a geometrical approach to curvature continuous joints of rational curves

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Computer Aided Geometric Design 10 ( 1993) 109-122 North-Holland 109 A geometrical approach to curvature continuous joints of rational curves Gerhard Geise Abteilung Mathematik, Institut ftir Geometrie, Technische Universitiit Dresden, MommsenstraJe 13. O-8027 Dresden, Germany Bert Jiittler Fachbereich Mathematik, SchlojJgartenstraJe 7, Technische Hochschule Darmstadt, W-6100 Darmstadt, Germany Received May 1992 Revised September 1992 Abstract Geise, G. and B. Jiittler, A geometrical approach to curvature continuous joints of rational curves, Computer Aided Geometric Design 10 (1993) 109-122. Rational Btzier curves are discussed from a projective-geometrical point of view. A projectively invariant Bezier representation of rational curves is presented. Geometric continuity of second and third order is characterized by special projective maps. These maps (certain perspective collineations) preserve curvature properties of a curve at a point. As an application, constructions for geometrically continuous joints of rational curves are derived. Keywords. Rational Bezier curves; projectively invariant Bezier representation; geometric continuity; rational geometric splines; conic splines; osculating parabola of a conic. Introduction This paper considers rational curves from a projective-geometrical point of view. First, in Section 1, some fundamentals from projective geometry are presented. Then, in Section 2, a projectively invariant Bezier representation of rational curves is derived. This representation allows to discuss properties of rational curves with help of projective maps. In recent years, such a representation was found by Farin [ Farin ‘831. Boehm has observed, that similar considerations were made already in 1870 ( [Haase 18701). The proof of the projective invariance is analogous to that of the affine invariance in the polynomial case: a projectively invariant geometric realization of the de Casteljau algorithm must be found. This paper presents a realization based on Brianchon’s theorem. Correspondence to: B. Juttler, c/o Prof. Dr. J. Hoschek, Fachbereich Mathematik, SchloBgartenstraBe 7, Technische Hochschule Darmstadt, W-6100 Darmstadt, Germany. Email: [email protected]. 0167-8396/93/S 06.00 @ 1993 - Elsevier Science Publishers B.V. All rights reserved

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Computer Aided Geometric Design 10 ( 1993) 109-122 North-Holland

109

A geometrical approach to curvature continuous joints of rational curves

Gerhard Geise Abteilung Mathematik, Institut ftir Geometrie, Technische Universitiit Dresden, MommsenstraJe 13. O-8027 Dresden, Germany

Bert Jiittler

Fachbereich Mathematik, SchlojJgartenstraJe 7, Technische Hochschule Darmstadt, W-6100 Darmstadt, Germany

Received May 1992 Revised September 1992

Abstract

Geise, G. and B. Jiittler, A geometrical approach to curvature continuous joints of rational curves, Computer Aided Geometric Design 10 (1993) 109-122.

Rational Btzier curves are discussed from a projective-geometrical point of view. A projectively invariant Bezier representation of rational curves is presented. Geometric continuity of second and third order is characterized by special projective maps. These maps (certain perspective collineations) preserve curvature properties of a curve at a point. As an application, constructions for geometrically continuous joints of rational curves are derived.

Keywords. Rational Bezier curves; projectively invariant Bezier representation; geometric continuity; rational geometric splines; conic splines; osculating parabola of a conic.

Introduction

This paper considers rational curves from a projective-geometrical point of view. First, in Section 1, some fundamentals from projective geometry are presented. Then, in Section 2, a projectively invariant Bezier representation of rational curves is derived. This representation allows to discuss properties of rational curves with help of projective maps.

In recent years, such a representation was found by Farin [ Farin ‘831. Boehm has observed, that similar considerations were made already in 1870 ( [Haase 18701).

The proof of the projective invariance is analogous to that of the affine invariance in the polynomial case: a projectively invariant geometric realization of the de Casteljau algorithm must be found. This paper presents a realization based on Brianchon’s theorem.

