@ ~ComputerGraphics, Volume23, Number3, July 1989

Ray Tracing Deterministic 3-D Fractals
J o h n C. H a r t * , D a n i e l J. S a n d i n * , L o u i s H. K a u f f m a n t*Electronic Visualization Laboratory
tDept, of Mathematics, Statistics and Computer Science
University of Illinois at Chicago

Abstract
As shown in 1982, Julia sets of quadratic functions
as well as many other deterministic fractals exist
in spaces of higher dimensionality than the complex
plane. Originally a boundary-tracking algorithm wasused to view these structures but required a large
amount of storage space to operate. By ray tracingthese objects, the storage facilities of a graphics workstation frame buffer are sufficient. A short discussionof a specific set of 3-D deterministic fractals precedes
a full description of a ray-tracing algorithm applied
to these objects. A comparison with the boundarytracking method and applications to other 3-D deterministic fractals are also included.
CR Categories and Subject Descriptors:
1.3.7 [Computer Graphics]:
Three Dimensional
Graphics and Realism - - Color, shading, shadowing
and texture.
G e n e r a l T e r m s : Algorithms, Theory.
A d d i t i o n a l K e y w o r d s a n d P h r a s e s : fractal,
quaternions, distance estimate, ray tracing, surface
determination.

1

dard 2-D frame buffer techniques have provided sufficient visual information about the structures sincemost of the research has concentrated on the dynamics of complex variables. However, recent investigations into higher-dimensionM dynamical systems[14,15,3,5,17] have shown the need for 3-D visualization tools that will give researchers a better understanding of these objects.
One such method is ray tracing, but this method
is prohibitively slow unless an efficient ray-surface intersection computation is used. While these functions
are available for Euclidean surfaces, they do not exist(yet) for fractal ones. However, using an unusual construction called the unbounding volume, made possible by a recent advance in the study of dynamics,
swift ray tracing of these deterministic fractal objects
is possible.
Prior to the description of the algorithm, a specificfamily of 3-D deterministic fractals, quaternion Julia
sets, is outlined. The generation algorithm is thendeveloped using this family as example. Rendering
procedures specific to fractal surfaces are then discussed. Finally, the algorithm is compared with its
predecessor [14] and applications to other families of
3-D deterministic fractal objects are shown.

Introduction
Dynamics
nions

in

the

Quater-Computer graphics has greatly aided the investigation of the dynamics of iterative functions. Stan-2

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The dynamics of quadratic functions have been observed mainly in the complex plane. However, as
shown first in 1982 [14], they exist in the 4-D spaceof the quaternions as well. A discussion of the special properties of Julia sets in the quaternions, forwhich the ray-tracing algorithm was developed to visualize, is preceded by an introduction to dynamics
and quaternion algebra.

©1989

ACM-O-89791-312-4/89/O07/0289$00.75289

,,(S,GGRAPH'89, Boston, 31 July-4 August, 1989

2.1

Dynamicsof Quadratic

Functions

The examples used in this paper are derived from thequadratic function

f (z) =+

(1)

where z is the iterated variable and p is a constantparameter of the equation.
The dynamics of a function f are expressed as then-fold application of function f to an initial valuez. The result is denoted as fn (z) and should not be
confused with simply raising the result of f ( z ) to thenth power.
The resulting value fn(z) is used to classify the
initial point z depending on its attraction to infinity.
Two sets may be constructed under this classification.
The filled-in Julia set/C~ and the Mandelbrot set .M.
D e f i n i t i o n 1 /C~, = {z: lira f ~ ( z ) 7 4 oo}rl---* Cx:)

Definition 2.h4 = { p : li~noof ; (zc) -74 oo, f~(zc) --- 0}
Note that z¢ is the critical point of the function.
There is only one critical point of eq. (1) and it isalways 0. Several critical points are common for polynomials of degree 3 or greater.
The interesting property that Julia and Mandelbrot sets share is that they are both fractal [12] possessing detail at every level of magnification.
2.2

