Blending curves. Albert Wiltsche

Transcription

1 Journal for Geometry and Graphics Volume 9 (2005), No. 1, Blenng curves Albert Wiltsche Institute of Geometry, Graz University of Technology Kopernikusgasse 24, A-8010 Graz, Austria wiltsche@tugraz.at Abstract. Two arbitrarily given curves k 1 (t) and k 2 (t) are blended to a third curve b(t) so that b joins k 1 and k 2 in given points A 1 and B 2 C l - and C m - continuously, respectively. In order to meet this objective we use polynomial functions α lm (t) for the blenng process. The Casteljau algorithm for curves is used in a special way to build the blended curve b(t). Furthermore we can use our construction to generate interpolating spline curves. Key Words: Spline curves, Hermite interpolation, interpolation MSC 2000: 68U05 1. Introduction Our aim is to find a simple method to join two curves k 1 (t) and k 2 (t) by a curve b(t) as shown in Fig. 1. For both curves k 1 and k 2 we first choose a parameter interval [t 0, t 1 ] which choice affects the final curve b. Then we assume t 0 = 0 and t 1 = 1 to simplify the notation which can always be achieved by a simple parameter transformation. The curve b(t) shall start with the parameter value t 0 = 0 at the point k 1 (0) = A 1 on k 1 and end with the parameter value t 1 = 1 at the point k 2 (1) = B 2 on k 2. So we have b(0) = k 1 (0) = A 1, k 1 (1) = A 2, b(1) = k 2 (1) = B 2, k 2 (0) = B 1. For b(t) we require d i (dt) b(0) = i d i (dt) b(1) = i (dt) i k 1(0), i = 0,..., l, (dt) i k 2(1), i = 0,..., m. (1) ISSN /$ 2.50 c 2005 Heldermann Verlag

2 68 A. Wiltsche: Blenng curves B 2 A 2 k 2 (t) k 1 (t) b(t) B 1 A 1 Figure 1: Two curves k 1 (t) and k 2 (t) are linearly blended to a third curve b(t) 2. Linear blenng Our construction shall be a linear blenng process between the curves k 1 (t) and k 2 (t) (see Fig. 1). Therefore we will use a function α lm (t) and the following setup to find a suitable curve b(t): b(t) = α lm (t) k 1 (t) + [1 α lm (t)] k 2 (t), t [0..1] R. (2) The function α lm (t) has to be chosen so that (1) holds. 1 First we calculate the derivatives of the curve b(t) b(t) = α lm (t) [k 1 (t) k 2 (t)] + k 2 (t) ḃ(t) = α lm (t) [k 1 (t) k 2 (t)] + α lm (t) [ k 1 (t) k 2 (t)] + k 2 (t) b(t) = αlm (t) [k 1 (t) k 2 (t)] + 2 α lm (t) [ k 1 (t) k 2 (t)] + + α lm (t) [ k 1 (t) k 2 (t)] + k 2 (t). (3) j ( ) [ ] j d j i d i (dt) b(t) = j i (dt) α lm(t) j i (dt) k 1(t) i (dt) k 2(t) + dj i (dt) k 2(t). j If we compare (3) with (1) we find the following contions for α lm (t): α lm (0) = 1 α lm (1) = 0 (4) (dt) α lm(0) j = 0, 1 j l, (dt) α lm(1) j = 0, 1 j m. Remark 1 In order to keep the construction simple we will use polynomial functions α lm (t). Instead of that one can also use rational or transcendent functions as it is shown for instance 1 The subscripts l and m of α lm incate the order of continuity of b, k 1 and b, k 2 at the respective points A 1 and B 2.

