Computer aided contact lens design and fabrication using spline surfaces

A method of computer-aided contact lens design and fabrication uses spline-based mathematical surfaces without restrictions of rotational symmetry. The spline encompasses any piecewise function with any associated constraints of smoothness or continuity. The method comprises some or all of the following steps: data acquisition, three-dimensional mathematical surface model construction, posterior surface description, ray tracing for anterior surface, and peripheral edge system (PES) design. The result is a mathematical or algorithmic description of a contact lens. Based on the more powerful mathematical representation of splines, these contact lenses can have posterior surfaces that provide a good fit to corneas having complicated shapes. This enables the design and fabrication of lenses (including soft lenses) with good optics for irregularly shaped corneas.

FIELD OF THE INVENTION
 This invention relates generally to the design and fabrication of contact
 lenses, and in particular to a method using spline-based mathematical
 surfaces without restrictions of rotational symmetry.
 BACKGROUND OF THE INVENTION
 Besides the obvious cosmetic aspect, contact lenses generally offer
 improved visual acuity compared to spectacles. In some cases, the
 difference is dramatic. For example, in the case of the corneal pathology
 of keratoconus (a corneal condition in which there is local region of high
 curvature), contact lenses can often succeed in providing excellent visual
 acuity (perhaps 20/20 using the standard Snellen eye chart) whereas
 spectacles are not able to provide more than a minor improvement over the
 uncorrected vision. In addition to improved visual acuity, contact lenses
 are also indicated for other diverse purposes, such as a medicine delivery
 system or as a "bandage" for protection of the cornea after erosion,
 trauma, or surgery.
 Current contact lenses have shapes formed from relatively simple
 geometries, mostly spherical of different radii, or conic sections, etc.
 Consequently, there are many limitations, including poor fit for corneas
 with complex shape (such as might be found in keratoconus or in
 post-surgical corneas), edges that are uncomfortable, limited optical
 correction, limited ballasting/stabilization designs (for orientation of
 non-rotationally symmetric lenses), etc.
 The most complex designs of the anterior or posterior surface of a contact
 lens are based on two or three surface zones. They are usually spherical,
 or sometimes so-called "aspherical". Although "aspherical" literally means
 "not a sphere", this term is used more narrowly in the contact lens field
 to refer to what mathematicians call surfaces of revolution of conic
 sections or toric surfaces.
 Most contact lenses are either lathed directly, or molded from molds that
 were produced from pins and inserts that were lathed or ground. The
 lathing and grinding technology that is commonplace in the contact lens
 industry produces rotationally-symmetric lenses (except for toric lenses
 for astigmatism which are generally produced by a rather ad-hoc "crimping"
 method) and the shapes are fairly simple geometrically. In order to
 realize more general shapes, more sophisticated fabrication techniques are
 necessary. Computer numerical control (CNC) machining moves a cutting tool
 along a path on a part according to a mathematical model. The concept is
 that the computer instructs the machine how to make the complex shape, and
 the machine is capable of making such a shape. Furthermore, such high
 accuracy is achieved that the usual requirements of polishing the scallops
 (ridges) is greatly reduced or even eliminated. Some CNC machines can
 produce contact lenses that are non-rotationally symmetric.
 There is a widely-held precept that making custom shapes is impractical and
 costly--that "one size fits all" is the only economical method. The advent
 of CNC machining shatters some deeply-held beliefs about manufacturing. In
 the traditional manufacturing process, the notions of mass production and
 economies of scale are predicated on the assumption of producing many
 identical copies of a product; but these ideas date from the industrial
 revolution. "Any color as long as it is black" is a Model T concept.
 Nowadays, automobiles are manufactured efficiently despite the fact that
 each one that rolls out of the factory door has a unique permutation of a
 dizzying array of options. Mass customization can be realized by
 integrating computers into the manufacturing process such that each
 contact lens can be automatically produced to custom specifications; the
 computer simply uses the particular set of values of the parameters of the
 mathematical model for each unit. Concerns about minimizing the number of
 different stock keeping units (SKU's) could be a thing of the past by
 embracing concepts of just in time manufacturing.
 Although CNC machines enable the fabrication of complex surfaces, and the
 variation from one unit to the next, they require that a powerful
 mathematical model be used. In the evolution from traditional manual
 machining to automated CNC technology, all details must be specified
 competely and precisely.
 PRIOR ART
 U.S. Pat. No. 5,452,031, Sep. 19, 1995, to Christopher A. Ducharme of
 Boston Eye Technology, Inc. perpetuates the classic concept of a surface
 of revolution of a curve and is restricted to rotationally symmetric
 surfaces. Ducharme uses the word "spline" in a narrow sense, which will be
 further explained later, of a piecewise cubic function with continuity of
 position and the first two derivatives, and that patent does not include
 any broader mathematical specifications. He describes the use of the
 spline only for the peripheral zone, not for the optic zone. Furthermore,
 Ducharme does not address the determination of the optics for the lens in
 the case of a spline-based design. Although he mentions thickness, he does
 not provide any information that would enable someone to design a lens to
 achieve desired optics. He describes only a spline that joins the optic
 zone with C.sup.1 (first derivative) continuity. Ducharme's discussion is
 limited to C.sup.1 continuity with the optic zone because it is simply
 implementing the so-called "clamped" spline, and provides no indication
 whatsoever of more sophisticated mathematics to transcend such limitations
 on the level of continuity. His discussion is limited to polynomials.
