APPLICATION OF ELLIPSES

Little is really known of the curvature of stratigraphic surfaces, cylindrical or noncylindrical, first 

because complete profiles of folds are rare and second 

because the geometry of actual folds has not been 

studied. The traces of the stratigraphic surfaces of 

some folds, however, are available in natural outcrops, 

either in profiles normal to the axial line or in profiles 

that do not depart materially from this orientation, so 

that they could be transformed with slight error into 

the desired plane. The geometrical measurement and 

analytical classification of such traces should constitute 

a noteworthy step in an understanding of the mechanics of folding and would also serve as a guide or 

check for some of the experimental work now being 

done on the deformation of rocks. Most folds, however, have to be reconstructed without the aid of such 

complete data, for commonly only a vertical profile is 

available, on which are charted a set of dips, together 

with lithologic data.

Ellipses have been shown to be particularly suitable 

for the approximate delineation of stratigraphic traces 

in profiles selected to show the maximum or minimum 

apical curvature; and four general types of cylindrical 

and noncylindrical folds have been illustrated, mainly 

by the use of elliptic arcs. A large number of semiellipses are required in this work. Various methods

are known for drawing ellipses, including the use of 

articulated linkages. The well-known method of 

drawing an ellipse with a string of constant length, 

anchored at the two foci, is impracticable, because it is 

difficult to prepare a string of a stated length and still 

more difficult to avoid stretching the string while 

drawing the ellipse. All ellipses illustrated in this 

paper were drawn by means of the trammel of Archimedes. This method, though discovered 2,200 years 

ago, remains still the most practical means for pointto-point elliptic charting. The ellipse is constructed 

as a glissette, by sliding a bar of length (a+b) between 2 coordinate axes, with the point of projection 

at the junction of the 2 semiaxes. Kectangular coordinates are used for simplicity, but an ellipse will still 

be produced if the axes are oblique. The mechanism 

and method are shown in figure 21^. A strip of lucite 

3 centimeters wide and 51 centimeters long is perforated by 101 holes of size 60, that are 0.5 centimeter 

apart. Holes of this size permit the insertion of a 

sharply pointed hard pencil. It is possible with this 

instrument to construct 2,450 ellipses with eccentricities ranging from 0.999 to 0.199; but if the holes 

were drilled 0.2 centimeter apart, 16,500 ellipses could 

be constructed. In figure 21F the major semiaxis, 

with a length of 30 units, is laid off on the strip of 

lucite as OA; the minor semiaxis, with a length of 14 

units, as OB. The resulting ellipse has an eccentricity 

of k =0.884, as determined by the equation 

The points on the ellipse are laid off as closely as seems 

necessary for accurate sketching of its locus.

A limited number of semiellipses have been prepared 

in graphical form for structural reference. These were 

drawn originally with major semiaxes of 20 centimeters 

and minor semiaxes ranging from 1 to 19 centimeters, 

and this provided 19 prolate semiellipses, as shown in 

figure 22. These were also drawn in oblate form, as 

shown in figure 23. But such figures, in order to be 

usable, must also be available at different scales; and 

the semiellipses of figures 22 and 23 were reproduced at 

reduced scales, resulting in 18 additional figures. 

These, however, could not be presented in this paper; 

and therefore those wishing to use these methods will 

have to provide a series of reductions of figures 22 and 

23. The scale of the profile must be adjusted to fit a 

set of such semiellipses, unless the geologist wishes to 

prepare other semiellipses of predetermined eccentricity 

and scale to fit a profile of fixed size.

The utilization of these ellipses and elliptic arcs will 

depend upon the type of fold that is to be reconstructed. 

If the fold is known to have parallel or approximately
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 113

.312

22. Nineteen prolate semiefflpses with eccentricities ranging from fc=0.312 to Jfc=0.099. The major semiaxes have the same length.

parallel surfaces, in whole or in part, these curves will 

not apply; instead the reconstruction will be accomplished by application of the method of evolute and 

involutes. But if the fold has nonparallel surfaces, one 

must first determine its departure from parallelism, and 

then select the genus of the nonparallel fold. The first 

decision requires an evaluation of the localization and 

degree of the stratigraphic thickening and thinning, as 

revealed by the available geologic data. The second 

decision will have to be made experimentally by fitting

the structural data to the simplest curvilinear traces 

that will serve the required purposes.

Let a vertical profile be given on which are plotted 

a series of dips. If the plane of the profile is oblique 

to the axial plane or surface, another plane profile may 

be drawn that intersects the original and is normal to 

the axial plane or surface. The given dips may be 

transformed in position and magnitude from the first 

to the second plane with little distortion, by well-known 

methods that require no description. The dips are thus
114 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

-999

FIGURE 23. Nineteen oblate semlelllpses with eccentricities ranging from Jfc=0.312 to fc=0.999. The major semlaxes have the same length.

finally assembled on the desired profile, in their proper 

relation to original separation and to relief. These 

dips, however, may not have the optimum spacing to 

yield the best structural interpretation. The best 

structural spacing is generally a uniform one, but unequal spacing may be required to place the dips at 

contacts between beds, or for other reasons. Any desired spacing may be obtained by use of a method given 

by the writer in an earlier publication (1947). Thus a 

profile of suitable character and orientation is now 

ready to be used in the reconstruction of a fold. 