3 A. Wiltsche: Blenng curves 69 in [5], [10] and [11]. In [5], [10], and [12] blenng functions are also used to blend two surfaces to a third one. Szilvási [11] even applies a special variant of the Coons method in order to blend four surfaces to a fifth one. Remark 2 In (4) we see the input data of a Hermite interpolation problem to the real parameter values t 0 = 0 < t 1 = 1 (see [6, p. 15], [7, pp. 4 11]). Therefore we know that there exists exactly one polynomial function α lm (t) with polynomial degree satisfying (4). deg α lm (t) l + m + 1 Although there exists a Hermite interpolation formula to determine α lm (t) (see [1]) we will use the Bernstein-polynomials Bi n (t) = ( ) n i (1 t) n i t i }... Bernstein-polynomials (5) i, n N 0, n i, t R, B0 0 := 1 The Bernstein-polynomials form a basis of the vector space of all polynomials of order n. Hence we can write α lm (t) in the form α lm (t) = l+m+1 λ i B l+m+1 i (t), λ i R. (6) Due to the well known properties of the Bernstein-polynomials and their derivatives at t = 0 and t = 1 we obtain by straight forward computation λ 0 =... = λ l = 1, λ l+1 =... = λ l+m+1 = 0. Hence the unique polynomial that solves (4) can be written in the form We summarize in α lm (t) = l Theorem 1 Given two parametric curves k 1, k 2 with B l+m+1 i (t) (7) and k 1 (t) k 2 (t), t R, k 1 C n 1, k 2 C n 2, n 1, n 2 N 0 k 1 (0) = A 1, k 1 (1) = A 2, k 2 (0) = B 1, k 2 (1) = B 2, A 1 B 1, A 2 B 2. Then there exists exactly one polynomial function α lm (t) with deg α lm (t) l + m + 1, so that for the curve the contions (1) hold. α lm (t) can be written in the form α lm (t) = b(t) = α lm (t) k 1 (t) + [1 α lm (t)] k 2 (t) l B l+m+1 i (t), 0 l n 1, 0 m n 2.

4 70 A. Wiltsche: Blenng curves For the special case that the two curves k 1 and k 2 have common points A 1 = B 1 and A 2 = B 2 and common derivatives at this points we state the following theorem: Theorem 2 Given two parametric curves k 1, k 2 with and k 1 (t) k 2 (t), t R, k 1 C n 1, k 2 C n 2, n 1, n 2 N 0 k 1 (0) = A 1 = B 1 = k 2 (0) k 1 (1) = A 2 = B 2 = k 2 (1) d i (dt) i k 1(0) = d i (dt) i k 1(1) = Then there exists exactly one polynomial function (dt) i k 2(0), 1 i l 1 and 1 l 1 < l n 1 (dt) i k 2(1), 1 i m 1 and 1 m 1 < m n 2 (8) α λµ (t) := α l l1 1,m m 1 1(t) and deg α λµ (t) l l 1 + m m 1 1 = λ + µ + 1, so that for the curve b(t) = α λµ (t) k 1 (t) + [1 α λµ (t)] k 2 (t) the contions (1) hold. α λµ (t) can be written in the form Proof: α λµ (t) = λ B λ+µ+1 i (t), λ = l l 1 1, µ = m m 1 1. (9) If we compare (8) with (3) we find the following contions for α λµ (t): α λµ (0) = 1, α λµ (1) = 0, (10) (dt) α λµ(0) j = 0, 1 j l l 1 1, (dt) α λµ(1) j = 0, 1 j m m 1 1. The contions (10) are again the input data of a Hermite interpolation problem. So we know that there exists exactly one polynomial function with deg α λµ (t) l l 1 + m m 1 1 = λ + µ + 1 which solves our problem. Analogously to Theorem 1 the polynomial α λµ (t) can be expressed in the form (9) with the help of the Bernstein polynomials. Remark 3 With the linear parameter transformation t(τ) = τ τ 0 τ 1 τ 0, τ 0 τ τ 1 one can adapt an arbitrary parameter interval [τ 0, τ 1 ] to [0, 1].