 Ducharme assumes that the lens must be lathe cut.
 A related international application, PCT/US94/10354, published Mar. 16,
 1995, of David M. Lieberman is restricted to contact lenses smaller than
 the diameter of the cornea. It also only addresses the peripheral portion
 of the posterior surface. Lieberman does not provide details of the shape.
 He discusses a simple "brute force" collection of point data with no
 higher-order mathematical structure developed. He refers to the use of
 1500 points to describe the surface. He does not include any mathematical
 methodology. This lack of mathematical model results in a representation
 that is unnecessarily large and inefficient.
 A related U.S. Pat. No. 5,114,628, May 19, 1992, to Peter Hofer, Peter
 Hagmann, Gunther Krieg, and Eberhard Vaas of Ciba-Geigy Corporation in
 Germany shows the manufacture of individually fitted contact lenses from
 corneal topography but does not provide details of the shape, and does not
 present any higher-order mathematical description. Like the Lieberman
 patent, the Hofer et al patent does not include any mathematical
 methodology.
 Note that even a combination of these prior-art references does not provide
 the physical features of the present application.
 OBJECTS AND ADVANTAGES OF THE INVENTION
 Accordingly, several objects and advantages of this invention are a more
 powerful mathematical representation to enable more complex shape
 description, without restrictions of rotational symmetry, for contact
 lenses to improve fit, optics, patient comfort, and corneal health.
 The very general spline formulations used in this method incorporate many
 features, including (but not limited to):
 continuity of arbitrarily high order
 geometric continuity as well as parametric continuity
 shape parameters, if desired
 elimination of rotational symmetry restriction
 spline-based optical zone (not constrained to be spherical)
 ability to embed exact spherical zones
 eccentrically-located optical zone
 complex-shaped tear layer gap ("mismatch")
 capability to have non-circular periphery
 The novel techniques presented here enable the design and fabrication of
 contact lenses that transcend the state of the art. Based on the more
 powerful mathematical representation of splines, these contact lenses can
 have posterior surfaces that provide a good fit to corneas of complicated
 shapes. This enables the design of lenses (including soft lenses) with
 good optics for irregularly shaped corneas.
 Further objects and advantages of this invention will become apparent from
 a consideration of the drawings and ensuing description.

DRAWING REFERENCE NUMERALS
 41. annular border
 42. radial line
 43. center
 44. periphery
 45. four-sided regions
 46. space curve
 47. triangular surface elements
 48. vertex
 49. hexagonal surface element
 50. radius of spherical zone
 51. center of curvature
 52. axis of symmetry
 53. conic (elliptical) zones
 54. sharp joint
 55. transition zone
 56. anterior surface
 57. posterior surface
 58. peripheral edge system (PES)
 59. arbitrary surface element
 SUMMARY OF THE INVENTION
 This invention provides a general mathematical description, without
 restrictions of rotational symmetry, of the anterior surface (including
 the optic zone), posterior surface, and peripheral edge system (PES) of a
 contact lens. FIG. 21 shows contact lenses using spline-based mathematical
 geometry to describe the anterior surface, posterior surface, and
 peripheral edge system (PES). In each of FIGS. 21a, 21b, 21c, and 21d, a
 contact lens is depicted in exploded view, showing the constituent parts
 of the anterior surface 56, posterior surface 57, and peripheral edge
 system (PES) 58. Each of these components is divided into a collection of
 smaller pieces, joined together with mathematical constraints of
 smoothness. Note that the decomposition into smaller pieces can be done in
 a different manner for each of these components.
 DETAILED DESCRIPTION OF THE INVENTION
 In this application, we use more powerful and general spline-based
 mathematical geometry to describe the anterior surface (including the
 optic zone), posterior surface, and peripheral edge system (PES) of a
 contact lens. Splines can represent very general and complex shapes in a
 compact and efficient manner. The word "spline" was originally used to
 refer to a plastic or wooden lath that is flexible and is used by a
 draftsperson to produce a smooth curve through a set of points. This
 physical spline can be modeled mathematically, producing a piecewise cubic
 function with continuity of position and of the first two derivatives;
 this mathematical model is also referred to as a "spline".
 Splines are a rich area of mathematics and there is a wide variety of
 different kinds of splines, each possessing advantages and disadvantages.
 In the present application, the word "spline" is used to refer to a wider
 class of mathematical functions than what would arise from the
 mathematical modeling of the physical device. Specifically, the word
 "spline" refers to any piecewise function with any associated constraints
 of smoothness or continuity (not necessarily measured simply as continuity
 of derivatives). The pieces themselves are not limited to cubic
 polynomials; they may be polynomials of any degree, Zernike polynomials
 (used in optics), or even non-polynomial functions (examples include, but
 are not limited to, rational, trigonometric, exponential, hyperbolic
 trigonometric, other transcendental functions, Fourier series, wavelets,
 etc.).
 As an example, without limitation, a useful surface representation is the
 biquintic B-spline surface, expressed as a weighted average of basis
 functions:
 ##EQU1##
 where V.sub.ij are control vertices and B.sub.ij (u, v) are basis
 functions, which are piecewise polynomials of degree five in each of u and
 v. (Boldface is used to indicate that the function is vector-valued.) The
 polynomials are derived so as to be continuous up to the fourth derivative
 at their boundaries [Bartels, Beatty & Barsky, 1987].
