Representation and deformation of developable surfaces

Representation and deformation of developable surfaces P Redont

The purpose of the paper is to contribute to the study of developable surfaces in computer-aided design by proposing a means of specifying and controlling them. A discrete representation for a developable is derived from the orientation of the tangent plane along a geodesic. The original surface is thus approximated by a smooth developable surface which consists of pieces of circular cones. This representation easily allows deformation of the developable. computer-aided design, developable surfaces, surface modelling, geometry

The role of developable surfaces in CAD and computer graphics applications has already been pointed out by Huffman ~. According to this author, "a developable surface offers a complexity that is in a very real sense exactly midway between that of a completely general surface and that of a plane surface. Consequently, developable surfaces constitute a class that might be ideally suited to be both richer than that of plane surfaces and more tractable analytically than that of totally arbitrary surfaces". A developable surface can be visualized as the surface obtained when smoothly bending tin foil or a paper sheet. However, a developable surface can be rolled out (developed) onto a plane without tearing, creasing or stretching. Thus developable surfaces are highly pliable surfaces that can simulate flexible 3D objects such as objects made of sheets of metal, paper, plywood, leather, fabric or similar materials. In this paper, a method for specifying and controlling developable surfaces is proposed. First, some rapid mathematical developments show how a developable surface can be characterized by the orientation of its tangent plane along a geodesic (plus one initial condition). This amounts to parametrizing the spherical indicatrix of the surface with the arc length of the geodesic. Then, based on this characterization, the discrete representation of a developable surface is investigated. A natural approximation of the spherical indicatrix with arcs of circles and of its Laboratoire d'Analyse Numerique, Universit~ des Sciences et Techniques du Languedoc, Montpellier, France

volume 21 number 1 jan/feb 1989

parametrization with suitable functions leads to a discretization of the surface with pieces of circular cones. The feasibility of the representation is discussed. Finally the practicability of the method is illustrated by the realization of an interactive graphical editor for developable surfaces. Easy and realistic deformations of surfaces are allowed. In particular, the surfaces can readily be flattened out to display their 2D patterns.

MATHEMATICAL DEVELOPMENTS Definitiens Here, facts about surfaces, and especially developable surfaces, are briefly discussed. A detailed treatment is given in Hilbert and Cohn-Vossen2; Faux and Pratt ~ also contains sections on developable surfaces. There are at least three ways of defining developable surfaces: • A developable is any surface that can be obtained by bending a plane. 'Bending' is a transformation that preserves arclengths, and hence angles, of all curves drawn on the surface; two surfaces that can be transformed into each other by bending are described as being 'applicable' to each other. • A developable surface is the envelope of a oneparameter family of planes. Each tangent plane touches the surface along a straight line, called a generator or ruling, and is the same at every point of this straight line; clearly the totality of the generators covers the entire surface. While planar surfaces and completely general surfaces respectively have a zero-parameter and a two-parameter family of tangent planes, developable surfaces have a one-parameter family of tangent planes; equivalently stated, they have a one-parameter family of normals. • The surface swept out by all the tangent lines to a space curve is a developable surface. Considered in this light, the surface is called a tangential developable, and the curve is its cuspidal edge (or edge of regression). The tangents to the cuspidal edge are the generators of the surface. So that the class of tangential developables exhausts that of developables, it is necessary to acid to it those of generalized cones, generalized cylinders, and the plane. A geodesic is a remarkable curve on a surface. It is

0010-4485/89/010013-08 $03.00 © 1989 (Butterworth & Co) Publishers L(d

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analytically defined by the property that in every point its osculating plane is perpendicular to the tangent plane to the surface. Geodesic lines can be visualized as shortest, straightest or frontal lines. Bending between two surfaces preserves geodesics. Consequently, the geodesics of a developable surface are the images under bending of the straight lines of a plane region. See Hilbert and Cohn-Vosen 2 for more detailed discussion of geodesic properties. Any point on.a surface where a normal can be defined may be mapped onto a unit sphere as the endpoint of the radius equipollent to the outward unit normal. This process, called spherical mapping, locally characterizes the shape of the surface. The image of a surface or of a curve drawn on it is referred to as the spherical or gaussian indicatrix of the surface or of the curve. Since a developable surface has a one-parameter family of normals, its spherical image is a curve. It coincides with the spherical indicatrix of any curve lying on the surface and crossing all the generators. Clearly the spherical indicatrix of a generalized cylinder is an arc of a circle of the unit sphere, and that of a circular cone is an arc of a circle. A cusp on the spherical indicatrix corresponds to a change of convexity of the developable.