5 A. Wiltsche: Blenng curves 71 Remark 4 The curve b(t) lies within the convex hull of the the two curve segments k 1 (t), k 2 (t), t [t 0, t 1 ] and its construction is affinely invariant. Remark 5 In the neighborhood of the points b(0) and b(1) the curve b sticks closely to k 1 and k 2, respectively. So you can imagine its shape before constructing the curve b(t). Remark 6 One can use line segments as tangents and circles of curvature at the points A 1 and B 2 to generate C 1 - and C 2 -continuous blenng curves. Fig. 2 shows our blenng construction between two line segments and a circle. The blenng function a lm must be of order l + m + 1 = = 4. So the blenng curves b 1, b 2 are of order six. l 2 c D 2 b 1 B 1 A 2 B 2 = C 1 b 2 C 2 D 1 A 1 l 1 Figure 2: Two blenng curves b 1, b 2 joining a circle c with the line segments l 1, l 2. The curves b 1, b 2 are C 1 -continuous at A 1 and D 2 and C 2 -continuous at B 2 = C 1. Remark 7 With our construction we can also build blenng curves which connect space curves. Remark 8 The advantage of using polynomial blenng functions α lm is that the blenng curve b is polynomial if the curves k 1 and k 2 are so. A sadvantage of the uniquely defined blenng functions α lm is the lack of design parameters. If we want to have more scope of design in the construction of blenng curves we have to use polynomial functions with degree > l + m + 1. If we prescribe a degree l + m d then a simple consideration shows that every possible polynomial function ᾱ lm satisfying (4) can be written in the form ᾱ lm (t) = l d Bi n (t) + λ i Bl+i(t), n (11) i=1 n = l + m d, d N, λ i R.

6 72 A. Wiltsche: Blenng curves This endows the user with d design parameters λ 1,..., λ d. The blenng curve b can now be described by b(t) = ᾱ lm (t)k 1 + [1 ᾱ lm (t)] k 2 (t) = = l d Bi n k 1 (t) + Bl+i n [λ i k 1 (t) + (1 λ i ) k 2 (t)] + i=1 n i=l+d+1 B n i k 2 (t) (12) Remark 9 In order to guarantee that the blenng curve b lies within the convex hull of the two curve segments k 1 (t), k 2 (t), t [t 0, t 1 ], one has to choose 0 λ i 1. Fig. 3 shows the uniquely defined cubic blenng function α 11 (t) = 2t 3 3t and two quartic blenng functions for C 1 -continuously blenng at t = 0 and t = 1. The quartic functions ᾱ lm depend on one design parameter λ: ᾱ 11 (t) = B 4 0(t) + B 4 1(t) + λ B 4 2(t). In the figure we set λ = 0.2 and λ = = 0:9 11 (t) = 0:2 0 1 Figure 3: The uniquely defined blenng function α 11 (t) and two quartic blenng functions with the design parameters λ = 0.2 and λ = 0.9 for C 1 -continuous blenng at t = 0 and t = 1 3. Algorithm 1. If we consider the representations of the uniquely defined function α lm (t) in eq. (7) and the blenng curve b in eq. (2) we see that we can evaluate the curve point b(t) at one fixed parameter value t with the help of the Casteljau-algorithm (see for instance [4] or [9]). As shown in Fig. 4 we construct the point b(t) as the curve point of a (l + m + 1) th -order Bézier curve to the Bézier points k 1 (t) = P 0 =... = P l and P l+1 =... = P l+m+1 = k 2 (t)