The fold is known or assumed to be nonparallel. If

the nature of the axial plane or surface is not already 

known, this information cannot be deduced from any 

assemblage of dips along a profile. But some idea of 

the dip of the axial plane or the configuration of an 

axial surface may be gained by the following method. 

Any assemblage of dips along a profile may be reconstructed, rightly or wrongly, as a parallel fold, which 

necessarily has an axial plane. One may therefore assume as a first approximation that the fold is parallel 

and then construct such a fold by the method of evolute 

and involutes. The approximate tilt of an axial plane 

is thus determined, to serve as a guide in reconstructing
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 115

the nonparallel fold. This axial plane may be usable 

as deduced, but more commonly it will need to be modified materially in fitting elliptic arcs to the given dips 

of a nonparallel fold.

The construction of parallel folds by the method of 

evolute and involutes has been described by the writer 

in earlier publications, (1940, 1944, 1947, 1948), and 

needs no repetition. It should be stated, however, that 

discontinuities in the evolute are to be expected, not 

only where the dips along a profile pass through values 

of 0° and 90° but also at other places where the increment of dip becomes markedly irregular. Thus a set 

of dips that is increasing gradually from west to east 

may suddenly begin to increase more slowly or more 

rapidly, or to decrease. Any of these conditions may 

cause an interruption in the formation of an envelope 

that will result in a discontinuity in the evolute. Such 

discontinuities, however, will not interfere with the 

generation of the involutes.

The nonparallel fold is reconstructed by selecting 

semiellipses or smaller elliptic arcs that will fit the real 

and interpolated dips along a profile and will come 

closest to fitting the dip of the axial plane deduced from 

an analogous parallel fold. A single semiellipse from 

a series of reductions of figures 22 or 23, moved in a 

direction parallel to its vertical axis and repeated at

equal or unequal intervals, as determined by the stratigraphy, will represent the traces of a congruent similar 

fold, of the type shown originally by Van Hise. But a 

series of semiellipses of the same eccentricity at different scales, or arcs thereof, will represent the traces 

of a noncongruent similar fold; and a selection of semiellipses or elliptic arcs of different eccentricity will lead 

to the construction of the cross section of a cognate 

fold. If all these possibilities fail, a composite fold 

must be constructed. The magnitude and localization 

of the stratigraphic thickening or thinning is further 

controlled by a choice of vertical or horizontal translation, or both, either proportional or differential. Semiellipses with the eccentricities shown in figures 22 or 23 

may be inadequate for the desired curvature; if they 

are inadequate, other semiellipses or elliptic arcs may 

be drawn by the method above described.

A transverse geologic profile of a cognate fold, constructed in the manner described above, is illustrated 

in figure 24. Six beds are shown with 12 dips that 

occur either naturally, or have been interpolated, at 

equal intervals along the contacts of the strata. These 

beds are known to be thicker at the trough of the syncline than along its flanks; and the geologic mapping 

shows that the thicknesses on one flank are greater than 

on the other. More than one solution of these requirerH-s

v v N I

\ \ x x 0,'

v \ x<3

FIGURE 24. Transverse geologic profile of a cognate syncllne with a slightly curved axial surface. Constructed from dips and contacts

plotted on a section normal to the axial line.
116 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

ments may be obtained, but the one illustrated is produced by 6 elliptic arcs, having eccentricities of 0.909, 

0.831, 0.793, 0.703, 0.774, and 0.786. In the preparation 

of this profile, the tilt of a tentative axial plane was 

first determined. Six arcs from a series of semiellipses 

of the type shown in figure 22 were then selected and 

oriented collinearly with their major axes along the 

trace of a tentative axial plane and their minor axes 

parallel at equally spaced intervals. These arcs were 

next translated differentially along the line of their 

major axes to fit stratigraphic intervals assumed to 

exist at the trough of the fold. The traces were then 

translated differentially in a direction parallel to the 

minor axes, in an attempt to fit the given dips and contacts. Failing in this, elliptic arcs of different eccentricity were used, on a trial and error basis, until the 

dips were finally duplicated as tangents to the elliptic 

arcs. This construction produced a slightly curved 

axial surface, as shown by its trace in figure 24.