Characterization

Consider a developable surface (D) distinct from a plane in 3D Euclidean space. A one-parameter family of vectors X(s), the endpoints of which lie on (D), can be identified with a curve (X) on the surface. It is assumed

that (X) is some geodesic line of (D), different from a ruling, and that s is its arc length. The spherical indicatrix (N) of developable (D) can likewise be identified with a one-parameter family of unit vectors N(o-) where ~ is the arclength of (N). To every point X(s) on the geodesic there is associated a point N(~) on the spherical indicatrix by the spherical mapping which thus defines arc length o- as a function of arc length s. Let a(s) denote the angle between geodesic (X) and the generator that intersects (X) at point X(s): 0 < a(s)< x. It is now briefly shown how developable (D) can be recovered from the spherical indicatrix (N), the function cT(s) and some value a0 = a(So) of angle a. Note that geodesic (X)itself is not assumed to be known. Calculations involve classical results about curves drawn on a surface (see any textbook on geometry of surfaces, e.g. Julia 4, Do Carmo~). Let (T, G, N) be the Darboux-Ribaucour trihedron of the spherical indicatrix (N): T is tangent to (N), G is the geodesic normal to (N); that is, the normal lying in the tangent plane to the unit sphere; and N is normal both to (N) and the unit sphere (see Figure 1). Vector G is also the direction of the generator that passes through point X(s) of the geodesic and the following equations hold dT T---=0 do" --=dG - P c T do-

(1)

G

X(s)

T

Figure 1. Developable ( D ) with geodesic ( X ), and the spherical indicatrix ( N ) with Darboux-Ribaucour trihedren at point Nf o-)

14

computer-aided design

where Pc is the geodesic curvature of the spherical indicatrix at arc length 0-. However, vectors dX/ds and d2X/ds 2 are given by dX = sin(a)T + cos(a)G ds

(2)

--

I

ds 2

~ss 1" + -~-S G ] + sin(a) "~s

dss G

and the following equation holds d2X Ida do'] T ' - ~ s2 = cos(a) dss -- Pc -~s But vector d2X/ds 2 is perpendicular to vector T (this is a consequence of (X) being a geodesic); hence da ds

d0Pc -~s

0

(3)

(equation cos(a)= 0 yields curves orthogonal to the generators; these are not geodesics, except if (D) is a generalized cylinder, but then Pc = 0). Equation (3), together with a(s0)= ao, fully determines angle a as a function of s. A mere quadrature then provides X(s) up to a vector constant, in view of equation (2). Up to an irrelevant translation vector, the following representation for surface (D)is obtained (s, t)-~ X(s) + tG(0-(s))

da d-~ = PC

cos(a) G(0-(s)) da/ds

dS d2a/ds 2 - - = 2 sin(a) + - cos(a) ds da/ds

Cone

(7)

where k and I are two a priori unknown parameters. Therefore, in view of equation (3), 0- satisfies do-

-k

Pc(o-) d-s = 1 + (ks + I) 2

(8)

This first-order ordinary differential equation, together with conditions

(9) should provide constants k, / and function 0-(s). Further, in view of equation (2), dX/ds is given by dX 1" (ks + / ) G d~- = (1 + (ks + I)2) 1/2 ~ (1 + (ks + I)2) 112 Using equations (1), (3) and (7), dG/ds is given by:

Furthermore it can be shown that the cuspidal edge (E) of (D), its arc length S and its radius of curvature R~ are given, respectively, by:

RE--

a(s) = arccotan(ks + I)

0-(sl) = o"1, o'(s2)= o'2, arccotan(ks0 + / ) = a0

The developable is retrieved together with its geodesic (X). Conversely it can be checked that the surface so defined is developable indeed, which completes the claim. Note that by the way the following equation for angle a as a function of 0- is obtained

