Abstract
We consider the problem of interpolating a finite set of observations at given time instant. In this paper, we introduce a new method to compute the optimal intermediate control points that define a \(C^{2}\) interpolating Bézier curve. We prove this concept for interpolating data points belonging to a Riemannian symmetric spaces. The main property of the proposed method is that the control points minimize the mean square acceleration. Moreover, potential applications of fitting smooth paths on Riemannian manifold include applications in robotics, animations, graphics, and medical studies.
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References
Barr, A., Currin, B., Gabriel, S., Hughes, J.: Smooth interpolation of orientations with angular velocity constraints using quaternions. ACM SIGGRAPH Comput. Graph. 26, 313–320 (1992)
Popiel, T., Noakes, L.: Bézier curves and C2 interpolation in Riemannian manifolds. J. Approx. Theory 148(2), 111–127 (2007)
Fang, Y., Hsieh, C., Kim, M., Chang, J., Woo, T.: Real time motion fairing with unit quaternions. Comput. Aided Des. 30, 191–198 (1998)
Zefran, M., Kumar, V., Croke, C.: On the generation of smooth three-dimensional rigid body motions. IEEE Trans. Robot. Autom. 14, 576–589 (1998)
Camarinha, M., Silva Leite, F., Crouch, P.: Splines of class \(C^k\) on non-Euclidean spaces. IMA J. Math. Control Inf. 12(4), 399–410 (1995)
Arnould, A., Gousenbourger, P.-Y., Samir, C., Absil, P.-A., Canis, M.: Fitting smooth paths on Riemannian manifolds: endometrial surface reconstruction and preoperative MRI-based navigation. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2015. LNCS, vol. 9389, pp. 491–498. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25040-3_53
Shoemake, K.: Animating rotation with quaternion curves. ACM SIGGRAPH 85(19), 245–254 (1985)
Hart, J., Francis, G., Kaufman, L.: Visualizing quaternion rotation. ACM Trans. Graph. 13(3), 256–276 (1994)
Ge, Q., Ravani, B.: Geometric construction of Bézier motions. ASME J. Mech. Des. 116, 749–755 (1994)
Nielson, G., Heiland, R.: Animated rotationsusing quaternions and splines on a 4D sphere. Program. Comput. Softw. 18(4), 145–154 (1992)
Samir, C., Adouani, I.: \(C^1\) interpolating Bézier path on Riemannian manifolds, with applications to 3D shape space. Appl. Math. Comput. 348, 371–384 (2019)
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces (1978)
Adams, F.: Lectures on Lie Groups (1982)
Acknowledgement
This research was partially supported by The National Center for Scientific Research as CNRS PRIME Grant and the I-Site Clermont Auvergne project.
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Samir, C., Adouani, I. (2019). Bézier Curves and \(C^{2}\) Interpolation in Riemannian Symmetric Spaces. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_61
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DOI: https://doi.org/10.1007/978-3-030-26980-7_61
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