The reconstruction of nonparallel folds, by the 

methods described in this paper, is evidently accomplished by an experimental and empirical technique 

that cannot yield a unique structural solution. These 

methods thus differ fundamentally from the method 

of evolute and involutes that produces one and only 

one reconstruction of a parallel fold. What, then, are 

the stratigraphic advantages ? A geologist may indeed 

prepare a structural section of a nonparallel fold that 

is not made in accord with any established plan; or he 

may try to adjust the structural data to fit conventional 

curves that correspond approximately to the stratigraphic traces. The utilization of elliptic arcs yields 

at least a solution that is amenable to measurement and 

checking; and it is the thesis of this paper that a conventional though empirical solution of this kind is preferable to stratigraphic conjecture. One of the measurements that is facilitated by the use of predetermined 

curves is that of stratigraphic thickness, now to be 

described.

STRATIGRAPHIC THICKNESS

Stratigraphic thickness, or other stratigraphic dimensions, can be measured from a structural profile 

either graphically or by numerical computation. The 

stratigraphic thickness of a nonparallel fold must first 

be defined, however, before it can be measured by either 

method. The utilization of predetermined curves in 

a profile, such as elliptic arcs, facilitates both the definition and the measurements. Assume that the section 

of a stratum within a fold is bounded by two semiellipses, or by arcs thereof. The stratigraphic thickness may then be defined as the area of the section delineated in the profile, divided by the length of another 

elliptic arc that lies midway between the two bounding arcs. Such a median arc can be constructed either 

graphically or according to an analytic formula.

A graphic measurement of the stratigraphic thickness requires two machines; a line integrator for the 

measurement of the median arc and a polar planimeter 

for measurement of the area between two bounding 

elliptic arcs. If such instruments are available and 

the geologist prefers the graphic method, the resulting 

measurement of thickness will doubtless be within the 

limits of accuracy of the constructed profile. But if 

an analytical measurement is preferred, certain methods need to be described.

The area of an ellipse is given by the simple expression irtib; and as an ellipse is bisymmetrical, the areas 

of the semiellipse and of the quarter ellipse are obvious. 

Many of the illustrations of nonparallel folds, howover, utilize elliptic arcs shorter than a quarter ellipse, 

which may either be coaxial or noncoaxial. If the two 

arcs are coaxial, the area between them is obtained by 

two integrations and a subtraction of the resulting integrals. The general case has to do with an area between two elliptic arcs, one of which has been translated both vertically and horizontally with regard to 

the other. In other words, if both arcs are treated as 

central conies, their origins are different. This condition is illustrated in figure 25.

Let the arc PQ be represented by the equation

fs+fi=l, referred to an origin at 0ij and let the arc

J£2 V2

RT be represented as -&+-#=* *> be referred to an origin 

at 02. Also let $1 be the area between a segment of the 

arc PQ^ the ordinates os=c and x d, and the a? axis; 

and let $2 be the area between a segment of the arc RT, 

the ordinates X=m and X=n, and the X axis. If the 

ordinates of any point in the two systems of coordinates 

are yr and TV, the required area is 

depending upon the relative positions of the two origins. 

Writing the equation of the arc PQ as 

we have &= (*' -(a?-xz)^dx

Jd 0.

Si - x > |_£Larc'sin-| I 

a L « « Q Jd

/S1 =Arc(02_ c2) M_ rf(02_d2)M +02 (&TC sin--arc sinza L \ a a

The area £2 is similarly found, and from & and $2 

the required area B is readily computed.

No exact formula exists for the perimeter of an 

ellipse. Soreau (1921) has presented an empirical
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 117

0,

25. Diagram showing measurement of the area between two noncoaxial elliptic arcs.

formula that gives a close approximation to the perimeter, as follows:

CTT b 77-1 

sin Cw if "a+6

where p is the perimeter, and a and b are respectively 

the semimajor and semiminor axes. A still closer approximation is obtained if 

6 q+0.036 

a' 0.97o + 1.096

The perimeter may also be expressed as an infinite algebraic series in the eccentricity (&), but this converges 

so slowly for large values of the eccentricity as to be 

practically useless. The perimeter is most accurately 

determined by an evaluation of the elliptic integral of 

the second kind, called by Legendre the E function.

The derivation of the length of an elliptic arc follows. Elliptic arcs are measured clockwise from the 

minor semiaxis, as shown in figure 26. Let 

s=QP, an elliptic arc 

a=OR=OA, the major semiaxis

b=OQ=OB, the minor semiaxis

(ut b2\X &=.; L-, the eccentricity

^=the eccentric angle

0=the minor eccentric angle

x=a sin <f>

y=b cos 0

0=arc sin k, the modular angle

The length of any plane curve that is parametrically 

defined as ss=ae(u) and y y(u) is given by the integral 

For an ellipse, let 

Then 

x=a sin 0, and y=6~cos <t>

dx -j-=a cos , dy , . and -^= 6 sm

= [a2 cos2
118 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

which, with the limits of integration inserted, may be 

written as 

sin2 s=a \ (l 

Jo

The length of a quarter ellipse, within the limits

0=0 is p/4=

FIGURE 26. Diagram showing measurement of the length of an elliptic 

arc. The minor eccentric angle, #, and the modular angle, 9, are the 

arguments used in modern tables of elliptic functions.