The

i j = k; j k =i; k i = j ,(3)

ik = - j ,

(4)

and

ji =-k;kj

revealing an unfortunate side effect of the quaternions: noncommutative multiplication.
2.3

Julia

Sets

in theQuaternions

By using the rules of quaternion algebra, eq. (1) canbe iterated in the quaternions and Julia sets may becomputed. Since the complex plane is a subset ofthe quaternions, the same complex Julia sets exist in
the quaternions but often have extensions outside thecomplex plane. In fact, if ~ has an imaginary component, then the extensions are nontrivial, containing
more information than their complex subsets.
A subset of these extensions can be visualized in
3-D by examining the intersection of the 4-space witha 3-space such as that spanned by 1 , i , j at Ok. It
should be mentioned that the Julia sets of eq. (1) inthe 3-space spanned by i t j , k at 0 are always concentric spheres centered at the origin [8].
An interesting property about quaternion Julia sets
is that given two complex Julia sets differing only bya rotation about the origin, their supersets in 3-D
may have completely different shapes. The rotationof the Julia set in the complex plane is computed byincorporating the homeomorphism

Quaternions

The values z and p are commonly defined as realor complex. However, these values may be definedin any algebraic system closed under addition andmultiplication. One such system, the quaternions [7],possesses the additionM benefit of having four dimensions.
D e f i n i t i o n 3 A quaternion value q is a four-tuple
consisting of one real part and three imaginaries

q = ql + qii + qjj + qkk
where i~ j~ k are imaginary units,i = j2 = k: = -1.

(2)

Algebraic operations can be defined in the quaternions by treating the quaternion values as polynomials of three variables i, j , k. For example, the coefficients of the sum of two quaternion values may
be found by adding their corresponding coefficients.
290

Quaternion multiplication is also similar to polynomial multiplication but with the special cases

g0(z) =

(5)

into the iterated function such that

L,,e(z) =(6)which suffices to rotate the positive real axis into thepositive imaginary axis and so forth in a counterclockwise manner about the origin in the complex
plane. The resulting Julia set is merely rotated in
the complex plane but appears quite differently in thequaternions since its intersection with the imaginary
3-space is changed. See [15,8] for details.

3

Ray

Tracing

3-D

Julia

Sets

Ray tracing is one of the more realistic methods ofrendering objects. Easily accounting for hidden surfaces and self-shadowing, it also provides a methodfor displaying reflection, transparency and refraction.
Mathematical objects may be ray traced by detecting

, i ~ Computer Graphics, Volume 23, Number 3, July 1989

estimated in time proportional to the time it takes todetermine if the point is external to the object.
The lower bound of the distance from a point external to the deterministic fractal set is given asd(z) =

sinhG(z)2ea(')lG'(z)llogG(z),(7)where G(z) is the electrostatic potential at point z
and Gr(z) is the gradient of this potential. For thequadratic family, the approximationd(z) =. If' (z)l logfn(z)21/'"(Z)I
Figure 1: A quaternion Julia set before and after aquarter turn in the complex plane.

their boundaries during a ray-casting step and rendering the surfaces by allowing the ray to be deflected
off to a light source.
A naive method to ray trace a quaternion Julia set
is to sample each point at a given resolution alongeach ray. This is not an entirely ridiculous method
since it is the basis of some volumetric rendering algorithms [10]. However, it is not practical when applied
to fractal objects since each point's classification may
rely on a large number of function iterations. Withthe use of a new ray-tracing mechanism, the amountof sample points per ray is greatly reduced.
3.1