7 A. Wiltsche: Blenng curves 73 A 2 B 1 k 2 (t) = P l+1 =; : : : ; = P l+m+1 k 1 (t) = P 0 =; : : : ; = P l b(t) A 1 B 2 Figure 4: The construction of a curve point b(t) for a fixed parameter value t with the help of the Casteljau-algorithm 2. If we want to use design parameters for the construction of our blenng curve b we can apply the Casteljau-algorithm again (see eq. (12)). The curve point b(t) for one fixed parameter value t can be evaluated as the curve point of a (l + m d) th -order Bézier curve to the Bézier points P 0 =... = P l = k 1 (t), P l+d+1 =... = P l+m+1+d = k 2 (t) P l+i = λ i k 1 (t) + [1 λ i ] k 2 (t), λ i R, i = 1,..., d, If we choose 0 λ i 1 then the Bézier point P l+i lies in between the straight line segment P 1 P l+m+1+d. A 2 B 1 ¹P l+1 k 2 (t) = P l+2 =; : : : ; = P l+m+2 k 1 (t) = P 0 =; : : : ; = P l b 1 b 2 P l+1 A 1 B 2 Figure 5: Two alternative curves b 1, b 2 to the blenng curve b of Fig. 4 (deg b i = deg b + 1, b 2 is drawn grey). The adtional design parameter is λ = 0.9 for b 1 and λ = 0.1 for b 2. The corresponng Bézier points P l+1, P l+1 are marked by quadrangles 4. Examples We can use our construction and Theorem 2 to build spline curves which interpolate given points P 0,..., P k at corresponng parameter values t 0,..., t k. 1. A well known interpolant is the Overhauser spline (see [2, 3, 4, 8]). It uses parabolas k i and k i+1 which interpolate the point triples (P i 1, P i, P i+1 ) and (P i, P i+1, P i+2 ), respectively. Then the curves k i and k i+1 are blended by the linear function α 00 (t) = t t i t i+1 t i to b i (see Fig. 6). 2. With the help of the polynomial α 11 (t) we can improve Example 1 to build a C 2 - continuous interpolating spline curve of order 5. Our spline curve depends only on a

8 74 A. Wiltsche: Blenng curves P i P i+2 k i b i k i+1 P i 1 P i+1 Figure 6: The C 1 -continuous Overhauser spline few points. Changing one point P i influences only the spline curve between P i 2 to P i+2 (see Fig. 7). P i 3 P i+3 P i 1 P i+1 b i 1 b i b i+1 k i 2 k i+2 b i 2 P i P i 2 P i+2 Figure 7: Parabolas blended to a C 2 -continuous spline 5. Conclusion In this paper we have shown the construction of polynomial functions α lm (t) aimed to blend two arbitrarily given parametric curves k 1 (t) and k 2 (t) to a third curve b(t). It turned out that the function α lm (t) can be described in a very simple form by means of Bernstein-polynomials. This fact enables us to use the de Casteljau-algorithm to generate the points of the blenng curve b(t). Furthermore we can use our construction to build interpolating spline curves. References [1] I. Beresin, N. Shidkow: Numerische Methoden 1. Hochschulbücher für Mathematik, Bd. 70, VEB Deutscher Verlag der Wissenschaften, Berlin [2] J. Brewer, D. Anderson: Visual interaction with Overhauser curves and surfaces. Computer Graphics 11(2), (1977). [3] W. Degen: The Shape of the Overhauser Spline. Computing Suppl. 10, (1995). [4] G. Farin: Curves and Surfaces for CAGD. 5 th ed., Morgan Kaufmann Publishers, San Francisco [5] E. Hartmann: Parametric G n blenng of curves and surfaces. Visual Computer 17, 1 13 (2001).

9 A. Wiltsche: Blenng curves 75 [6] R.J.Y. McLeod, M.L. Baart: Geometry and interpolation of curves and surfaces. University Press, Cambridge [7] G. Nuernberger: Approximation by Spline Functions. Springer Verlag, [8] A. Overhauser: Analytic definition of curves and surfaces by parabolic blenng. Technical Report, Ford Motor Company, [9] H. Prautzsch, W. Boehm, M. Paluszny: Bézier and B-Spline Techniques. Springer Verlag, [10] M. Szilvásy-Nagy, T.P. Vendel: Generating curves and swept surfaces by blended circles. Comput.-Aided Geom. Design 17, (2000). [11] M. Szilvásy-Nagy: Construction of flexible blenng surfaces. Third Croatian Congress of Mathematics, preprint, Split [12] A. Wiltsche: A polynomial tool for blenng surfaces. Grazer Mathematische Berichte, to appear (2005). Received October 30, 2004; final form May 2, 2005