The quadrature of the E function cannot be performed in terms of any of the elementary functions; 

but it may be computed numerically to any required 

degree of accuracy; this makes it as available for 

practical work as other tabulated functions. The 

values of E and of the other elliptic functions have 

been computed to 12 decimals by the Spenceleys (1947). 

As two variables, <£ and 0, comprise the arguments, a 

double interpolation is required. The angular values 

of <f> must also be converted to radians. By taking the 

values of E (<£, 6) from these tables, the lengths of 

elliptic arcs are not difficult to obtain, though some 

arithmetical computation is necessary. It is obvious 

from figure 26 that the length of any elliptic arc is 

the distance measured clockwise from the minor semiaxis to the farthest terminal point of the arc, minus 

the corresponding distance to its nearest terminal 

point.

The lengths of semiellipses and quarter ellipses, however, may be computed and tabulated. Thus the writer 

has computed the lengths of 99 semiellipses, with major 

semiaxes of 10 and minor semiaxes ranging from 9.9

to 0.1; and for reference the corresponding eccentricities are also given. In the reconstruction of a fold, it 

may be feasible to utilize semiellipses within this range 

of centimeters, inches, or other units of length. If 

possible, the semiperimeters may be read directly from 

the table. If, however, it is necessary to assume other 

lengths for the semiaxes, the corresponding semiperimeters may still be obtained from the table with 

a trivial amount of interpolation, and without the computation of eccentricity. Thus the semiperimeter of 

an ellipse with semiaxes of a=17 and 5=13 is 1.7 times

10X13 

that of an ellipse with semiaxes of a 10 and 5= 

17

or 5=7.647. Interpolating in the table between 5=7.6 

and 5=7.7, it is found that the semiperimeter of such 

an ellipse is 27.84. Therefore the semiperimeter of an 

ellipse with semiaxes of &=17 and 5=13 is 27.84 X 

1.7=47.33. The table of semiperimeters follows.

Semiperimeters of ellipses, &=10

b

9.9

9.8

9.7

9.6

9.5

9.4

9.3

9.2

9.1

9.0

8.9

8.8

8.7

8.6

8.5

8.4

8.3

8.2

8.1

8.0

7.9

7.8

7.7

7.6

7.5

7.4

7.3

7.2

7.1

7.0

6.9

6.8

6.7

k

0. 1411

.1990

.2431

.2800

.3122

.3412

.3676

.3919

.4146

.4350

.4560

.4750

.4931

.5103

.5268

.5426

.5578

.5724

.5864

.6000

.6131

.6258

.6380

.6499

.6614

.6726

.6834

.6940

.7042

.7141

.7238

.7332

.7424

p/2

31.26

31.10

30.95

30.79

30.64

30.48

30.33

30.17

30.02

29.87

29.71

29.56

29.41

29.26

29.11

28.9fi

28.81

28.66

28.51

28.36

28.22

28.07

27.92

27.77

27.63

27.48

27.34

27.20

27.05

26.91

26.77

26.63

26.49

b

6.6

6.5

6.4

6.3

6.2

6.1

6.0

5.9

5.8

5.7

5.6

5.5

5.4

5.3

5.2

5.1

5.0

4.9

4.8

4.7

46

4.5

4.4

4.3

4.2

4.1

4.0

3.9

3.8

3.7

3.6

3.5

3.4

k

0. 7513

.7599

.7684

.7766

.7846

.7924

.8000

.8074

.8146

.8216

.8285

.8352

.8417

.8480

.8542

.8602

.8660

.8717

.8773

.8827

.8879

.8930

.8980

.9028

.9075

.9121

.9165

.9208

.9250

.9290

.9330

.9367

.9404

P/2

26.35

26.21

26.07

25.93

25.80

25.66

25. 53

25.39

25.06

25.13

24.99

2486

2473

2460

2448

24.35

24.22

24.10

23.97

23.85

23.72

23.60

23.48

23.36

23.25

23.13

23.01

22.90

22.79

22.67

22.56

22.45

22.35

b

3.3

3.2

3.1

3.0

2.9

2.8

2.7

2.6

2.5

2.4

2.3

2.2

2.1

2.0

1.9

1.8

1.7

1.6

1.5

1.4

1.3

1.2

1.1

1.0

.9

.8

.7

.6

.5

.4

.3

.2

.1

k

0.9440

.9474

.9507

.9539

.9570

.9600

.9629

.9656

.9682

.9708

.9732

.9755

.9777

.9798

.9818

.9837

.9854

.9871

.9887

.9902

.9915

.9928

.9939

.9950

.9959

.9967

.9975

.9982

.9987

.9992

.9995

.9998

.9999

p/2

22.24

22.14

22.03

21.93

21.83

21.73

21.63

21.54

21.45

21.36

21.27

21.18

21.09

21.01

20.93

20.85

20.78

20.70

20.63

20.56

20.50

20.44

20.38

20.32

20.27

20.22

20.17

20.13

20.10

20.07

20.04

20.02

20.01

ANALYSIS OF FOLDS

The analytical classification and measurement of 

folds can be done only for complete or partial flexures, 

whose stratigraphic traces or parts thereof are visible 

and mappable in satisfactory profiles. Two general alternatives exist for approaching this problem. The 