Unbounding

Volumes

One method of increasing the speed of ray tracing is
the use of bounding volumes. A bounding volume,usually a sphere or ellipsoid, envelopes several surfaces such that if a ray does not intersect the bounding volume it does not intersect the surfaces corttairted
in it.
Bounding volumes are quite useful in hastening ray
tracing of stochastically-defined fractal surfaces [9,4].
Unfortunately their application to deterministic fractals has not been as successful. However, with thediscovery of the distance estimate, we can increase
the speed of ray tracing deterministic fractals using

uubounding volumes.
Unbounding volumes are defined as volumes that
do not contain any part of the object. Thus, given
any point outside the object, the ideal bounding volume is the largest volume that does not intersect theobject centered at the point. If this volume is a spherethen its radius is the distance to the object. Givena point and a deterministic fractal object, its exact
distance cannot be computed efficiently but it can be(8)

is sufficiently accurate [13,6]. See [17] for the computation of f'(z).
By using a distance estimate we can define an unbounding volume of a deterministic fractal set as asphere that is guaranteed not to intersect 1 nor contain the set in question. Since the distance estimate
may be much smaller than the distance along the ray
to the object, several repeated distance calculations
must be made as the ray is traversed from eye to surface.

3.2

Ray

Traversal

Given the set of unbounding spheres completely surrounding an object, a ray is traversed from the eye
through the projection plane to the object, testingand incrementing at each point along the ray. By incrementing by the radius of the unbounding sphere,
we leap along the ray until we approach the surface.
As the current point on the ray approaches the surface, the unbounding spheres get smaller and smaller.
To hasten convergence, a small number c is defined as
the minimum ray increment. This increment should
be set to give the best depth resolution with respect
to the resolution of the projection plane.
The ray traversal equation may be stated inductively given the eye position r0 and a point in the
projection plane p~,u as
rn+l = r , + m m a x d ( r , ) , e.

(9)where m is a unit slope vector of the raym -P~'~ - r0
IP=,y - r0l(10)

1 W i t h t h e e x c e p t i o n of a single p o i n t .
291

'89, Boston, 31 July-4 August, 1989

ro

~i

r2

|¢ ¢
r 3 r 4 r5

Figure 2: Ray traversal using unbounding spheres.

3.3

Thin Objects

Often the extensions of K: into 3-D are very thin, suchas when ~ is a dendrite. This creates the possibility
that incrementing by ~ may traverse the ray completely through the object.
This problem has also had manifestations in the
2-D study of these images such as the complex Mandelbrot set. To show that the "islands" off the main
continent of .M are connected to it, the Mandelbrot
set may be defined computationally as.A4, = {z: d(z) < e}(11)where d 0 is the distance estimate as defined in eq. (7).
The result is the "hairy" Maudelbrot set [13] revealing
its dendritic structures 2.
A similar technique is used to ray trace dendritic
sheets in 3-D. By terminating ray traversal when
the distance is less than the minimum ray increment,
thickness is added to the object while maintaining its
structure and detail.

3.4

Avoiding Bad Distance Estimates

When the approximation to eq. (7) is used, it is inaccurate when z is far from the set. This can result in
exaggerated distance estimates which could possibly
push the ray far into the interior of the object.
To avoid these bad estimates a single bounding volume may be used to contain the fractal set if it can bebounded. Another alternative is to set a maximum
distance to increment along the ray.

4

Rendering

2 N o t e that these h a i r s m a y b e seen v e r y clearly as t h e set

292

4.1

NormalApproximation

One reason fractal lines, such as the border of/C, areso interesting is that they are nondifferentiable. The
slope at any point is undefined because closer examination shows that the point has different surroundings. Hence, when expanded to surfaces, 3-D fractal
surfaces are nondifferentiable and thus have no exact
normal defined.
In order to realistically render these surfaces ashading model must be used which requires the definition of a surface normal. Normals may be approximated by examining a point's relationship with its
surroundings. Two approximations have been foundto work quite well: Z-buffer neighbors, previously discussed in [14], and the gradient.
4.1.1

Z-buffer Neighbors

As shown in [14], the surface normal of a fractal surface may be approximated as the cross product of
two vectors embedded in the surface. Given a bufferof visible z-values Z we can define three points

x
Y

=
=

{~, 0, z~+~,~ - z~,~}{0,e,z~,v+c-z~,u]

(12)
(13)

0

-=

{0,0, Z ~ , y - Z z , ~ }(14)where e is the width of an element in the z-buffer.
The surface normal may then be approximated as thenormal of the plane defined by these three points.