Correspondence to: B. Juttler, c/o Prof. Dr. J. Hoschek, Fachbereich Mathematik, SchloBgartenstraBe 7, Technische Hochschule Darmstadt, W-6100 Darmstadt, Germany. Email: [email protected].

0167-8396/93/S 06.00 @ 1993 - Elsevier Science Publishers B.V. All rights reserved

110 G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints

Section 3 contains a characterization of geometric continuity of second and third order by special projective maps (perspective collineations). Geometric continuity of second resp. third order corresponds to a bundle resp. pencil of tonics (see [Geise & Nestler ‘9 1 ] ). The tonics of this bundle or pencil and the given curve(s) have a three- or four-point contact, respectively. This bundle resp. pencil can be generated by special perspective collineations. So, geometric continuity of second and third order can be characterized by these maps.

A corollary of this characterization is an invariance property of perspective collineations: certain perspective collineations preserve the curvature of a curve at a point. With help of this invariance property, a construction of second-order continuous joints of rational curves is derived in Section 4.

The ratio IC~/ICZ of curvatures of two curves which join with tangent continuity at a common point is a projective invariant. This is a result of classical projective-differential geometry [Bol ‘501, [Smith 18691. (An analogous assertion holds for the ratio of tor- sions.) In [Pottmann ‘9 1 ] and [Goodman ‘9 11, this ratio is expressed in terms of cross ratios of control and weight points and so a projectively invariant algorithm for curvature continuous joints of rational curves is found.

In this paper, a direct geometrical construction of curvature continuous joints of rational curves is derived with help of the invariance property of perspective collineations. This construction is applied to an interpolating second-order continuous conic spline. Furthermore, the construction of the osculating parabola of a conic and conditions for the existence of an interpolating third order continuous conic spline are presented.

It is quite surprising, that solely using constructive-geometrical methods we can obtain such far-reaching results.

1. Fundamentals

In this section we present some fundamentals from projective geometry. For further details, the reader is referred to [ Coxeter ‘641 or [ Blaschke ‘541.

The following considerations require that we are working in the projectively closed real

Euclidean plane. Its points (a, 6, c, . . . ) and lines (a^, b^, 2, . . . ) are described by homoge-

neous coordinate vectors from R3. The point p p* I = 0. The Cartesian coordinate vectors of (finite) points are a b c _)2_).*** They are obtained by removing the 0th components and then dividing by them:

PO wherep = p1 . 0 P2

(1)

The symbols A and v denote the intersection of lines and the connection of points, respectively. The coordinate vectors of intersection points and of connecting lines can be computed by the usual vector product.

A nondegenerated linear map of R3

n : lR3 + R3 : n(P) = AP for points p,

n(T) = (A-‘)*i for lines? (2)

(where A is a nonsingular 3 x 3-matrix) is called a projective collineation of the projectively closed real Euclidean plane. It maps points to points and lines to lines.

G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints 111

Fig. 1. A perspective collineation.

A projective collineation R is called a perspective collineation, if it preserves a line a^ pointwise (i.e. all points of 3 are fixed points of n), and a point z linewise (i.e. all lines through z are fixed lines of n). The line a^ is the axis, the point z is the centre of n. (If zTa^ # 0, then the matrix A of a perspective collineation has exactly two real eigenvalues: a single and a double one, where all eigenspaces are nondegenerated.)