 Splines provide a mechanism by which the smoothness between adjacent zones
 can be accurately specified. The idea is that the pieces are
 mathematically "stitched" together in such a way that where they join is
 imperceptible. Furthermore, this can be done with complete mathematical
 specification and fine precision. In some cases, deliberate
 discontinuities can be locally introduced if desired (using, for example,
 multiple knots in nonuniform B-splines).
 The constraints of smoothness or continuity can take various forms, and may
 be relatively simple or complicated. The most common case is that of
 continuity of position and of the first two derivatives. A more general
 case is that of continuity of position and of the first n derivatives, for
 some integer n (where the derivatives may be scalar-valued or
 vector-valued). However, the constraints can be much more complicated,
 being expressed by some set of equations (which may be vector-valued). For
 example, a more complicated analogue of the continuity of position and of
 the first two derivatives is continuity of position, unit tangent vector,
 and curvature vector. The surface analogue of this continuity for curves
 would involve continuity of the tangent plane and osculating sphere. This
 approach is sometimes referred to as second order geometric continuity
 [Barsky & DeRose, 1995]. Still more complicated constraints can describe
 higher order continuity. Furthermore, there are constraints known as
 "Frenet frame continuity"; this is discussed in [Hohmeyer & Barsky, 1989].
 As an example of the form of the constraints, the following are the
 constraints of fourth order geometric continuity (denoted by G.sup.4) for
 curves:
 ##EQU2##
 where .beta.2, .beta.3, and .beta.4 are arbitrary, but .beta.1 is
 constrained to be positive, where superscript (i) denotes the i.sup.th
 parametric derivative, and where q(u), u .epsilon.[0,1], and r(t), t
 .epsilon.[0, 1] denote two parametrizations meeting at a common point.
 In the present application, each piece or zone forms part of a surface. It
 is important to note that there are no requirements or assumptions of
 symmetry. Furthermore, this approach does not have the usual restriction
 that the periphery of the lens be circular. For example, an interesting
 design would be an oval shape that might facilitate orientation and
 stabilization for a non-rotationally symmetric lens. Another possibility
 is to include a truncated portion which would also be useful to help
 orientation and stabilization.
 A special case of this formulation is a surface of revolution, that is, a
 surface formed by rotating a curve about an axis. Even though this is
 included in the present formulation, it is a special case. More generally,
 the present formulation can describe surfaces without constraints of
 symmetry.
 DECOMPOSITION OF THE SURFACE--DESCRIPTION OF FIGS. 1 TO 8
 As described in the section entitled "Summary of the Invention", the
 contact lens comprises the constituent parts of the anterior surface,
 posterior surface, and peripheral edge system (PES). Each of these
 components is divided into a collection of smaller pieces, joined together
 with mathematical constraints of smoothness. Note that the decomposition
 into smaller pieces can be done in a different manner for each of these
 components; that is, there is no requirement that all decompositions be of
 the same form for all the components.
 There are many possibilities regarding the decomposition of each surface
 into smaller pieces. This is know as the "topology" (not "topography") of
 the surface. This describes the connectivity of the adjacent pieces,
 without limiting the kind of mathematical surface formulation being used
 to specify each piece. FIGS. 1 through 8 show, without limitation, some
 possible arrangements of the pieces:
 (1) FIG. 1 shows the pieces as a sequence of annular regions 41, that is,
 rings.
 (2a) FIG. 2a shows the aforementioned annuli further subdivided by
 superimposing a set of radial lines 42, emanating from the center 43, and
 terminating at the periphery 44, thereby forming a collection of
 four-sided regions 45, each bounded by two arcs 41 from the rings and two
 straight lines 42 from the radii.
 (2b) FIG. 2b shows that the set of radial lines 42 discussed in (2a) need
 not emanate from the center, in this figure, all the radial lines start at
 the same annular border 41.
 (2c) FIG. 2c shows that the set of radial lines 42 discussed in (2a) need
 not emanate from the center nor all start at the same annular border; in
 this figure, the radial lines emanate from various borders 41, and
 terminate at the periphery 44.
 (2d) FIG. 2d shows that the set of radial lines discussed in (2a) need not
 emanate from the center, nor start at the same annular border, not
 continue all the way to the periphery; that is, each radial line 42 may be
 between any pair of rings 41.
 (3) FIG. 3 shows a grid of four-sided regions 45, where each region is
 bounded by four space curves 46, and where each space curve extends across
 the entire surface.
 (4a) FIG. 4a shows a variant of the preceding one (FIG. 3) where the space
 curves 46 in one direction are not required to extend across the entire
 surface so that the number and size of regions can be adaptive to shape,
 but where the number of such regions increases in each successive strip.
 (4b) FIG. 4b shows a variant of FIG. 4a where the number of such regions in
 each strip is independent of the number in the adjacent strips; that is,
 the space curves 46 in one direction may be between any pair of space
 curves in the other direction.
 (5) FIG. 5 shows a collection of equilateral triangular surface elements
 47, forming a hexagonal grid where all interior vertices 48 have exactly
 six edges emanating from them.
 (6) FIG. 6 shows a collection of general triangular surface elements 47,
 that is, where each region is bounded by three space curves 46, and where
 an arbitrary number of edges may emanate from each vertex 48.
 (7) FIG. 7 shows a collection of triangular 47 and hexagonal 49 surface
 elements, that is, where each region is bounded by either three or six
 space curves 46, and four edges emanate from each vertex 48.
 (8) FIG. 8 shows a collection of regions, where each region can have an
 arbitrary number of sides 59 (at least three, of course).
