2 The De Casteljau algorithm revisited

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1 A new geometric algorithm to generate spline curves Rui C. Rodrigues Departamento de Física e Matemática Instituto Superior de Engenharia Coimbra, Portugal ruicr@isec.pt F. Silva Leite Departamento de Matemática Universidade de Coimbra 3000 Coimbra, Portugal fleite@mat.uc.pt Abstract This paper presents a new geometric algorithm to construct a C k -smooth spline curve that interpolates a given set of data on a complete Riemannian manifold. Although based on a modification of the De Casteljau procedure, our algorithm is implemented in three steps only, independently of the required degree of smoothness, and therefore introduces a significant reduction in complexity. The key role is played by the choice of an appropriate smoothing function. Another important feature relies on the fact that the calculation of each spline segment depends only on local data, as in the De Casteljau procedure. This is particularly useful in applications, since any change in the data at a particular instant of time only requires the recalculation of two segments of the spline. Mathematics Subject Classification 2000: 41A15, 53C22, 65D17. Keywords: Splines in R n. 1 Introduction We address the problem of generating a C k -smooth k 1) curve s : [a,b] R M, on a complete Riemannian manifold M, which fulfills a set of interpolation conditions of the form st i ) = p i and ṡt i ) = v i, 1) for a given partition : a = t 0 < t 1 < < t m = b of the time interval [a,b], given points p i on M and vectors v i tangent to M at p i, i = 0,1,...,m. This problem arises in many engineering applications, in particular when M is a Lie group or a sphere. Our algorithm is performed in three steps only, no matter the degree of smoothness required, and is based on the De Casteljau construction of a C 1 -smooth cubic spline. We review this construction in section 2. The new geometric algorithm is first presented for Euclidean spaces. Although it can be generalized to complete Riemannian manifolds, only the Lie group case is presented here, in section 4. Due to space limitations many details have been omitted and will appear in a forthcoming paper. 2 The De Casteljau algorithm revisited The classical De Casteljau algorithm is a geometric procedure, based on successive linear interpolations, that is used to construct parameterized polynomial curves of degree d in R n, from a sequence of d + 1 given points in R n. The resulting curve will connect the first point and the last point. The other points are only used to implement the algorithm and for that 1

2 reason are called control points. See Farin [3] for details. This algorithm can also be used to generate a curve that fulfills all the interpolation conditions 1) above. For instance, from the given points p 0, p 1 and the given vectors v 0, v 1 two control points, x 0 and x 1, are uniquely determined by x 0 = p v 0 and x 1 = p v 1. In this case the De Casteljau algorithm is performed in three steps and the resulting curve will be a cubic polynomial. For simplicity we may consider the time interval [0,1] instead of [t 0,t 1 ]. In the first step the line segments α 0 t) = p 0 + t x 0 p 0 ), α 1 t) = x 0 + t x 1 x 0 ), α 2 t) = x 1 + t p 1 x 1 ) are computed. Then, two curves are generated from convex combinations of the curves constructed in the previous step β 0 t) = 1 t)α 0 t) + tα 1 t), β 1 t) = 1 t)α 1 t) + tα 2 t). These two curves are finally combined in a similar way to generate the polynomial curve given by s 0 t) = 1 t)β 0 t) + tβ 1 t) = 1 t) 2 α 0 t) + 2t1 t)α 1 t) + t 2 α 2 t). Repeating the De Casteljau construction for each other subinterval [t i,t i+1 ], one obtains a spline curve t st) in R n which is the result of the concatenation of all spline segments t s i t) and has the following final form t t st) = s i i t i+1 t i ), t [t i,t i+1 ], i = 0,1...,m 1. The spline curve is locally a cubic polynomial, satisfies the interpolation conditions 1), but is only C 1 -smooth. Remark 2.1 To generate a C k -smooth curve using this algorithm, one needs to prescribe k derivatives at each instant t i. The construction of each spline segment t s i t) will require 2k control points and the computation of 2k + 1)k + 1) curves performed in 2k + 1 steps. It is clear that the computational cost of this algorithm increases substantially with k. The De Casteljau algorithm has been generalized for complete Riemannian manifolds [4] and [2]). The idea was to replace the linear interpolation procedure in the classical case by geodesic interpolation. The resulting curve has the same degree of smoothness as in the Euclidean case, but the implementation of the algorithm is much harder even for low dimensional cases. 3 The new geometric algorithm The new geometric algorithm proposed here is based on a modification of the De Casteljau algorithm, but has the ability of generating a spline curve in three steps only, with any required degree of smoothness. This surprising property of our algorithm is due to the role played by a smoothing function which will be predefined as soon as the degree of smoothness is chosen. We now show how to calculate the curve t s i t) which connects the point p i at t = 0 to p i+1 at t = 1 with initial and final velocities v i and v i+1. As in the De Casteljau algorithm we use two control points x i = p i + v i, x i+1 = p i+1 v i+1 2