A perspective collineation is uniquely determined by the axis 3, the centre z and one pair @, n (p ) ) consisting of a point p and its image under n (where the line p V n (p ) passes through z, and both axis and centre are neither incident with p nor with n(p) ). The image n (x ) of an arbitrary point x under rr can be constructed by (cf. Fig. 1)

1. q:=iiA (xvp), 2. R(X) := (z vx) A (q v n(p)). (3)

Let a = alp + cx2q, b = j31p + pg, f = qlp + 924 and x = <lp + &q be four points on a line p V q (a,, . . . , t2 E R). The real number

(4)

is called the cross ratio of the four points. It is invariant under projective maps. If all points are finite, it can be expressed in terms of signed (oriented) Euclidean distances dist(., .):

cr(a b f x) .= dfstkx) .dist(f ,b) 9,) .

dist(f ,x) .dist(a,b)’ (5)

The four points a, b, f, x are said to be in harmonic position if their cross ratio is equal to 1, -1 or 2.

If the last point x runs through 7, eq. (4) defines a projective scale on the line i, i.e. a

bijective map 7 + R u {co}.

Let B be a symmetric nonsingular 3 x 3-matrix. The set of all points x satisfying xTBn = 0 forms a conic.

A conic is uniquely determined by five of its tangents. Six tangents Ti (i = 1,. . . ,6)

of a conic are connected by Brianchon’s theorem: the three lines (?i A & ) V & A?[ )

(where (i, j,k,l) E {(1,2,4,5), (2,3,5,6), (3,4,6,1)}) intersect in one point, the so- called Brianchon’s point.

112 G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints

projective scde on b2vb3:

Fig. 2. The BCzier polygon of a rational curve.

Let two projective scales on two lines 3 and g be given. The system of lines connecting corresponding points of both scales (i.e. points which correspond to the same real number) envelops a conic or all lines pass through one point (Steiner’s generation of tonics). This fact yields the parametric representation of a conic as a rational curve of degree 2.

2. Rational curves and their BCzier polygon

A projectively invariant Bkzier representation of rational curves is presented in this section. As a first application, linear-fractional parameter transformations of these curves are discussed.

2. I. Interior and exterior weight points

Let

x(t) = kB/(t)bi, t E LO, 11 i=O

(with the Bernstein polynomials B/ ( t ) = (y) t’ ( 1 - t ) n-i ) be the homogeneous coordinates of a given rational BCzier curve of degree ~1. Usually, the control points bi E R3 are written in the inhomogeneous form

(7)

with weights wi E R. Obviously, this description excludes points at infinity. In addition to the control points, interior weight points Wi,i+l = bi + bi+l and exterior weight points fi.i+ 1 = bi - bi+ I are introduced (i = 0,. . . , n - 1). The control and weight points form the Bgzier polygon of the given curve (see Fig. 2). The interior weight points have been introduced in [Farin ‘831. The point wi,i+l divides the line segment bi V bi+l of the Bkzier polygon by the ratio Wi+l : wi. The four points bi, bi+l, Wi,i+l and fi,i+l are in harmonic position. They define a projective scale on the line bi V bi+l:

bi V bi+l +~u{~),

PI--+ Cr(bi,bi+l,f i.i+l,P)- (8)

G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints 113

Fig. 3. The double-step construction.

With respect to this scale, the four points bi, wi.i+t, bi+ I and f i.i+, correspond to 0, i, 1 and 00, respectively.

2.2. Geometric realizations of the de Casteljau algorithm

With help of the projective scales on the lines of the Bezier polygon, geometric realizations of the de Casteljau algorithm can be derived. Farin has developed such a realization using constant cross ratios [ Farin ‘831. Another one is based on Brianchon’s theorem. (Similar considerations can already be found in [Haase 1870]! That paper discusses the existence of a rational parametric representation of an algebraic curve. Then the parameter of this representation is shown to have a projective-geometrical meaning. A de Casteljau-like construction based on Brianchon’s theorem is derived in the end.)