 There are many kinds of mathematical functions that can be used to describe
 each region, including (but not limited to): constant radii (circular and
 spherical), conic and quadric, polynomials containing only even-powered
 terms, polynomials containing only odd-powered terms, full polynomials of
 a specified, but arbitrary, order, Zernike polynomials (used in optics),
 transcendental functions such as exponentials, trigonometric functions,
 and hyperbolic trigonometric functions, rational functions, Fourier
 series, and wavelets.
 Given the mathematical description representing the posterior surface of
 the contact lens, the next step is to use ray tracing to determine the
 anterior surface of the lens such that the lens will provide the optics to
 correct the patient's vision. Note that the anterior surface is also a
 general spline surface with no assumptions of symmetry.
 MODELING THE EDGE
 Another use of the spline model is in the modeling of the edge of the
 contact lens. The primary source of discomfort is the interaction between
 the eyelid and edge of the lens. Consequently, the importance of good edge
 design is evident. Nonetheless, edge design is rather ad hoc and
 heuristic. The new approach to edge design uses spline mathematics with
 precise and repeatable control over the shape of the edge.
 Having determined the anterior and posterior surfaces of the contact lens,
 a mathematical formulation is constructed to form the peripheral edge
 system (PES) of the contact lens to meet these surfaces with appropriate
 continuity.
 There is no requirement or assumption that the anterior and posterior
 surfaces of the contact lens have the same diameters where they meet the
 peripheral edge system (PES); that is, it may be wider in either the
 anterior or posterior portion. For example, it may be useful to have a
 larger posterior surface diameter so as to match the corneal contour;
 alternatively, a larger anterior surface might serve optically to handle
 off-axis, oblique rays in the periphery.
 Since there are no requirements or assumptions of symmetry, this peripheral
 edge system (PES) is itself a general surface in three dimensions. A
 special case of this formulation is a surface of revolution, where the
 edge has a constant cross-sectional curve shape.
 The computation of the peripheral edge system (PES) can take into account
 the edge lift, specified as either the axial edge lift (AEL) or radial
 edge lift (REL) It can also use a new quantity defined here as tip lift
 (TL), the distance between the very tip of the edge and the tangent to the
 posterior surface. A large tip lift places the extreme point further from
 the posterior surface (and notionally the cornea).
 FIG. 9 shows an example, without limitation, of the design of a spline
 edge; the shape is specified by a mathematical description which is
 complete and precise, as well as repeatable. In this example, the edge is
 guaranteed to maintain continuous slope and curvature with the peripheral
 curve of the posterior surface as well as with the anterior surface.
 In this example, all the edges shown are defined by uniform cubic B-spline
 curves in the plane and then rotated; however, the method is more general
 and would permit a non-rotationally symmetric edge surface in three
 dimensions.
 In this example, the B-spline is specified by m+1 control vertices in the
 following manner: Three vertices control the position, first, and second
 derivative of the edge where it meets the anterior surface. These vertices
 are completely constrained by the position and shape of the anterior
 surface. Another three vertices control the meeting with the posterior
 surface, in the same way. Then, three more vertices control the position
 and derivative of the extreme tip of the edge. They are partially
 constrained to ensure that the point remains the maximal point.
 In particular, for this example, these control vertices are determined in
 the following manner. The edge curve, Q(u), comprises the curve segments
 Q.sub.3 (u), . . . , Q.sub.m (u).
 For the beginning of the ith segment (i=3, . . . , m),
 ##EQU3##
 ##EQU4##
 where V.sub.i are control vertices.
 The entire edge curve starts at Q.sub.3 (0), which is given by:
 ##EQU5##
 ##EQU6##
 ##EQU7##
 This can be equated to the position (denoted A), and first and second
 derivatives (denoted A.sup.1 and A.sup.2, respectively), of the anterior
 surface at the join point yielding the following conditions:
EQU Q.sub.3 (0)=A (5.1)
 ##EQU8##
 ##EQU9##
 Equating (4) and (5) yields three equations that can be solved for V.sub.0,
 V.sub.1, and V.sub.2.
 Similar equations for where the edge curve joins the posterior surface (at
 Q.sub.m (1)) can be established and solved for V.sub.m-2, V.sub.m-1, and
 V.sub.m.
 The vertices V.sub.m/2-1, V.sub.m/2, and V.sub.m/2+1 control the tip. The
 vertex V.sub.m/2 is completely defined by the width, tip lift (TL) and
 coordinate system local to the edge. In the local system,
EQU V.sub.m/2 =(width, 0, TL). (6.1)
 This can be transformed into a global system if required.
 To ensure that this is the maximum point, the x- and y-coordinates in the
 local system of vertex V.sub.m/2-1 are the same as the corresponding
 coordinates of V.sub.m/2+1 ; the z-coordinate is used to control the
 derivative. Hence:
EQU V.sub.m/2-1 =(width, 0, TL+d) (6.2)
EQU V.sub.m/2+1 =(width, 0, TL-d) (6.3)
 where d is some measure of the derivative.
 The second derivative is 0 because the vertices V.sub.m/2-1, V.sub.m/2, and
 V.sub.m/2+1 are collinear. More general constraints on V.sub.m/2-1 and
 V.sub.m/2+1 can incorporate tip curvature.
