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An inclusive Conformal Geometric Algebra GPU animation interpolation and deformation algorithm

Abstract

In the last years, Geometric Algebra with its Euclidean, Homogeneous and Conformal models attracts the research interest in many areas of Computer Science and Engineering and particularly in Computer Graphics as it is shown that they can produce more efficient and smooth results than other algebras. In this paper, we present an all-inclusive algorithm for real-time animation interpolation and GPU-based geometric skinning of animated, deformable virtual characters using the Conformal model of Geometric Algebra (CGA). We compare our method with standard quaternions, linear algebra matrices and dual-quaternions blending and skinning algorithms and we illustrate how our CGA-GPU inclusive skinning algorithm can provide as smooth and more efficient results as state-of-the-art previous methods. Furthermore, the elements of CGA that handle transformations (CGA motors) can support translation, rotation and dilation(uniform scaling) of joints under a single, GPU-supported mathematical framework and avoid conversion between different mathematical representations in contrast to quaternions and dual-quaternions that support only rotation and rotation–translation, respectively. Hence, our main novelty is the replacement of different types of algebras, and their in-between conversions between CPU and GPU, such as linear algebra matrices, quaternions, dual-quaternions and Euler angles for animation interpolation and skinning with a single mathematical representation, the CGA motors which can optimally handle the composition of translation, rotation and scaling joint transformations and interpolations. Employing latest CGA code generators, we provide a sample implementation of our algorithm running natively in a vertex shader program on modern GPUs for typical deformable virtual character simulations.

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Acknowledgments

The research leading to these results has received funding from the European Union People Programme (FP7-PEOPLE-2013-ITN) under Grant agreement \(n^O\) 608013.

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Correspondence to George Papagiannakis.

Appendix: CGA-GPU inclusive algorithm

Appendix: CGA-GPU inclusive algorithm

CLUCalc implementation for animation blending with CGA motors approach

figuref

Vertex shader

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Papaefthymiou, M., Hildenbrand, D. & Papagiannakis, G. An inclusive Conformal Geometric Algebra GPU animation interpolation and deformation algorithm. Vis Comput 32, 751–759 (2016). https://doi.org/10.1007/s00371-016-1270-8

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Keywords

  • Geometric Algebra
  • Conformal model
  • Virtual Reality
  • Animation Blending
  • Animation
  • Skinning
  • GPU-based skinning
  • Virtual Character Simulation