The basic step of the de Casteljau algorithm is replaced by the double step:

b j bf+’

I

bi+1 H bi+= I

= B,z(t)bj + B;(t)bj+, + B;(t)bj+2. b’:+’

6+2

I+1

(9)

Rational curves of degree 2 are tonics. A tangent element of a curve is “tangent plus point of contact”. The conic (9) is determined by the two tangent elements “ (bi V bi+ ’ )

plus b!” and “(b{+= v bjz:) plus bj+2” and by the tangent b{+’ v b{I:. The point bf+=

is the point of contact of the conic and its tangent bi+’ V 6:::. It can be constructed by applying Brianchon’s theorem to three tangent elements. These tangent elements are considered as double tangents. The points of contact figure as intersection points of these double tangents (Fig. 3):

1. p = (bj+’ v b;+=) A (bj v b:‘f;)

2. I bj+= = (bj+l VP) A (bf+’ v bj’f;)

(10)

The point p is Brianchon’s point. With help of this construction, the de Casteljau algorithm can be realized geometrically

as follows.

114 G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints

Fig. 4. A geometric realization of the de Casteljau algorithm.

Let the parameter t E [0, 1 ] be given.

1. by:= bj (i = O,...,n).

2. Find the points bf on b~Vb~+, from t = cr(bp,bp,,,fi,i+l,bf)! (i = O,...,n - 1).

3. Find bf from the above construction ( IO)! (j = 2,. . . , n; i = 0,. . . , n - j).

4. r(t) := 6,” (see Fig. 4).

(11)

(The degenerated case of several b{ being collinear has to be excluded in construc- tion ( 10). It can be handled by mapping a nondegenerated situation to the degenerated one with help of a degenerated projective map.)

Both realizations are invariant under projective maps. Thus we have proved the following theorem.

Theorem 2.1. The relationship between rational curves and their B&er polygon is projec- tively invariant.

The control and weight points form a projectively invariant BCzier representation of a rational curve. This fact is the basis of the considerations in the next sections. Analogous assertions hold for dual BCzier curves (see [Hoschek ‘831) and Bezier surfaces (cf. [ Jiittler ‘921). (In the case of BCzier surfaces, each mesh of the Bezier polygon net carries four interior (resp. exterior) weight points, but these weight points cannot be chosen arbitrarily: they have to be coplanar.)

2.3, Linear-fractional parameter transformations

A rational BCzier curve can be reparameterized by a linear-fractional parameter trans- formation

t*(t) = to . t

to + (t - 1) . (20 - 1) (to E R u {CCJ}) (12)

(cf. e.g. [ Farin & Worsey ‘9 I], [Patterson ‘86 ] ) . This transformation preserves the con- trol points bi. It can be formulated in a geometric way, as follows. Let wT,~+~ and f;i+i be the points corresponding to a given parameter value to and to to/ (2to - 1) with respect to the projective scale (8) on bi V bi+l, respectively. The control points bi (i = 0,. . . , n)

G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints 115

f’ 01 (, f 23

f 1.2 bl WI.2 %2 ,,b2 f;2

Wil W0.l

4l

f 01 r--l W2.3

Wi.3

b3

CT3

Fig. 5. A linear-fractional parameter transformation.

and the new weight points wL+ 1 and f;i+, (i = 0,. . . , n - 1) describe the reparameter- ized Bezier curve (Fig. 5 ) . For to = co, the above transformation (12) permutes interior and exterior weight points and yields the complementary segment of the given curve.

3. Geometric continuity and perspective collineations

Now, a characterization of geometric continuity of second and third order by special projective maps is derived.

Certain perspective collineations preserve the curvature of a given curve at a point.

Lemma 3.1. Letr = n(t) beacurve, ~0 = x(to) oneofitspointsand?= x(to)Vi(to) the tangent at q. The curve n is assumed to be at least four times continuously differentiable at t = to. Let a^ be an arbitrary line ( # ?j through no and let z be an arbitrary point (f x0) on ?, and let .ti), ntii) and nciii) be perspective collineations

(i) with centre x0 and axis 2, (ii) with centre z and axis ? and

(iii) with centre x0 and axis i-, respectively. Then the curves x (t), zci) (x (t) ) and n cii)(x (t)) have a three-point contact (i.e. a G*-joint) and the curves x (t ) and n ciii) (x(t)) even have a four-point contact (i.e. a G3-joint) at xg

Figures 6, 7 and 8 show the perspective collineations n”), nui) and nuii) in the case of rational quadratic BCzier curves. Additionally, t = 0, i.e. no = be is chosen.