mapped arcs representing stratigraphic traces may be 

fitted empirically to preselected curves, using some system of coordinates; or these arcs may be represented 

analytically in terms of invariants of such curves, 

without reference to any system of coordinates. As no 

folds are simple geometric structures, the analysis of 

stratigraphic traces by either method will invariably
CLASSIFICATION, DELINEATIONj AND MEASUREMENT OP NONPARALLEL FOLDS 119

result in approximations, but these can be made as accurate as the work requires. Such analytical methods 

may be useful in certain practical applications, such as 

subsurface geologic or geophysical prospecting; but 

the principal application will probably be in the classification of folds and in theoretical studies relating to the 

mechanics of folding.

The conies, being relatively simple curves of the second degree, are first considered in the analytical representation of stratigraphic traces. Any 5 points, of 

which 3 or more are not collinear, may be fitted to an 

ellipse or to a hyperbola; 4 points to a parabola; and 3 

points to a circle. Mertie (1948) has shown that 5 

points on a stratigraphic trace may be fitted graphically 

to an ellipse by the application of Brianchon's theorem. 

Analytically the procedure is as follows. Assume a 

random origin of rectangular coordinates, and select 5 

points on the mapped stratigraphic trace, preferably 

about equally spaced along the arc. The coordinates 

of these points are then measured in terms of the selected origin. The equation of a conic passing through 

these 5 points is expressed as follows:

xy

X?

y\

2/2 

1/4

=0

where (»i, yO, (a?2 , y2 ), (a?8 , y8 ), (a?4 , 2/4), and (a?5 , y5 ) 

are the coordinates of the 5 points. This determinant 

is reducible order by order by either of the 2 methods 

of condensation given by Muir and Metzler (1933). 

It becomes finally the general equation of the conic 

Ax*+Bxy+ Cy*+Dx+ Ey+F =0

in which the 6 constants are numerically determined. 

The discriminant of the general conic is 

If A ?£ 0, the conic is an ellipse, hyperbola, or parabola, 

depending respectively whether BZ 4AC is less, 

greater, or equal to zero. The equation is unlikely to 

represent a parabola, as it is not possible in general to 

pass a parabola through 5 random points. The equation of the ellipse or hyperbola may now be simplified 

to one whose axes coincide with the axes of coordinates.

The coordinates of the central conies are «= B*" 

and 

2AE-BD

The resulting simplified equation will have the form 

of 

where 

If by any chance the general equation of the conic 

should represent a parabola, a conversion to the standard form yz=2mx may be accomplished by equally 

well-known methods of analytical geometry.

The simplified equation of the ellipse or hyperbola 

may need to be further modified to fit more closely the 

entire stratigraphic trace. To accomplish this objective, measure and record the coordinates of a considerable number of equally spaced points, including 

the five original points, in terms of the new origin of 

coordinates. An equal number of residual equations 

are then derived, having the general form of 

and the angle between the old and new systems of coordinates is 0=1/2 arc tan A-C

which are linear in A' and B' '. From these, two normal equations will be obtained, which may readily be 

solved for A' and B'. Computation of the residuals 

will determine the closeness of fit.

The fitting of stratigraphic traces to preselected 

curves is a problem that differs from the fitting of observed data to empirical curves, as described in many 

textbooks. Observed measurements are automatically charted in relation to an origin of coordinates. 

Stratigraphic traces as premapped arcs do not determine or suggest where the origin of coordinates should 

be placed. In the method of fitting traces to conies, 

as outlined above, random origin is assumed, but the 

resulting equation of the conic contains the information needed for a close placement of the simplest origin 

of coordinates. This method cannot be used advantageously for cubics or algebraic curves of higher degree. Thus, nine points are required to define >a cubic, 

and the resulting equation would have to be written 

as a determinant of the tenth order. The labor involved in clearing such a determinant is prohibitive; 

the analysis that would be required for reducing the 

equation to its simplest form, by the transformation of 

coordinates, is difficult; and the final least square adjustment of nine coefficients would likewise be impracticable. Neither can the method be applied to transcendental curves.