 In this example, all the control vertices for the edge are expressed in a
 coordinate system that is defined by the position and tangent at the join
 with the posterior surface. The coordinate system is defined by the radial
 unit tangent vector to the posterior surface (pointing out from the
 center) (x-axis), a second "vertical" vector (z-axis) perpendicular to the
 radial unit tangent vector and in the plane of the edge curve, and a third
 vector (y-axis) perpendicular to these two, forming a right handed set.
 The origin is located at the join point with the posterior surface. The
 control vertices are expressed in this frame so that they move as the
 posterior surface is manipulated. This is also a very convenient way to
 define the tip position. The "width" of the edge is the maximum
 x-coordinate obtained by the edge curve. The "tip lift" and "tip
 curvature" are the z-coordinate and curvature, respectively, of the point
 with the the maximum x-coordinate.
 This coordinate system in three dimensions enables a well-defined
 generalization from the usual planar edge context to three dimensions.
 Furthermore, by defining the mapping in terms of the posterior surface
 parameters, it is suitable for any posterior surface, without any
 assumptions about the shape.
 This method enables the evaluation of the edge shape at every point around
 the lens, based on the local coordinate system there. The plane of the
 edge curve is not fixed in space but is determined in three dimensions
 from the posterior surface.
 The edges shown in FIG. 9 all have m=10 and each has a different width and
 tip lift, giving a variety of appearances:
 elongated edge (FIG. 9(i))
 low edge with small tip lift and anterior surface inwards (FIG. 9(ii))
 blunt edge (FIG. 9(iii))
 Korb style edge [Korb, 1970] (large tip lift) (FIG. 9(iv))
 Mandell style edge [Mandell, 1988] (FIG. 9(v))
 Mandell style edge rotated (FIG. 9(vi))
 MAIN STEPS IN THE METHOD
 The new method provides a much wider class of shapes to be available to
 describe the anterior surface (including the optic zone), posterior
 surface, and peripheral edge system (PES) of a contact lens. Moreover, the
 new method enables contact lenses to be designed and fabricated so as to
 be a custom fit for patients whose corneas have complex or subtle shapes.
 Furthermore, rather than "guessing" with trial lenses from a limited set of
 available stock shapes, the new method enables contact lenses to be
 designed and fabricated so as to be a custom fit for patients whose
 corneas have complex or subtle shapes. The new method comprises some or
 all of the following steps (as shown in FIG. 22):
 (1) Data acquisition. Data is obtained about the patient's cornea using a
 corneal topographic mapping system, sometimes called a "videokeratograph".
 There are, at present, a dozen or so such systems that are commercially
 available, as well as several others undergoing research and development.
 They are based on several different principles, but all share the intent
 to provide corneal shape information.
 (2) Three-dimensional mathematical surface model construction. From the
 data obtained in (1), an accurate three-dimensional mathematical surface
 model of the patient's cornea is constructed. Note that the mathematical
 surface model has no requirements or assumptions of symmetry. (This model
 would generally also be a spline-based cornea model although
 non-spline-based corneas models could also be used.)
 (3) Posterior surface description. From the mathematical model of the
 patient's cornea in (2), a mathematical description representing the
 posterior surface of the contact lens is developed. The calculation allows
 the insertion of a tear layer between the cornea surface and posterior
 surface. (This is particularly important in the case of rigid lenses.)
 This is done by adding "offset function" to the mathematical
 representation of the anterior surface cornea to yield the mathematical
 formulation for the posterior surface of the contact lens. As mentioned
 above, there are also no requirements or assumptions of symmetry, not even
 that the "footprint" of the contact lens be circular.
 The lens should not fit tightly on the cornea, but instead there should be
 some movement of the lens on the eye. The shape of the posterior surface
 should not be identical to that of the cornea, but there should be a
 "lens-cornea relationship" involving a "mismatch" between the corneal
 surface and the back surface of the lens so as to have a tear layer
 between the cornea and contact lens. For a rigid lens, the tear layer
 varies from 10 to 20 microns at the center and 50 to 100 microns at the
 edge. It is much thinner and there is less movement in the case of soft
 lenses compared to rigid lenses. The movement of the lens and the volume
 of the tear layer play important roles in helping provide oxygen to the
 cornea during the wearing of a rigid contact lens. The thickness of the
 tear layer varies at different points under the contact lens. The optimal
 posterior surface shape depends on many factors including eyelid forces,
 surface tension, and tear viscosity. This involves the complex issues of
 how the contact lens moves and where it will rest on the eye. For a rigid
 lens, there is oscillation from an extremum position at the upper eyelid
 during a blink to an equilibrium position where the lens comes to rest
 after the eye has been open for at least 5 seconds. After the blink, the
 lens slowly drifts back to a balance-of-forces position. For a soft lens,
 there is less movement, a fact which can be exploited in creating soft
 lenses whose posterior surfaces more closely correspond to the corneal
 contour.
 (4) Ray tracing for anterior surface. Given the posterior surface of the
 contact lens, ray tracing is used to define the anterior surface of the
 lens such that the lens will provide the optics to correct the patient's
 refractive error. The ray tracing yields a set of points on the anterior
 surface, and then a general spline surface (with no assumptions of
 symmetry) is fit to these points.
 For example, but without limitation, the anterior surface can be
 calculated from the thickness of the contact lens, specified in the
 direction of the refracted ray inside the contact lens, expressed in terms
 of the coordinates of the posterior surface, in the case where the object
 is taken to be at infinity (incoming rays are all parallel) [Klein &
 Barsky, 1995].