E(s) = X(s)

previous section it is no longer assumed that the-arc length 0- of the spherical indicatrix is a given function of the arc length s of the geodesic. Only two values 0-1 = 0-(sl) and 0-2 = 0-(s2) are supposed to be known (that is, the images N(0-1) and N(0-2) under the spherical mapping of two distinct points X(sl) and X(s2) of the geodesic are known). The angle a0 between the geodesic and the generator which intersects (X) at some point X(s0) is still supposed to be known. The spherical indicatrix (N) of the cone fully defines the cone up to a translation. If the origin of the space and the apex of the cone are chosen to coincide, then (D) can be parametrized by (0-, t ) ~ tG(0-). The concern here, however, is to retrieve the cone with its geodesic (X) from data (N), 0-1 = o'(s~), 0-2 = 0-(s2), and ao = a(s0), a result which will prove to be useful later. Considering the plane image of the cone under bending, the following expression for the angle a at point X(s) between the geodesic (X) and the generator is easily derived

dS/ds da/ds

(4) (5) (6)

case

The preceding result is now particularized to the case of the cone. The same notation is retained. Consider a generalized cone (D) with spherical indicatrix (N) and let (X) be some geodesic of (D). In contrast to the

volume 21 number 1 january/february 1989

dG

kT

ds

1 + (ks + I) 2

Thus dX/ds can be analytically integrated (up to an implicit constant vector) X(S) = ~1 (1 + (ks + /)2)1/2G(0-(s))

(10)

and the equation above allows retrieval of cone (D) together with its geodesic (X). A parametrization for (D) can be given by (s, t)--* X(s) + tG(s)) In the case of a circular cone, 0-(s) can be made explicit as a function of s. The spherical indicatrix of the circular cone is obviously an arc of a circle on the unit sphere.

15

Let ~ be the angle between the axis of/the cone and any of its generators. The geodesic curvature of the indicatrix is constant: Pc = tan(~). Because of equation (8), dG/ds satisfies: ddr -k d~- tan(~) = 1 + (ks + I)2 Hence, dr = % -- cotan(~)arctan(ks + I) where dr0 is an integration constant. Constants dr0, k, and / are determined by conditions (9).

DISCRETE REPRESENTATION OF DEVELOPABLE SURFACE A simple approximation to a developable surface is required which allows rapid and easy visualization, control and processing of the surface. Here, the idea basic for the discrete representation of a developable in unveiled; the feasibility of the method is discussed later. According to the preceding sections a developable surface (D) can be defined by: • its spherical indicatrix (N) on the unit sphere • the correspondence s~dr(s) under the spherical mapping between the arc length s of some geodesic (X) of (D) and the arc length dr of the indicatrix • the angle a0 between the geodesic and the generator intersecting (X) at some point X(s0) Yet, in the real world, objects are usually discretized: • indicatrix (N) is approximated by a sequence of n + 1 points M0, M2, ..., M2n on the unit sphere (only for convenience are the indices of the points supposed to be even) • M2~ is the spherical image of some point X(s2,) of the geodesic: the orientation of the tangent plane has been tracked along geodesic (X) at discrete points of known arc lengths s2, (i = 0. . . . . n) Whatever the original data, it is proposed that it be discretized in the following manner: • first, the spherical indicatrix (N)is approximated by a curve (C) built up with arcs of circles (C,)(i = 0..... 2n); • second, the arc length ~ on (C), which is meant to approximate dr, is parameterized on each (C,) by a function of the form 3 , - cotan(~A)arctan(k,s +/,). The approximation to a developable (D) described here is built with pieces of circular cones.

Approximating spherical indicatrix Consider n + 1 points M0, M2, ..., M2n on the unit sphere together with tangent directions to, t2. . . . . t2n which are meant to build a discretized spherical

16

indicatrix of developable (D). If tangents t2, are not part .of the original data they can be estimated in various ways from points M2,. A simple method for finding an arc-of-circle interpolation of points M2, on the sphere which respects tangents t2, is proposed. It is necessary to recall some facts about the stereographic projection 2. Consider the figure made by a sphere and the horizontal plane tangent to its south pole. The stereograhic projection maps the sphere deprived of its north pole onto the plane. The image of a point on the sphere is the intersection between the plane and the straight line defined by the point and the north pole. The stereographic projection possesses the interesting qualit~y of preserving angles and circles. In view of that property, the interpolation problem on the sphere readily reduces to the same problem in the plane. Let P2, and u2, be the stereographic images of M2, and t2,. The following question in the plane must now be answered: given two points P2, and P2,+a with tangents u2, and u2~+2, how can they be interpolated simply with a curve built up with arcs of circles which respects tangents u2, and u2,+2 at points P~, and P2,+2Solutions to this problem have already been described by several authors 6-1°. Here their ideas are presented briefly; refer to the original works for details. Points P2, and P2,+2 can be connected with two arcs of circles D2,+1 and D2,+2 meeting tangentially at, say, point P2,+1, such that D2,+1 is tangent to u2, at point P2, and D2,+2 is tangent to u2,+2 at point P2,+2. Of necessity point P2,+~ lies on the circle passing through P2,, P2,+2 and the incentre of the triangle formed by P~,, P2,+2 and u2,, u2,+2, or on the circle passing through P2,, P2,+2 and two excentres of the same triangle. The position of P2,+~ may be chosen according to some smoothness criterion and depends on whether curve (D2,+1, D2,+2) is to have an inflection point at P2,+~. Now let points M2,+~ and arcs C, be the images under the inverse stereographic projection of points P2,+ ~ and of arcs D,. The curve (C) constituted with arcs of circles C, interpolates the initial points M2, and tangents t2, on the unit sphere. Curve (C) may have inflection points or cusps (at M~,), but there is no point with a discontinuous tangent (corner point).