3 and three steps. First, we define three line segments: l i t) = p i + t x i p i ) = p i + tv i, c i t) = p i + t p i+1 p i ), r i t) = x i+1 + t p i+1 x i+1 ) = p i+1 v i+1 ) + tv i+1 2) which will play the role of left, center and right component of the spline segment t s i t). Notice that the left and the right components satisfy l i 0) = p i, r i 1) = p i+1, l i 0) = v i, ṙ i 1) = v i+1, l j) i 0) = 0, r j) i 1) = 0, j 2. 3) Next, we introduce a smooth real-valued function φ : [0,1] [0,1] satisfying φ0) = 0, φ1) = 1, φ j) 0) = 0, φ j) 1) = 0, j = 1,2,...,k 1, 4) and compute two new curves from convex combinations of the ones previously constructed: a i t) = 1 φt))l i t) + φt)c i t), b i t) = 1 φt))c i t) + φt)r i t). Finally, we perform the following convex combination to generate the curve: s i t) = 1 φt))a i t) + φt)b i t) = 1 φt)) 2 l i t) + 2φt)1 φt))c i t) + φt) 2 r i t). 5) Remark 3.1 One possible choice for the function φ satisfying all the conditions 4) is the following polynomial of degree 2k 1 k 1 φt) = γ j=0 where a k+j = 1) k 1 ) j k 1 j and γ 1 a k+j = k + j. j=0 a k+j k + j tk+j Theorem 3.1 If φ is a real-valued C k -smooth function satisfying 4), then 1. the spline segment t s i t) defined by 2) and 5) satisfies the following conditions s i 0) = p i, s i 1) = p i+1, ṡ i 0) = v i, ṡ i 1) = v i+1, 6) s j) i 0) = 0, s j) i 1) = 0, j = 2,...,k; 2. the resulting spline curve t st) given by t t st) = s i i t i+1 t i ), t [t i,t i+1 ], i = 0,1...,m 1 is C k -smooth and satisfies the interpolation conditions 1). 3

4 Proof. Applying Leibniz s formula for the jth derivative of a product to equation 5) we get j 1 s j) i t) = l=0 j l) φ j l) t)b i t) a i t)) l) + 1 φt)) + φt) j 1 l=0 j 1 l=0 + 1 φt)) 2 l j) i j l) φ j l) t)c i t) l i t)) l) j l) φ j l) t)r i t) c i t)) l) t) + 21 φt))φt)c j) i t) + φt) 2 r j) i t). Since φ satisfies 4) and l i 0) = c i 0), a i 0) = b i 0) and c i 1) = r i 1), a i 1) = b i 1) we get s j) i 0) = l j) i 0), s j) i 1) = r j) i 1), j = 0,1,...,k. The boundary conditions 6) follow from 3). The second part of the theorem is a consequence of conditions 6). Remark We note that the spline t st) will not be C k+1 -smooth, except for some degenerated cases. 2. If we want a C 1 -smooth curve then we may take φt) = t and, in this case, the spline generated by our algorithm coincides with the one produced by the De Casteljau algorithm. 3. If we want a C k -smooth curve for k 2), our algorithm produces a spline which also satisfies s j) t i ) = 0, i = 0,1...,m, for j = 2,...,k. If these conditions were initially prescribed with conditions 1) in the problem statement, the De Casteljau algorithm could also be used to solve the problem. But as already observed in remark 2.1 the complexity of this algorithm increases substantially with k while the complexity of our algorithm doesn t depend on the value of k. 4. It is clear that the degree of smoothness of the spline generated by the new algorithm only depends on the choice of a function satisfying conditions 4). For that reason we call φ a smoothing function. 4 The new algorithm on Lie groups In this section we combine the ideas used to implement the new algorithm in R n with those used to generalize the De Casteljau algorithm to Riemannian manifolds see details in [1]). Geodesic arcs play the role of straight line segments and when the manifold is a compact and connected Lie group G, equipped with a left and right-invariant Riemannian metric, geodesics are easily expressed in terms of one-parameter subgroups. The elements in the Lie algebra L of G will be represented by capital letters. Similarly to the Euclidean case, the construction of the spline curve that solves the initial problem is local, so that the details will be presented only for the construction of the spline segment t s i t) that joins two given points in G, p i at t = 0) and p i+1 at t = 1), with prescribed velocities v i = V i p i and v i+1 = V i+1 p i+1, where V i and V i+1 belong to the Lie algebra of G. The smoothing function is the same as in the previous case. We now describe the three basic steps to obtain the required spline segment t s i t). 4