 The lens thickness, denoted t.sub.p, at an arbitrary point, P, on the
 posterior surface is:
 ##EQU10##
 where n.sub.lens is the index of refraction of the contact lens, t.sub.ref
 is the thickness of the contact lens along a reference ray, n.sub.eye is
 the composite index of refraction of the eye (combination of indices of
 refraction of the vitreous humor, crystalline lens, aqueous humor, cornea,
 and tear film), f is distance from the point where the reference ray hits
 the posterior surface to the focal point, d is the distance from the focal
 point to the point P on the posterior surface, n.sub.air is the index of
 refraction of air, and .theta. is the angle with respect to the incoming
 direction of the ray within the lens. This angle is calculated by tracing
 a ray from the focal point to the posterior surface and then using Snell's
 Law for general surfaces [Welford, 1986].
 Note that the ray tracing approach to generate the anterior surface to
 achieve the optics avoids the usual assumption that the optical zone of
 the lens is spherical. Furthermore, in some cases, perfect spherical zones
 can be locally embedded in the spline surface shape (using, for example,
 rational splines).
 (5) Peripheral edge system (PES) design. Given the posterior and anterior
 surfaces of the contact lens, a mathematical formulation of the edge is
 defined to meet these surfaces with appropriate continuity. An example,
 without limitation, of the derivation of the edge was given above.
 DECOMPOSITION OF THE SURFACES OF THE LENS--DESCRIPTION OF FIGS. 10 TO 17
 The new method provides a much wider class of shapes to be available to
 describe the anterior surface (including the optic zone), posterior
 surface, and peripheral edge system (PES) 58 of a contact lens. FIGS. 10
 through 17 show, without limitation, some possible spline-based contact
 lenses. These lens correspond to the arrangements illustrated in FIGS. 1
 through 8, respectively.
 (1) FIG. 10 shows the pieces as a sequence of annular regions (rings) 41.
 (2a) FIG. 11a shows the aforementioned annuli further subdivided by adding
 a set of radial lines 42, emanating from the center 43, and terminating at
 the periphery 44, thereby forming a collection of four-sided regions 45,
 each bounded by two arcs 41 from the rings and two straight lines 42 from
 the radii.
 (2b) FIG. 11b shows the set of radial lines 42 discussed in 2a emanating
 from the first annular border 41 rather than from the center.
 (2c) FIG. 11c shows the set of radial lines 42 discussed in (2a) emanating
 from various annular borders 41, and terminating at the periphery 44.
 (2d) FIG. 11d shows the set of radial lines 42 discussed in (2a) emanating
 from various annular borders and terminating between various annular
 borders 41.
 (3) FIG. 12 shows a grid of four-sided regions 45, where each region is
 bounded by four space curves 46, and where each space curve extends across
 the entire surface.
 (4a) FIG. 13a shows a variant of the preceding one (3) where the space
 curves 46 in one direction are not required to extend across the entire
 surface so that the number and size of regions can be adaptive to shape,
 but where the number of such regions increases in each successive strip.
 (4b) FIG. 13b shows a variant of (4a) where the number of such regions in
 each strip is independent of the number in the adjacent strips; that is,
 the space curves 46 in one direction may be between any pair of space
 curves in the other direction.
 (5) FIG. 14 shows a collection of equilateral triangular surface elements
 47, forming an hexagonal grid where all interior vertices 48 have exactly
 six edges emanating from them.
 (6) FIG. 15 shows a collection of general triangular surface elements 47,
 that is, where each region is bounded by three space curves 46, and where
 an arbitrary number of edges may emanate from each vertex 48.
 (7) FIG. 16 shows a collection of triangular 47 and hexagonal 49 surface
 elements, that is, where each region is bounded by either three or six
 space curves 46, and four edges emanate from each vertex 48.
 (8) FIG. 17 shows a collection of regions, where each region can have an
 arbitrary number (at least three) of sides 59.
 COMPUTERS AND NETWORKS
 This method allows for spline-based contact lens design with or without
 corneal topography information being available. That is, this method is
 also applicable in the absence of precise information about the corneal
 contour. Even with nothing more than standard slit-lamp biomicroscopy, a
 posterior surface could be designed using this method.
 The result of this method is a mathematical or algorithmic description of a
 contact lens. This can be used for data compression, transfer, exchange,
 conversion, formatting, etc. and for driving computer numerical control
 (CNC) manufacturing devices including, but not limited to, lathes,
 grinding and milling machines, molding equipment, and lasers.
 Such data can be used directly, or transferred over telephone lines via
 modems, or via computer networks. This approach enables patients to have
 their corneal topography analyzed in one location and a contact lens
 fabricated simultaneously or subsequently at a remote location. Both the
 corneal topography and the contact lens design could be displayed on the
 computer screen. FIG. 18 illustrates a pilot system for such a
 multi-window display. Clockwise, from the upper left, the windows show a
 corneal topographic map, a spline-based contact lens design, a simulated
 fluorescein pattern of that lens design on the patient's cornea, and a
 three-dimensional surface representation of the cornea color-encoded with
 the value of sphere.
 SMOOTH TRANSITION ZONE
 Another important application of spline-based contact lens design is the
 calculation of a smooth transition zone for the smoothing of the junction
 between two adjacent zones of the lens. The new approach involves splines,
 geometric continuity, and shape parameters to control the flattening of
 the shape of the transition zone of a contact lens.
 In practice, this problem of discontinuities at the junction between zones
 is addressed by polishing; however, this process alters the specifications
 of the surface shape in an unknown, unpredictable, and unrepeatable way.