Approximating spherical mapping At the end of the previous step, 2n arcs of circles (C,) have been constructed, which interpolate the initial points M2,, and build a curve (C) that approximates the spherical indicatrix (N). The spherical mapping assigns some point N(dr) on the spherical indicatrix to point X(s) on the geodesic. An approximation to the spherical mapping as a function of arc length s is required, such that the image of (X)is curve (C). This can be achieved by defining the arc length ~ of (C) as a function of s in such a way that point C(~)lies near point N(dr). In view of the preceding sections it is natural to define piecewise on each arc (C,) as a function of the form ~i-cotan(~,)arctan(k~s +/,), where 3,, k,, and /, are unknown constants, and ~, is defined by arc (C,) (cos ~, is the radius of arc of circle (Ci)). Note that thus 2n patches of circular cones (K,) are defined bounded by

computer-aided design

generators and with a segment of one particular geodesic (-=,) drawn on each cone. Now let constants k~ and /, be determined. By definition, ~(s2,) should be the arc length on (C) at point M2,. Now to which value s~,+~, the arc length of intermediate point M2,+1, should be assigned has to be decided (only even values s2, and M2, are initially given). It is proposed that s2,+1 be chosen according to the following linear interpolation 52,* 1

s,, length(C,,+ ~) tan(l~ ,,. ,) + s..,~, length(C2,+1)tan(~,,+ ~) length(C,,+ ~)tan(~:, + ,) + length(C2,~,) tan(~2i + 2) Thus • has to satisfy

recursions: one increasing and the other decreasing from index j to compute all constants k~ and/~. Finally has been expressed as a function of s. Thus (C) together with its parametrization by ~(s) defines an approximation to the spherical mapping.

Representing the developable At the end of the previous two steps, a family of circular cones (K,) has been constructed, each with a geodesic segment (-=,) distinguished. Each cone, however, is defined up to a translation. Supposing that the apex of cone (K,) coincides with the space origin, the geodesic segment (_=,)is given by (remembering equation (10)) 1

=-,(s)=~(l+(k,s+ly)~/2r(3(s))

length(C,) = - cotan(~,)[arctan(k,s, + I,)

s,_l<<.s<,%s,

- arctan(k,s,_, +/,)] i = 1..... 2n (11) The aim is to assemble cones (K,) and (K,÷I) along a common generator to build a smooth surface (K) with one particular geodesic (-=) consisting of segments (--=,). It is expected that curve (-=) will be differentiable at the joint between (K,) and (K,+~), so it is written that the angles of (-=,) and (-=,+~) with the generator common to (K,) and (K,+~) are equal arccotan(k,s, +/,) = arccotan(k,+ ls, + I,+1) i = 1 ..... 2 n - - 1

(12)

Moreover, after initial hypotheses, the angle between geodesic (X) of surface (D) and the generator crossing (X) at the point of arc length so is known: a(so)= a0. Thus surface (K) is constrained to be such that the angle of (-=~) with the first generator of (K~) also be equal to ao arcotan(kls0 +/1) = a0