Proof. Instead of the occurring curves, their osculating tonics at xc (i.e. the unique tonics having a five-point contact, cf. [Bol ‘501) can be considered. (In [Geise & Nestler ‘911, BCzier representations of the tonics having three-, four- or five-point contacts with a given curve are derived.) Obviously, it is sufficient to prove the above assertion for tonics x.

The set of the images rtu) (x) of the given conic x under all perspective collineations of type (i) forms a pencil of tonics. All tonics of such a pencil pass through four common points (which may be complex or multiple). Here, these four points are fixed points under the collineations rcu). Thus, they have to lie on the axis ii‘.

The conic x and the axis a^ intersect in exactly two points: in x0 and a second one p. The second intersection point p cannot be a multiple point of the pencil of tonics as the

116 G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints

Fig. 6. The perspective collineation n(‘): p H p*.

Fig. 7. The perspective collineation x(“): p ++ p’. Fig. 8. The perspective collineation 71 tiii): p H p *.

perspective collineations rrci) do not preserve the tangent of x at p. Thus, xc is a triple point of the pencil of tonics: all tonics of the pencil have a three-point contact at xc.

The proof of case (ii) is dual to the above. Analogous deductions prove case (iii). 0

If in case (ii) the infinite point of the tangent 7 is chosen as centre z, our lemma yields a well-known result from the theory of polynomial BCzier curves: the control point b2 of a polynomial Bezier curve can be shifted along a parallel to be V 211 without influencing the curvature of the curve at be.

In case of tonics, the converse of Lemma 3.1 holds as well. Thus we have the following theorem.

Theorem 3.2. TWO curves x(t) and Y(T) have at a common point x (to) = y (TO) a G2-joint (a G3-joint), if and only if their osculating tonics are connected by a perspective collineation of type (i) or (ii) (of type (iii)).

Now, the perspective collineations xu), rrcii), rrtiii) will be used for the construction of

geometrically continuous joints between rational Bezier curves.

G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints 117

Fig. 9. The first perspective collineation. Fig. 10. The second perspective collineation.

4. Geometrically continuous joints between BCzier curves

With help of the perspective collineations of Lemma 3.1, a construction of G2-joints between Bezier curves is derived in this section. As an example, interpolating conic splines with second-order continuity are discussed.

The third case of Lemma 3.1 allows to construct the osculating parabola of a conic (i.e. the unique parabola having a four-point contact). Furthermore, necessary and sufficient conditions for the existence of an interpolating conic spline with third order continuity can be formulated.

4.1. Construction of G2-joints

Let bi (i = -n,...,O) and Wj,j+l resp. fj,j+, (j = -n,...,-1) be the control points and the interior resp. exterior weight points of a given rational BCzier curve of degree n (n > 2). This curve is to be continued by a second rational Bezier curve of degree n at its point bo. The control and weight points of the second curve are bi (i = 0,. . . , n ) and UJj,j+t resp. fj,j+1 (j = 0 ,..., n - 1). The first control point 61 is assumed to lie on b_1 v bo: this guarantees the first order continuity of both curves at bo. The weight points wo,~, f o,, and 201.2, f 1,2 are considered unknown. The control points b-2, b-1 and bo resp. bo, bl and b2 are assumed to be not collinear.

Both curves should join with second order continuity at bo. Meeting this requirement will produce the unknown weight points wo.~, f o.l and w1,2, f 1,2. They can be con- structed as images of the given weight points u1_2._1, f _2,_ 1 and ~~-1.0, f _1.o under two perspective collineations:

- The first perspective collineation n ti) has the centre b. and the axis

3 = bov ((b_2vb_l)A (b,vbd) (13)

and it maps b_, into bl (see Fig. 9). The points n(‘)(b-2), bl = x(‘)(b_l) and 62 are collinear.