Higher plane curves may be used, however, if a 

random origin of coordinates is used. Many higher 

plane curves are available, some of which simulate 

parabolic, hyperbolic, or elliptic curvature, whereas 

others are exponential, logarithmic, trigonometric, or 

hyperbolic in nature. In particular, the polynomial 

y=a+ &#+<?#+ da?3 4- .... may be adjusted by least
120 SHORTER CONTRIBUTIONS TO GENERAL GEOLOGY

squares to fit any nonperiodic curve, and further 

diversity may be added by substituting for y any function of y. Such curve fitting, referred to an arbitrary 

origin of coordinates, may be useful in practical work 

such as underground exploration, but is of little value 

for the classification of stratigraphic traces or in other 

theoretical applications, because curves referred to different systems of coordinates are not comparable.

The utilization of invariants that are independent 

of any system of coordinates is a much better method 

for the classification of stratigraphic traces. Algebraic relations between such invariants are called intrinsic equations. Figure 27 illustrates these invariFIGDEB 27. Diagram illustrating angle of contingence (x), length of 

arc (a), and radius of curvature (R), which are used in formulating

intrinsic equations.

instead of the radius of curvature.

The curvature (K), where £=-«' may be used

ants, which constitute the length of an arc, s PiPz, 

the corresponding angle of contingence, x? and the 

radius of curvature, R=OP2 . The curvature K is the 

reciprocal of the radius of curvature. Any two of 

these variables will determine the shape of a curve, 

without locating it in <a plane. Equations in s and \ 

are called Whewell equations; those in s and R (or K) 

are called Cesaro equations; and those in x and R (or 

K) are called Euler equations. It is easier to measure 

lengths of arcs and angles of contingence than to 

measure R or K, for which reason Whewell equations 

are the most practical for the present purpose.

The method consists in measuring the lengths of 

arcs and the corresponding angles of contingence of 

stratigraphic traces in the profile of some actual fold, 

covering if possible the entire fold for each trace. The 

photograph of a section of a fold, transformed if necessary into a plane normal to an axial or an apical line, 

will serve as the original data. These measurements 

are then fitted to the intrinsic equation of some assumed curve by the method of least squares. The 

residual equations are first formed, from which the 

normal equations and the values of the constant parameters are determined by well-known methods. 

Computation of the residuals will determine how 

closely the assumed curve can be made to fit the stratigraphic traces. If the parameter or parameters of the 

equation are not linear and no algebraic transformation will lead to a linear relation, other least square 

methods are available for determining the values of 

the parameters. Such methods are clearly presented 

by Scarborough (1950).

Cartesian or other coordinates are not used in the 

methods above described, but when it is desired to utilize Whewell equations, it generally is necessary to 

derive them from conventional equations. The procedure for obtaining a Whewell equation from an explicit equation in cartesian coordinates is as follows: 

Let the given equation be 

»«=/(*) 

Thentan x=/(a;) 

But the differential of the arc of any plane curve is 

Therefore 

and

After this integration is performed, we have 

s=F(x)

Eliminating a? from equations 

tan *=f(x) 

and 

s=F(x) 

we obtain the desired Whewell equation 

Other methods are available for the conversion of implicit cartesian, polar, and parametric equations to 

Whewell equations.

Intrinsic equations of the hyperbola and ellipse are 

not well suited to this method, unless one is versed in 

the use of elliptic functions. The Whewell equation 

of the parabola 2/2 =4aa?, however, is 

s=a[sec % tan x+log (sec %+tan x)l

Probably the best application of Whewell equations in 

matching stratigraphic traces is in the use of higher 

plane curves, particularly transcendental curves, with 

arcs that have a general resemblance to the conies. A
CLASSIFICATION, DELINEATION, AND MEASUREMENT OF NONPARALLEL FOLDS 121

few such equations that might prove of interest, followed by their Whewell equations, are given below.

Parabolic catenary: y=a cosh (^j 

s=a tan x

Catenary of uniform strength: y= a log sec (-J

s=o log tan (^ 

Semicubical parabola: Say*=2a?

9s = 4o(sec3 x 1)

Cycloid: x=a vers-1 (|) + (2oy- y2)* 

s=4a sin x

Epicycloid: a; = (a+6) cos t+b cos |~ (a+6) -^J 

y=(a + b) sin f+6 sin j (a + 6)--^ J

Lo+26' xJ

5= - 46(o+b) .  - sin 

a

LITERATURE CITED

Hills, E. S., 1953, Outlines of structural geology: London, 

Methuen & Co., Ltd., 3d ed., p. 85-86.

Mertie, J. B., Jr., 1940, Stratigraphic measurements in parallel 

folds: Geol. Soc. America Bull., v. 51, p. 1107-1134.

 1944, Calculation of stratigraphic thickness in parallel 

folds: Am. Assoc. Petroleum Geologists Bull., v. 28, p. 

1376-1386.

 1947, Delineation of parallel folds and measurement of 

stratigraphic dimensions: Geol. Soc. America Bull., v. 58, 

p. 785, 779-802.