Fractal Surfaces

The deterministic family of fractals has provided computer graphics with the most complicated borders.
.Me- 2v[,

The surfaces defined by these borders in 3-D, although quite chaotic, often reveal the structure of theobject. A proper rendering of a fractal surface should
reveal its order while hinting at its chaos.
Since the surface of a fractal is infinitely convoluted, its normal can only be approximated. The approximated normal signifies the structure of the surface while at finer resolutions the light is scattered inall directions. Thus the surfaces should be diffusely
shaded using the Lambertian model.
Also, to achieve the most information from eachview, it is often better to use axle light instead ofambient light. By defining a point light source at theeyepoint, every viewable point on the object will receive light and thus even heavily-shadowed sections of
the object will reveal information about themselves.

4.1.2

Gradient Computation

The previous method is a useful normal determination tool if a Z-buffer is maintained during rendering.
Ray tracing does not require a Z-buffer so a normal~Computer Graphics, Volume 23, Number 3, July 1989

F0
Constant
LinearQuadratic

approximation method using a single point in 3-space
would be more useful.
This can be accomplished by computing the gradient of a point on the surface. The gradient may becomputed in a 3-D density map as

EffectInverse clarity
Even clarity
Exaggerated depth

Table i: Depth-cueing exponents and their effects on
renderings

N z = D~+c,u,z - D=-e,u,z
N y = D=,y+~,z - D ~ , y - e , z
N z = D=,y,z+e - D ~ , , y , z - e

05)

where D=,y,z is the density at the point x, y, z.
The density function of a deterministic fractal is defined on its exterior and can be any continuous function based loosely on the distance to the set. Twouseful functions are the potential G 0 and the estimated distance d 0. The latter should be used whenpossible since it is more closely associated with theactual distance although the former works when adistance estimate is not defined.
Other gradient functions may be defined based on
the number of samples taken. The 6-point gradient
may be augmented by adding samples from points
sharing edges producing an 18-point gradient. By
including points sharing corners, a 26-point gradientresults.
4.2

gives an even clarity appearance with close objects
appearing as detailed as far. Quadratic depth cueing gives an exaggerated depth cue, blurring distantsurfaces.
The c~ parameter is tuned to balance the equilibrium of order and chaos. Setting c~ too small willproduce a very noisy surface whereas a large o~ will
wash out detail. When defining o~ a good starting
point is to set it to an order of magnitude smaller
than the pixel width.

Clarity

Since these objects are fractal, they should reveal
more detail when closely inspected. However, if theminimum ray increment e is constant, the surfaces
will not reveal a finer structure when examined. A
variable-resolution system is required such that closer
sections of the object are defined at higher resolutions
as suggested in [2].
One method of increasing the depth resolution isto set c to a function of distance from the eye. The
clarity function P~,6 is definedr(d) =

Thus, by setting(16)

given d, the Euclidean distance from the eye to thecurrent location on the ray,d = Ira - rol.

Figure 3: The same Quaternion Julia set rendered
twice to show the difference between constant andlinear clarity.

(17)

The parameter 6 is a depth-cueing exponent and o~ isan empirical proportion defining the depth resolutionof the object.
Three effects are defined by varying the depthcueing exponent. When the clarity function is constant, inverse depth cueing occurs giving the appearance that farther objects have more detail. This may
seem useless but is quite adequate when viewing entire fractal objects from a distance. Linear clarity=- ,0l)(18)

a variable-resolution rendering system is constructed
allowing small details of the surfaces to be investigated without overcomputing the other visible surfaces. The increment e may also be used in the gradient computation as the distance along the axes tosample nearby densities.

5

Application
ministic

to

Other

Deter-

Fractals

The quaternions are convenient to observe 3-D dynamics since all three dimensions may be spanned
293

~~SIGGRAPH '89,Boston,31July-4August,1989
by a single variable. Other 3-D spaces may be constructed using multiple real or complex variables.
Complex Julia sets form a 3-D object when they arestacked [14,12]. The cubic connectedness locus is afour-dimensional object when its two parameters arecomplex [5,17]. Also, Iterated Function Systems may
be three-dimensional if they are specified with affine
transformations of three real variables [1].