- The second perspective collineation rrcii) has centre bl and axis b_l V bo and it maps n(‘)(b-2) into b2 (see Fig. 10).

The given curve and its image under 7rCii) o rcCi) join with second order continuity at their common point b. (see Lemma 3.1! ). The control and weight points of the image of the

118 G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints

given curve are bs, br, be, 7~(~~)(rr(‘)(b_3)), . . . and

We 1 = 7P) (7c(i) (f_l.o)),

WI:2 = n(ii)

fo.l = 71(ii) (i) (n (=I.0 I),

Wi) (f-2.-1 ) ), fr,* = .(ii) 0) (14) (n (w-2.-1)),... *

(The perspective collineations nCi), nCii) preserve the orientation of the given curve. Thus, the image of the complementary segment of the given curve is chosen as the new interior segment: interior and exterior weight points are permuted.)

The control and weight points bs, 64, . . . and 202.3, f 2.3, 203.4, f 3,4, . . . do not influence the curvature of the second curve at be. They can be chosen arbitrarily.

A possible realization of the whole construction is:

First perspective collineation (see Fig. 9):

1. ii:= brJv ((b_2Vb_r)A (biVb2)),

2. c:= (b-2Vbo)A (biV&),

3. /J := (f-2.-1 vbo) A (61 vb2),

4. wo.1 := (bOVbl)A(hV(a^A(f-l.OVf-2-l))).

Second perspective collineation (see Fig. 10):

Let 8 be an arbitrary line # ba V bl through bi!

5. p := (boVc) Ag^,

6. q := (boVb2) Ag^,

7. r:= (pVh)A (boVbl),

8. 2~1.2 := (q V r) A (bl V b2).

(15)

(A permutation of f and w yields the construction of the unknown exterior weight points. Figures 9 and 10 show the construction ( 15) in the case of rational quadratic curves, i.e. for n = 2.)

The solution of the problem (to find the unknown weight points ~0.1, f o.I and 201.2, f 1.2 ) is uniquely determined up to linear-fractional parameter transformations ( 12). This results e.g. from Boehm’s formula for the curvature K of a rational Bezier curve at its first control point (see [Boehm ‘871) or from the continuity constraints developed in [Degen ‘881. Thus we have the following theorem.

Theorem 4.1. The two given curves described by the control and weight points b-,, . . . , bo;

w-n.-n+l,f_n._n+l,...,w-~.O,f_l.O and bo,...,h; ~.l,fo.l,...,Wn-l.n,fn-l.n (where bo, b_, and b , are assumed to be collinear and b-2, b _ 1 and bo to be not collinear) have at bo a three-point contact (i.e. a G2-joint) if and only if the weight points ~00.1, f 0.1 and w~.~, f 1,2 result from the weight points w-1.0, f _1.o and w-2.-1, f -2.-1 by construction ( 15 ) and a linear-fractional parameter transformation ( 12 ).

In Fig. 11 (a), an interpolating G2-conic spline is constructed using the above construc- tion ( 15 ). The conic segments are described by rational Bezier curves of degree 2. The spline interpolates the four tangent elements “bo v b 1 plus bo”, “b 1 v b 3 plus b 2”, “b 3 V b 5 plus b4” and “b5 v b6 plus bb”. In Fig. 11, the lirst segment of the conic spline is given

G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints 119

(a) :b)

b0

Cd)

Fig. 11. The construction of an interpolating G*-conic spline. (a) One step of’ COnStruCtiOn Of the SPlinC.