 1948, Application of Brianchon's theorem to construction

of geologic profiles: Geol. Soc. America Bull., v. 59, p.

767-785. 

Muir, Thomas, and Metzler, W. H., 1933, A treatise on the theory

of determinants: New York, Longmans, Green & Co., p.

64-65, 67. 

Nevin, C. M., 1942, Principles of structural geology: New York,

John Wiley & Sons, Inc., p. 46-48. 

Scarborough, J. B., 1950, Numerical mathematical analysis:

Baltimore, The Jolfns Hopkins Press, 2d ed., p. 463-469. 

Soreau, Rodolphe, 1921, Nomographie ou traits des abaques:

Paris, Librairie Joseph Gilbert, p. 94.

Spenceley, G. W. and Spenceley, R. M., 1947, Smithsonian elliptic functions tables: Smithsonian Misc. Colln., v. 109, p.

1-366. 

Van Hise, Charles R., 1896, Principles of North American preCambrian geology: U. S. Geol. Survey 16th Ann. Rept., pt

1, p. 599.

INDEX

Analysis of folds..__.........._........_....._..........____... 118-121

Apical line of folds........................................................ 96,96,107

Application and construction of ellipses_______. ___________ 112-116 

Arches_______________________________________ 92 

Axes, major and minor...._____________________.._____ 97

Axial line.............. ................._............................. 95,97,99

Axial plane......_____._.___ 96,96,97,99,102,103,105,106,107,108,109,114

Axial surface--.....,... ................. 95,96,97,108,104,106,107,108,109,114,115

Bibliography.___________________________________ 121 

Biparted hyperboloid of revolution_____________.________ 107

Canoe-shaped domes . ........_____._____________ 95,96,107

Canoe-shaped folds. _____.__._________._ 95,96,107,109,110, 111, 112 

Cartesian sections_________________________________ 107 

Circles and circular arcs........._._______.._____ 92,93,96,99,110,111

Circular arcs...........-.-. ....-..................-... .... 92,93,96,99,110,111

concentric..__________________________________ 99 

Classification of folds..___________________.__________ 95-97 

Cognate curves__________________________________ 92,93 

Cognate domed surfaces . ______________.. 97, 111 

Cognate quaquaversal domes__________________________ 111 

Cognate folds.....________.__._____.__...___ 96,103-105,106,115

Composite curves- _______.__ 92,93

Composite domed surfaces________________________ 97,111-112

Composite domes________________________________ 109

Composite folds..._________________________ 96,104,106-106,108

Concentric folds._____________.___________________ 96,99

Congruent folds ________._ 95,115

Conic sections__________________________________ 92

circles and circular arcs____________________ 92,93,96,99,110,111

ellipses and elliptic arcs___________________________ 92,93,

95,99,101,102,104,107,108,109,112,113,114,115,116

hyperbola and hyperbolic arcs ...__.______________. 92,93 

parabola and parabolic arcs_________________________ 92,93 

Conies...___.__.__..__________..._______________ 119 

Congruent similar folds______________________________ 99 

Construction and application of ellipses___________________ 112-116 

Cupolas. . _____________ 95 

Curved axial surface._._....._._____________________ 96

Cylindrical folds.___.__........_._..._.__....._.......___.. 96-107

cognate... ..............................__............. 96,103-106,106,115

composite.................................................... 96,105,106,108

concentric folds.....__....._._.____.____________._ 96,99

congruent folds______________________________ 95,115 

nonparallel (tee alto under namet of particular foldt)...... 92,99,107,113,114,116

parallel..................... ....................... 96,96,97,99,101,104,106

plunging______._________.______________. 95,96,107 

pseudoparallel ..... . 101,103,106

similar............. ........................ 91,96,96, 97,99,101,102,106,115

supratenuous_______._.__.___.____________ 105

.................................................................... 96

canoe-shaped._________.___....______________ 95,96,107

cognate quaquaversal .... 111

composite. . ______.. 109 

configuration of ... . ..._______.. _ 109 

elongate............ .............__.................... 96, 96,107,111-112

principal sections of. __... 96,97,109 

quaquaversal cognate... __... 111 

Domed surfaces...______.____________.__ 97,107,109,110, 111, 112 

cognate. .. _____ 97,111 

composite__.____._ .......___________... 97,111-112

Ellipses, construction and application .... ________.. 112-116

Ellipses and elliptic arcs____ 92,93,95,99,101,102,104,107,109,112,113,114,115,116 

Ellipsoid, triaxial . .. . .. . .............................. 107

Ellipsoid of revolution.. . ______... . ...... 107

Page

Elliptic arcs, application of in reconstruction of nonparallel folds 97

measurement of areas under. 116

measurement of perimeters of _.. 116,117,118

Elliptic functions of conic sections..., ____.... 117,118

Elongate domes............................................... 95,96,107,108,111-112