 Instead of this traditional approach, this new spline method replaces the
 discontinuous junction with a specially designed transition zone inserted
 between the two zones, with the important property of joining both zones
 smoothly. Furthermore, we can precisely quantify the degree of smoothness
 where the transition zone joins each adjacent zone. The constraints of
 smoothness or continuity can take any of a wide variety of forms as was
 described above.
 Mathematically, the geometry of a conventional lens surface, at a junction,
 is discontinuous in slope or curvature. Usually, the adjacent zones are
 both spherical, but of different radii 50, and the centers of curvature 51
 lie along the axis of symmetry 52 of the lens (FIG. 19). This results in a
 discontinuity in slope and curvature at the junction. In the more complex
 case where one relaxes the constraint of co-axial centers of curvature, it
 is possible to achieve a continuous slope. It is a common misconception
 that a continuous slope is always sufficient for smoothness. Note,
 however, how the curvature will necessarily remain discontinuous at the
 junction.
 The transition zone is guaranteed to maintain a given level of continuity
 with the adjacent zones. As an example, but without limitation, FIG. 20
 (i) shows two conic (elliptical) zones 53. The sharp area where they join
 54 is replaced by a transition zone 55 which joins smoothly to both
 adjacent zones 53. In this example, this transition zone is a quintic
 polynomial. Furthermore, this formulation provides "shape parameters" that
 can be adjusted interactively to modify the shape in realtime [Barsky,
 1988]. Both the "width" and "flatness" shape parameters of the zone can be
 independently controlled by the user, if desired. FIGS. 20 (ii)-(iv) shows
 several possible transition zones. In each of these three figures, there
 are three alternative transition zones 55 corresponding to different
 "flatness" values. FIGS. 20 (ii), (iii), and (iv) show small, medium, and
 large "width" transition zones, respectively. In all cases, this
 transition zone is guaranteed to maintain continuous slope and curvature
 with the adjacent zones. In FIG. 20 (v), the transition zone is shown in
 the same grey level as the adjacent zones, illustrating that one cannot
 detect where the transition zone joins its neighboring zones.
 DESCRIPTION OF TICULAR PREFERRED EMBODIMENTS
 A primary application of spline-based contact lens design is the creation
 of contact lenses for keratoconus. These lenses need complex posterior
 surface shapes to accommodate the "cone". As an example, but without
 limitation, FIG. 21 shows such a keratoconus lens using spline-based
 mathematical geometry to describe the anterior surface 56, posterior
 surface 57, and peripheral edge system (PES) 58. The posterior surface has
 a local region of much higher curvature than the overall surface (minimum
 radius of curvature of 4.5 mm compared to overall radius of curvature of
 7.67 mm) and yet the surface is smooth and the curvature continuously
 varies across the surface. In FIG. 21a, the surfaces are decomposed into
 pieces corresponding to FIGS. 2b and 11b, that is, a sequence of annular
 regions, which are further subdivided by adding a set of radial lines 42,
 emanating from a given annular border 41 and terminating at the periphery
 44, thereby forming a collection of four-sided regions 45, each bounded by
 two arcs 41 from the rings and two straight lines 42 from the radii. In
 this figure, the radial lines divide each ring into six such regions. The
 annular boundaries on the anterior surface are at 4/9 mm, 12/9 mm, 20/9
 mm, 28/9 mm, and 4 mm. For the posterior surface, the annular boundaries
 are at 1 mm, 1.5 mm, 3 mm, 4.5 mm, 4.9 mm. In general, the number of
 annular boundaries on the posterior surface is independent of that on the
 anterior surface.
 FIGS. 21b and 21c are based on the arrangement corresponding to FIGS. 2c
 and 11c, that is, where the the set of radial lines 42 discussed in for
 the previous figure emanate from various annular borders 41, and
 terminating at the periphery 44. In FIG. 21b, the number of regions 45
 doubles in successive rings; specifically, there are 4, 8, 16, and 32
 regions in the first, second, third, and fourth rings, respectively. FIG.
 21c has the number of regions 45 increasing by a unit increment in
 successive rings; that is, there are 6, 7, 8, and 9 regions in the first,
 second, third, and fourth rings, respectively. The annular boundaries for
 both FIGS. 21b and 21c are the same as in FIG. 21a.
 FIG. 21d corresponds to the formation described for FIGS. 3 and 12, that
 is, a grid of four-sided regions 45, where each region is bounded by four
 space curves 46, and where each space curve extends across the entire
 surface. The annular boundaries are the same as for FIGS. 21a, b, and c.
 SUMMARY, RAMIFICATIONS, AND SCOPE
 Thus, the reader will see that we have provided a method of contact lens
 design and fabrication using spline-based mathematical surfaces without
 restrictions of rotational symmetry. Splines can represent very general
 and complex shapes in a compact and efficient manner.
 The novel techniques presented here enable the design and fabrication of
 contact lenses that transcend the state of the art. Based on the more
 powerful mathematical representation of splines, these contact lenses can
 have posterior surfaces that provide a good fit to corneas of complicated
 shapes. This enables the design of lenses (including soft lenses) with
 good optics for irregularly shaped corneas.
 The present application describes a method that can accommodate higher
 order continuity, geometric continuity as well as parametric continuity,
 shape parameters, elimination of rotational symmetry restriction,
 spline-based optical zone (not constrained to be spherical), ability to
 embed exact spherical zones, eccentrically-located optical zone,
 complex-shaped tear layer gap ("mismatch"), and the capability to have
 non-circular periphery.