(b) wI = 0.5. (c) wI = 1.0. (d) ~1 = 2.0.

in standard form (we = w2 = 1). The weight w1 of the first control point can be chosen arbitrarily. Figures 11 (b)-(d) illustrate the influence of wt : by increasing wt , the whole spline curve is pulled to the edges of the control polygon. The weight wt can be viewed as a global tension parameter of the curve.

4.2. The osculating parabola of a conic

Polynomial BCzier curves of degree 2 are parabolas (i.e. the infinite line is a tangent of these curves). Let a rational curve of degree 2 with the control and weight points bc, 6i, b2 and WO.~, fo.i, w~,~, f1,2 be given. The three control points be, bl and b2 are assumed to be not collinear.

The given curve is a parabola iff w2/wi = ml/w0 (cf. e.g. [Lee ‘871). This condition can be expressed in terms of ratios of the control and weight points:

d&t (h, ~1.2 1 dist(bl, wo.1) dist(w1.2,61) = dist(we.i,bo) *

(16)

In general, this condition is not fulfilled. Consider perspective collineations with centre bc and axis boVb 1. There exists a unique

perspective collineation rruii), mapping the given conic to a parabola (see Fig. 12 ):

120 G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints

Fig. 12. The construction of the osculating parabola of a conic.

1. Find the point p on bu v b 1 from dist (ba,p) = dist(wa.1, b i)!

2. Construct the parallel 2 to bc v b2 through p!

3. wi.2 := jr\ (b. v w,.2). (17)

4. b; := (bl V Wf.2) A (bo V b2).

5. b;; := bo, w;., := wo.l, b; := bl.

The image of the given curve under the perspective collineation rcciii) has the control and weight points b& b;, b; and w&, f&, Wi.2, f ;,2. Resulting from 1. and from the intercept theorems, it fulfills condition ( 16): The image curve is a parabola. The given curve and its image under rrciii) have a four-point contact at bo (see Lemma 3.1). Thus, the image curve is the osculating parabola of the given conic at bo.

It can be reparametrized by linear-fractional parameter transformations ( 12). One of the possible parametrizations is a polynomial one: The interior weight points are the midpoints of the segments and the exterior weight points are the infinite points of the lines of the Bezier polygon.

4.3. The existence of an interpolating G3-conic spline

Let a conic segment with the control and weight points b-2, b_l, bo and w-z-i,

f-2.-1, w- 1.~, f _ I.0 be given. This curve is to be continued by a second conic segment at its point bo. One point of this second conic segment is assumed to be known, it is chosen as control point b2. Both curves should have a four-point contact (i.e. a G3-joint) at their common point bo. The second conic segment is uniquely determined by these constraints.

Generally, the second segment may intersect the infinite line. Such intersection points (i.e. poles) are undesired in practical applications. How does the number of poles depend on the given point b2?

The answer to this question is given in Fig. 13: if b2 E Mi, then the second conic segment has exactly i poles (i = 0, 1,2). The auxiliary curve p is the osculating parabola of the given conic segment at bo.

This answer follows immediately by considering the perspective collineations with axis b_ I Vbo and centre bo. They generate all possible second segments. The osculating parabola p separates the solutions of our interpolation problem with respect to the number of its poles.

G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints 121

Fig. 13. The relation between bz and the number of poles.

Conclusion

The constructions concerning geometric continuity of second order can be directly generalized to nonplanar curves: they then take place in the osculating plane of the occurring curves. In [ Jiittler ‘921, geometric realizations of the algorithms for subdivision and degree elevation of rational BCzier curves are derived. Furthermore, the de Casteljau algorithm is shown to be a generalization of Steiner’s generation of tonics. In this sense, rational Bezier curves of arbitrary degree are generalized conks.

The authors believe that the constructive-geometric

( 1992), Die Konstruktive Geometrie der rationalen Bezierkurven und -flHchen, Diplomarbeit, TH Darmstadt.

122 G. Geise, B. Jiittler / Geometrical approach to curvature continuous joints

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