Evolute parallel curves............................. 91,92,99,107,109,113,114,115,116

Folds (see alto under namet of foldt), properties of 92,95,

96,97,102,103,105,106,107,108,109,115 

reconstruction of. ..................................... 97,112,113,114,115,116

Qlissettes.. ........................................................... 91,108,112

Horizontal translations.....____._______.......... 97,103,104,106,109

Hybrid stratigraphic traces_______.___.___....___. 96,99,104,106,106

Hybridized composite folds______________ . 106 

Hyperbola and hyperbolic arcs . . 92,93 

Hyperboloid, biparted...... .. 107

Introduction...._____._..._ __ ....... 91-92

Line integrator in measurement of stratigraphic thickness 116

Linear translation____________________________ 97,99,101,105

nonproportional___.__.. ........ 101-102

Measurements, stratigraphic thickness. 116,117,118

Noncylindrical folds...................................... ........ 96-97,107-112

cupolas. 96 

domes canoe-shaped 95,96,107 

cognate surfaces of..... 97, 111

composite surfaces of.. 97,111-112 

elongate.. ......................................... 95,96,107,108,111-112

parallel surfaces of._______.....____ 97,109,110

principal sections of.__________________- .. 96,97,109 

similar surfaces of. 97,110, 111 

oblate and prolate.._______...______ 110, 111, 113,114 

quaquaversal......................................... 95,97,107,109,110,111-112

Oblate and prolate noncylindrical folds ... .. .... 110, 111, 113,114

Parabola and parabolic arcs.. . ^ 92,93 

Paraboloid, elliptic. .. .._.................. ................... 107

Parallel curves.... ............................................. 92,93,95,104

evolute .................................... 92,99,107,109,113,114,115,116

involutes.. ............................. 92,97,99,107,109,113,114,115,116

Parallel domed surfaces__________________________ 97,109-110 

Parallel folds.... -.. . . 96,97-99

construction____ , 115 

Parallelism, Leibnitz' definition of.. _ 92,93 

Plane curves. 92-95

higher ................ ... .-............ . 92,93,99,106,119

Planimeter, polar U6 

Plunging fold.. ... . 95, 96,107

Prolate and oblate noncylindrical folds... 110, 111, 113,114 

Pseudoparallel curves .. ... - 93, 97,103

Quadric surfaces.. ............................ .... 107,109

biparted hyperboloid.--.. . ..... ...... 107, 111

ellipsoid, triaxial.. ......... ............... ................ 106,109

elliptic paraboloid_________________________ 107, 111 

oblate.. -.- - 110,111

principal sections of. . . .. 107 

prolate________... . 110,111 

surfaces of revolution... ......................... - 107,109,110, 111

surfaces parallel to . .... HO

123
124 INDEX

Page

Quaquaversal domes, cognate._______-_________________ 111 

Quaquaversal folds. . 95, 97,107,109,110,111-112

Quarter ellipses.____. __ 93,95, 96,105,106,108

Semiellipses..__________.________________________ 101 

central (coaxial). _--.__._..._._-..._.________.____ 95 

cognate... __............ ....................... 101,102,104, 108,118

mixed__..._........................... ............___..___ 96,105

noncoaxlal. _____________ _________________ 95, 99,117 

semiperimeters, table of__ _________________________ 118 

similar.......-............. ..........!....................... 95,99,101,108

Similar curves, definition oL___.__.__..________._________ 92, 93

congruent curves__________. _____________ 92, 93, 95, 99,101

Similar domed surfaces_;________________^_______ 97,110-111

Similar folds.._____.___......................... 95,96,97,99-103,105,106,115

Similar noncongruent ellipses__________________________ 101 

Simple folds._______.___.----.---...--.-........___...........__ 95-97

Stratigraphic thickness._____._...____......_.________ 116-118

Page 

Stratigraphic traces............ ............. ........... ......... 96,104,114

change of scale.___......._........_...-.--....-...-. 97,99,101,103

eccentricity. - - 97,99,101,103

hybrid ..... . 96,99,104,105,106

linear translation. .. ........ 95,97,99,101,102,103,105

mixed... ._ .. .._. 96,104

rotation of axes..__......-..----.-...-..---...----...- 93,95,97,104,106

unrelated_ ... - 96 

variables available _ 96,97

Summary of cylindrical folds..... . 106-107

Supratenuous fold-..--....... . 105

Symmetry and asymmetry, applied to cylindrical folds..__ ...___ 97,109

Thickness, Stratigraphic _.........__._ 92,95,101,103,105,107,116,117,118

Traces, Stratigraphic. (See Stratigraphic traces.)

Trammel of Archimedes______. ______.__.____..._____ 108,112

Vertical translations__.____.. __... . 97,104,105 

differential.-.. _....... 102

Vertical and horizontal translation _ __._______. ______ 105,109

Whewell equation... .... ... .. .. .. . . 120,121

o