5.1Julia Set Stacks

A Julia set stack may be specified in 3-D as a slice(i.e. zeroset) of the four dimensional space C × C
defined by the two complex variables z and c fromeq. 1. By looking at the z-planes and varying some
single-dimension function of c to define the third dimension, the Julia sets may be stacked to form a 3-D
object.
There currently is no proven distance underestimate for this set although some images may be generated using empirical formulas based on the Mandelbrot set distance estimate.

D e f i n i t i o n 4 C+n
D e f i n i t i o n 5 C- = {a, b: lim far,b(--a)-74 CX)}

A picture of this set may be found in [17], presumably
created using the technique outlined in [14].
At the moment, a distance estimate does not exist
for this set either. However, the cubic connectedness
locus has been proven to be connected [5] suggestingthat potential measurement and therefore distance estimation may be possible.
5.3

5.2

The Cubic Connectedness Locus

The cubic connectedness locus C is specified by the
cubic functionfa,b(z) = z a - 3a2z + b,

(19)

where z is a complex variable and a, b are complex
parameters. The parameter a is squared in the equation because the two critical points of the equation
are :t:a.
Since a and b are both complex, a double complex
plane is constructed that houses the four-dimensional
cubic connectedness locus. The locus in this case consists of two components, C+ and C-, based on thestatus of the appropriate critical point.
294

3-D Iterated

Function

Systems

The most useful forms of deterministic fractals are
Iterated Function Systems or IFS's. An IFS can becreated to simulate almost any form using the Collage
Theorem [1]. Then, given only the resulting set of
iterative equations, the form can be reconstructed.
Recently, deterministic IFS Julia and Mandelbrot
set functions have been discovered and their exteriors
have been categorized according to escape iterations
not unlike their quadratic counterparts [1,19]. Thissuggests that perhaps potential and distance measurements can be made on these sets as well.

6

Figure 4: Stack of Julia sets for Ira(c) = 0.

na{a, b: nli~noofa,b() "-~ (:x~}Comparison
Tracking

with

Boundary

The ray-tracing algorithm's predecessor, Boundary
Tracking [14], generates 3-D Julia sets by first locating a starting point on the boundary of the object and then recursively detecting its neighbors until the entire boundary is scanned. To converge,
this algorithm must constantly verify that neighboring points have not been previously tested, which requires the efficient storage of all previously generated
points. Thus, the Boundary Tracking algorithm runs
in object-space and therefore object-time a.
The main advantage of Boundary Tracking is that
objects are only generated once and may be repositioned as often as desired, requiring only re-rendering.
Using the ray-tracing technique, repositioning of theobject requires re-generation of the viewable sections of the object as well as re-rendering. However,since the image-time ray-tracing algorithm generates
these objects more efficiently than the object-time
Boundary Tracking algorithm, it is the best choicefor parameter-space animations such as varying/~ ineq. (1), 0 in eq. (5) or the k axis component of theviewable subspace of the quaternions.
3S¢e [20] for a discussion of image-space vs. object-space
and image-time vs. object-time.

@ ~The main disadvantage of Boundary Tracking isthat it requires storage of every point in the object.
This amount of storage can approach cubic proportions since the number of points defining a surface isO(rD), the resolution r of the surface raised to its
fraetal dimension D [12]. The ray tracing technique,using image-space, requires exactly r 2 + O(1) space 4,
the storage resources of a graphics workstation frame
buffer.
Another problem with Boundary Tracking is constant resolution.
However, a variable-resolution
Boundary Tracking algorithm was developed to create [16] by generating certain sections of the set already known to be closely examined in the fly-by
at higher resolutions. Although Boundary Tracking
saves computing time by generating the object only
once, the ray-tracing algorithm is the better choicefor fly-bys since its dynamic resolution allows a more
realistic inspection of the surface.
Finally, ray tracing allows certain special effects
such as reflections and refractions. The former produces the interesting effect of revealing only macroscopic images of its environment; the subtle details
are lost in the convoluted interreflections of the fractal surface. Refraction as well as simple transparency
should be avoided until a reliable internal distance estimate is developed to prevent minimum incrementsin the interior of the sets.