5 Step 1: We first construct the geodesic segments which are the left, center and right components and are defined by: l i t) = e tvi p i, c i t) = e twi p i, where W i = log p i+1 p 1 i ), 7) r i t) = e t 1)Vi+1 p i+1, The following boundary conditions are easily checked: l i 0) = p i, l i 1) = e Vi p i, l i 0) = V i p i, li 1) = V i e Vi p i, c i 0) = p i, c i 1) = p i+1 ċ i 0) = W i p i, ċ i 1) = W i e Wi p i, 8) r i 0) = e Vi+1 p i+1, r i 1) = p i+1, ṙ i 0) = V i+1 e Vi+1 p i+1, ṙ i 1) = V i+1 p i+1. Step 2: Now we define t a i t) and t b i t) by: a i t) = e φt)ait) l i t), where A i t) = log c i t)l 1 i t)), b i t) = e φt)bit) c i t), where B i t) = log r i t)c 1 i t)). The following alternative formulas for these two curves will simplify checking the boundary conditions of the spline curve at t = 1. 9) a i t) = e 1 φt))ait) c i t), b i t) = e 1 φt))bit) r i t). 10) Indeed, e 1 φt))ait) c i t) = e φt)ait) e Ait) c i t) = e φt)ait) e log cit) l 1 i t)) c i t) = e φt)ait) l i t)c 1 i t)c i t) = e φt)ait) l i t) = a i t), and similarly for b i. Now using the conditions 4) for the smoothing function φ and the boundary conditions 8), we obtain from 9) and 10) the following: a i 0) = b i 0) = p i, a i 1) = b i 1) = p i+1, ȧ i 0) = V i p i, ȧ i 1) = W i e Wi p i, 11) ḃ i 0) = W i p i, ḃ i 1) = V i+1 p i+1. 5

6 Step 3: or, similarly, Finally, we define the spline segment t s i t) by s i t) = e φt)sit) a i t), where S i t) = log b i t)a 1 i t)), s i t) = e φt)sit) e φt)ait) e tvi p i. 12) Again, an alternative formula for this last curve can be derived s i t) = e 1 φt))sit) e 1 φt))bit) e 1 t)vi+1 p i+1. 13) Before we show that the curve t s i t) satisfies the required boundary conditions, we derive two alternative expressions for the derivative, one from 12), another from 13). ) ṡ i t) = Ω L φsi t) + e φt)ad Sit) Ω L φai t) + e φt)ad Sit) e φt)ad Ait) V i s i t), 14) ṡ i t) = Ω L ψsi t) + e ψt)ad Sit) Ω L ψbi t) + e ψt)ad Sit) e ψt)ad Bit) V i+1 ) s i t). 15) In both cases we use the Campbell-Hausdorff formula e At) Bt)e At) = e ad At) Bt) = + j=0 ad j At)Bt)), 16) where ad denotes the adjoint operator on L defined by adab) = [A,B], and the following formula for the derivative of the exponencial, which may be found in Sattinger and Weaver [5] d e At)) 1 = Ω L dt At)e At), where Ω L At) = e u ad At) Ȧt)du. 17) To show that piecing together the spline segments, the resulting spline curve is C k -smooth, one needs to derive higher order covariant derivatives. The covariant derivative of a vector field along a curve in G a manifold imbedded in some high-dimensional Euclidean space R l ) may be viewed as a new vector field along that curve, which results from differentiating as a vector field along a curve in R l and then projecting it, at each point, onto the tangent space to G at that point. Details are rather technical and will be omitted in this paper. Nevertheless, we announce the following theorem which is a generalization to Lie groups of theorem 3.1. Theorem 4.1 If φ is a real-valued C k -smooth function satisfying 4), then 1. the spline segment t s i t) defined by 12) satisfies the following boundary conditions s i 0) = p i, s i 1) = p i+1, 0 ṡ i 0) = V i p i, ṡ i 1) = V i+1 p i+1, 18) D j ṡ i D 0) = 0, dtj j ṡ i 1) = 0, j = 1,...,k 1; dtj 2. the resulting spline curve t st) given by t t st) = s i i t i+1 t i ), t [t i,t i+1 ], i = 0,1...,m 1 is C k -smooth and satisfies the interpolation conditions 1). These results can be extended to other complete Riemannian manifolds, as long as explicit formulas for geodesic curves are known. This is the case for spheres. More details will appear in a forthcoming paper. 6

7 Acknowledgements This work was supported in part by ISR-Coimbra and project POSI/SRI/41618/2001. References [1] P. Crouch, G. Kun, and F. Silva Leite. De Casteljau algorithm for cubic polynomials on the rotation group. In Proceedings of the Second Portuguese Conference on Automatic Control, pages , Porto, Portugal, September [2] P. Crouch, G. Kun, and F. Silva Leite. The De Casteljau algorithm on Lie groups and spheres. J. Dynam. Control Systems, 53): , [3] Gerald Farin. Curves and surfaces for computer aided geometric design. Computer Science and Scientific Computing. Academic Press Inc., Boston, MA, third edition, [4] F. C. Park and B. Ravani. Bézier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME Journal of Mechanical Design, 117:36 40, [5] D. Sattinger and O. Weaver. Lie groups and algebras with applications to physics, geometry and mechanics. Appl. Math. Sci. 61. Springer-Verlag,