DISCUSSION

From equations (11) and (12) it is easy to derive the following linear system which gives k, and/, recursively from k, 1, /,_1

k,s, + I, = tan[arctan(k,_ is,_ 1 +/,_ 1) - tan(~,) length(C,)]

k,s,_l+l,=k,_ls,_l+l,_l

where F(3) is the unit geodesic normal to curve (C) (as well as the direction of the generator of cone (K,) passing through point -=,(s)). This equation evidently enables the display of cone (K,) as a locus of straight lines issuing from the origin and crossing curve (~=i). The aim is now to assemble all cones to build offe smooth surface with a geodesic constituted of segments (_=,). Let each cone (K,+I) successively be translated so that points .=.,(s,) and ~,+1(S,) coincide. Then two consecutive cones (K,) and (Ki+~) have the same generator through point .=.,(s~), the direction of which is given by the geodesic normal F(T(s,)) to curve (C) at point M, = C(3(s,)). Furthermore, cones (K~) and (K, + 1) have the same tangent plane along their common generator, the direction of which is given by the plane tangent to the unit sphere at point M,. In short, two consecutive cones are tangent. Thus a smooth discrete representation for our initial developable (D) is obtained, which is built up with circular cones.

i = 2. . . . . 2n i=2,...,

2n

The recurrence is initialized with

klsl + I1 = tan(x/2 - a0 - tan(~l) length(C~)) kiss, + /1 = cotan(ao) Once k, and /, are computed, constants 3, can be calculated to complete the definition of 3. In fact, these values are irrelevant and only ensure the continuity of function 3. For the sake of simplicity it is supposed that the angle ao corresponds to arc length so. If in fact a0 corresponds to some arc length s,, then there are two

volume 21 number 1 january/february 1989

The feasibility of a discrete representation for a developable surface based on the orientation of the tangent plane along a geodesic has been shown. This representation is simple. It yields circular cones, and this is certainly the most elementary way of approximating a developable with a smooth surface. This representation is flexible. The same geodesic line can be used to describe any bent configuration of the surface by respecifying angle a0 = a(So) and the orientation of the tangent plane along the line (since geodesics are preserved under a bending). This representation is realistic in the sense that the model surface patch is always exactly applicable onto the original surface patch; the length of the geodesic segment under consideration is (s2n- so) on both surfaces. Other representations are possible for a developable based, for example, on two plane sections, or on the orientation of the surface along a particular line (other than a geodesic: a plane section, a line of curvature, or the like). However, there is little chance to get at the simplicity and flexibility of the representation proposed in this paper; a plane section, for instance,

17

is not preserved under bending. In pra2ztice, however, a geodesic must be exhibited on the original developable, which is not necessarily a difficulty, as is now demonstrated. Suppose a developable is held in front of the eyes. If, in actual fact, the surface results from bending a plane region, then any straight line drawn on the surface before bending yields a geodesic. In particular, the edges of a plane patch are very often straight lines which provide remarkable geodesics after a bending. An interesting Case is the existence of a regular pattern on the plane patch. After bending, regularly spaced points on a geodesic can easily be found. This can be considered as a special aspect of the 'shape from regular pattern' problem 1~. In the case where no geodesic line can be found more or less obviously, then one can be tracked, at least over short distances, using a range sensor~2. Once a geodesic has been found, the orientation of the tangent plane may result from measures ~2. If no geodesic can be picked, then one can be computed from an analytical representation of the surface, which may have been deduced from some representation, and above all the arc length ~ of the spherical indicatrix can be related to the arc length s of the geodesic; this is now briefly shown. First consider a tangential developable. It can be defined by its cuspidal edge (E). Let S, RE, and T~ be the arc length, the radii of curvature and torsion of (E). The arc length ~r of the spherical indicatrix is given by d~

1

dS

T~

to observe, say, a rectangular sheet of metal that has • been bent. The user selects a parallel to the longer edges (a geodesic), and knows, by some means, the orientation of the tangent plane at discrete points on this line. The user enters the corresponding stereographic projections P0, P2. . . . . P2n. In this example, points on the geodesic are regularly spaced and n = 6. The program performs the arc-of-circle interpolation in the stereographic plane and displays the geodesic with intermediate points (see Figure 2(a)). Then the user enters the angle a0 = a(So) between the geodesic and one generator: in this example, a0 = arccotan(1/4) and s0=O. The program computes and displays the developable in the form of a solid sheet of material. The extreme generators of each cone composing the developable are drawn wherever they can be seen (Figure 2(b)).

On the other hand, the following can be successively derived from equations (5) and (6) da

1

dS

RE

ds = S cos(a)dS dS RE cos2(a) The three equations above implicitly define o" as a function of s through variable S and function a(S) (and two arbitrary constants involved in function a and in the primitive of cos(a)). Note that because of equation (4), the general expression for geodesics on a tangential developable is obtained

f cos(a)dS dE

X = E -I- ~

cos(a)

dS

Second consider a generalized cone. It can be defined by its spherical indicatrix N(~). Then equation (8) implicitly defines o" as a function of s, up to arbitrary constants k and I.