 The mathematical pieces of the surface are not limited to cubic
 polynomials; they may be polynomials of any degree, polynomials containing
 only even-powered terms, polynomials containing only odd-powered terms,
 full polynomials of a specified, but arbitrary, order, Zernike polynomials
 (used in optics), or even non-polynomial functions (examples include, but
 are not limited to, rational, trigonometric, exponential, hyperbolic
 trigonometric, other transcendental functions, Fourier series, wavelets,
 etc.) as well as constant radii (circular and spherical), conics and
 quadrics.
 The result of this method is a mathematical or algorithmic description of a
 contact lens. This can be used for data compression, transfer, exchange,
 conversion, formatting, etc. and for driving computer numerical control
 (CNC) manufacturing devices including, but not limited to, lathes,
 grinding and milling machines, molding equipment, and lasers.
 Although the description above contains many specificities, these should
 not be construed as limiting the scope of the invention but as merely
 providing illustrations of some of the presently preferred embodiments.
 Many ramifications are possible, and some further applications of
 spline-based contact lens design will now be discussed.
 The spline-based method has applications for the design and manufacture of
 both rigid and hydrogel (soft) contact lenses as well as for scleral
 contact lenses. It is appropriate for lenses that are either smaller,
 larger, or equal in size to the diameter of the cornea. The method has no
 limitation on the nature of the material. It has relevance to daily wear
 lenses, extended/flexible wear lenses, frequent/planned replacement
 lenses, daily/weekly disposable lenses lenses, aphakic lenses, prosthetic
 and therapeutic lenses, bifocal/multifocal lenses, toric lenses, and
 intra-ocular lenses.
 In addition to the above-mentioned calculation of the posterior surface,
 anterior surface (including the optic zone), peripheral edge system (PES),
 and smooth transition zone, there are many other applications of
 spline-based contact lens design. These include, but are not limited to,
 the following:
 computation of continuously varying thickness of a contact lens for prism
 ballasts for a non-rotationally symmetric lens
 computation of continuously varying thickness of a contact lens for optics
 for continuously varying index of refraction and/or for continuously
 varying materials
 use of splines to specify continuously varying optical power correction in
 a bifocal or multifocal lens; for example, varying radially out from the
 center or in the inferior/superior portions of the lens
 construction of "bosses" protruding from the anterior surface to interact
 with eyelid to provide stabilization for orientation of a non-rotationally
 symmetric lens
 etching of identifying marks on a lens for a wide variety of purposes,
 including (but not limited to): distinguishing left and right lenses,
 providing a mark to show correct orientation for a non-rotationally
 symmetric lens (e.g., at 6:00 o'clock position), marking a brand name,
 etc.
 specification of extreme toric lenses (called "ultratorics"). Special
 splines are used to specify shapes similar to tori, for anterior and/or
 posterior surfaces (for hydrogel or rigid lenses). This would be
 particularly well-suited to corneas having high toricity, as well as for
 the correction of residual astigmatism using such surface shapes for both
 the anterior and posterior surfaces.
 creation of custom disposable or frequent/planned replacement lenses.
 Although custom lenses are generally thought of as the antithesis of
 disposable or frequent/planned replacement lenses, this method uses the
 process of creating an inexpensive disposable mold of custom shape which
 is then used to mold a set of custom disposable or frequent/planned
 replacement lenses for a patient.
 creation of a hydrogel lens with a complex irregular shape on the posterior
 surface to approximate the cornea, a simple (perhaps spherical) anterior
 surface, using a material with low index of refraction to approximate that
 of the tears and cornea (possibly by using a high-water content material).
 Such a lens would be indicated for irregularly-shaped corneas that
 currently could only be served by rigid contact lenses, if at all.
 Currently, hydrogel lenses drape over the cornea and result in poor
 optical surfaces for irregularly shaped corneas. Spline mathematics would
 allow the determination of surface shape such that the hydrogel lens
 placed on the cornea would provide optical correction.
 creation of a contact lens that corrects the monochromatic aberrations of
 an eye. The anterior surface shape is determined to provide
 aberration-free optics and is represented as a spline surface.
 creation of a contact lens with complex posterior surface shapes to
 accommodate astigmatism that is severe or irregular (non-orthogonal), and
 any corneal distortions which might arise from keratoconus, pellucid
 marginal degeneration, ectasia such as keratoglobus, post-trauma,
 micropsia, pterygium, and scarring from ulcerative keratitis, etc.
 creation of a contact lens with complex posterior surface shapes to
 accommodate post-surgical corneas including penetrating keratoplasty (PK)
 grafts as well as corneal refractive surgery failures (for example, but
 not limited to, radial keratotomy (RK), photorefractive keratectomy (PRK),
 automated lamellar keratoplasty (ALK), and laser in-situ keratomileusis
 (LASIK)).
 creation of contact lens for use as an erodible mask for shape transfer in
 laser surgery such as PRK
 creation of a sequence of contact lenses of varying shapes for use in
 orthokeratology
 creation of contact lenses for use as molds in Precision Corneal Molding
 (PCM), Controlled Kerato-Reformation (CKR), and orthokeratology appended
 claims and their legal equivalents, rather than by the examples given.
 Thus, the scope of the invention should be determined by the appended
 claims and their legal equivalents, rather than by the examples given.