7

Conclusion

The research on this project began as a method ofvisualizing quaternion Julia sets in 3-D using theresources of a common computer graphics workstation. The first attempt relied on the Inverse Iteration
method of generating Julia sets [11,18] which operated in image-space and object-time but produced
less than satisfactory results due to inherent problemsof the process itself magnified by the addition of an
extra dimension [8]. The ray-tracing solution, being
forward iterative, does not experience the problemsof the Inverse Iteration method while still requiring
only image-space.
7.1

ParallelImplementation

ComputerGraphics,Volume 23, Number3, July 1989of the Pixel Machine, each processor connected only
to its portion of the frame buffer, dictates that imagespace, image-time algorithms will run at the most
optimal level.
The ray-tracing code is replicated into 64 equivalent programs running simultaneously as if in a race,
each generating and rendering its ~4th of the image.
Of course, the first operation of each program is tofind out which pixel with respect to the entire frame
buffer it is working on.
Currently, full screen images (1280 × 1024) takeabout an hour to generate. When positioning the object, lower image resolutions are used to create a moreinteractive environment. Also, reducing the depth
resolution will increase the speed of the algorithm.
7,2

Acknowledgments

The authors wish to thank Alan Norton and Charlie
Gunn for their communications, A T ~ T for its major grant to the Electronic Visualization Laboratory
which supported this research, Maggie Rawlings forher artistic advice with the illustrations, and Tom
DeFanti and Maxine Brown for their assistance withthe manuscript. This research would not have beenpossible without the efforts of the faculty, staff andstudents of the Electronic Visualization Laboratory.

References[1] Barnsley, M. F. Fractals Everywhere. Academic
Press, New York, 1988.
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Ray tracing deformed surfaces.

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[4] Bouville, C. Bounding ellipsoids for ray-fractal
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45-51.
Branner, B., and Hubbard, J. It. The iterationof cubic polynomials, Part h The global topologyof the parameter space. Acta Mathematica 160,
3 (1988), 143-206.

The current implementation of the algorithm runs on
an AT&T Pixel Machine with 64 parallel processors
each running at about 10 MFLOPS. The architecture[6] Fisher, Y.

4If the gradient normal approximation method is used, aZ-buffer is not required. The only other considerable amountof memory used is a small array the size of the maximum allowable iteration count used to optimize the computation of
the distance estimate [17].

[71 Hamilton, W. R. Elements of Quaternious, 3rd

The Science of Fractal Images.

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295

~~SIGGRAPH '89,Boston,31July-4August,1989[8] Hart, J. C. Image Space Algorithms for Visualizing Quaternion Julia Sets. Master's thesis,
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[10] Levoy, M. Display of surfaces from volume data.
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[11] Mandelbrot, B. B. Practal aspects of the iteration of z ~ Az(1 - z) for complex A and z. Annals New York Academy of Sciences 357 (1980),
249-259.
[12] Mandelbrot, B. B. The Fractal Geometry of Nature, 2nd ed. Freeman, San Francisco, 1982.

Figure 5: A dendritic quaternion Julia set, set in asea whose waves are periodic functions of the distanceestimate.

[13] Milnor, J. Computers in Geometry and Topology. Marcel Dekker, Inc., 1989, ch. Selfsimilarity and hairiness in the Mandelbrot set,
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[14] Norton, V. A. Generation and rendering of geometric fractals in 3-D. Computer Graphics 16, 3(1982), 61-67.
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[17] Peitgen, H. The Science of Fractal Images.
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Figure 6: Close up of the surface of the quaternion
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[18] Peitgen, H., and Richter, P . H . The Beauty ofFractals. Springer-Verlag, New York, 1986.
[19] Prusinkiewicz, P., and Sandness, G. Koch curveas attractors and repellers. [EEE Computer
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Figure 7: Close up of an interesting section of theJulia set for 0 = 110 °296