EDITOR A modest graphical interactive editor founded upon the above study has been written. A user is assumed

18

b Figure 2. (a) Stereographic projection of the approximated spherical irJdicatrix (C). Larger dots are initial points P2,;smaller dots are intermediate points P2,+ 7. The inner circle (radius 1) is the orthogonal projection of the equator; the outer circle (radius 2) is the stereographic projection ot the equator. ( b ) The developable associated to the spherical indicatrix of Figure 2(a)

computer-aided design

Flattening out the developable Flattening out a developable piece of material is an important operation that gives the plane images of curves drawn on the surface, and in particular the images of the edges. Thus a plane patch can be adequately manufactured to fit, after bending, a desirable 3D developable shape. Determining the 2D pattern of a 3D developable patch is straightforward with the discrete representation given here. All that is required

/

/

j,

/

'\

/

a

Figure 4. Controlling the detormation ot a developable by its spherical indicatrix

/ is the development of each circular cone. The following example illustrates the process. Figure 3(a) shows a developable with its section by the plane defined by the eye position and the two extremities of the geodesic segment used in defining the surface. Figure 3(b) shows the flattened developable with the plane image of the section (and the 2D pattern of the cones composing rhe developable).

Arbitrary bending

b Figure 3. (a) A developable with a plane cross section. (b) The unrolled developable of Figure 3(a) with the trace ot the cross section

volume 21 number 1 january/february 1989

Arbitrary bending of a developable patch can be readily carried out by modifying points M~ on the unit sphere (or their stereographic projections P~), arc lengths s~ or angle a0. They allow control of the developable. Some consistency in changing the control values is to be observed, of course, to avoid introduction of singularities due to the apexes of the cones inside the patch. Realistic deformations are enabled (indeed, surfaces are applicable to one another), and animation should be possible. Figure 4 shows how the spherical indicatrix, through its stereographic projection, can be

19

used to control the shape of the de,~lopable. The spherical indicatrix tends to retract arid the developable tends to flatten.

CONCLUSIONS The aim of this work has been to provide a discrete representation for developable surfaces based on the orientation of the tangent plane along a geodesic. The representation .simply consists of pieces of circular cones suitably connected along a common generator to build a smooth surface. This is the simplest representation for a developable, and yields a smooth surface, while a polyhedral approximation does not. It allows easy flattening out and bending of the surface. The above study may be successfully applied, for instance, to modelling metal foils.

ACKNOWLEDGEMENT The author is much indebted to Professor J Lemordant of Grenoble University for many valuable discussions.

REFERENCES 1 Huffman, D 'Curvature and creasing: a primer on paper' IEEETrans. Comput. Vol C25 No 10 (October 1976) pp 1010-1019 2 Hilbert r D and Cohn-Vossen, S Geometry and the imagination Chelsea, New York, NY, USA (1952) 3 Faux, I D and Pratt, M J Computational geometry

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tor design and manufacture Ellis Horwood, Chichester, UK (1980) 4 Julia, G Cours de g~om~trie Gauthiers-Villars, Paris, France (1941) 5 Do Carmo, M Differential geometry of curves and surfaces Prentice Hall, Englewood Cliffs, NJ, USA (1976) 6 Sabin, M A 'The use of piecewise forms for the numerical representation of shape' Dissertation Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary. Report 60/1977 ISBN 9633110351 (1977) 7 Shippey, G A 'Interpolation through a set of data points using circular arc segments' Internal Note Tech/IEG/67/15 Ferranti Ltd, Edinburgh, UK (1967) 8 Shippey, G A 'Piecewise approximation using circular arc segments' Internal Note Tech / IEG/ 67 / 3 Ferranti Ltd, Edinburgh, UK (1967) 9 Bolton, K M 'Birac curves' Comput.-Aided Des. Vol 7 No 2 (April 1975) pp 89-92 10 Sandel, G 'Geometry of compound curves' Z. Angew. Math. Mech. Vo117 No 5 (1937) pp 301-302 11 Ikeuchi, K 'Shape from regular pattern' Artificial Intelligence Vol 22 (1984) pp 49-75 12 Brown, M K 'The extraction of curved surface features with generic range sensors' Int. J. Robot. Res. Vol 5 No 1 (Spring 1986) pp 3-18

computer-aided design