Geometric modeling in shape space

Martin Kilian*, Niloy Mitra, Helmut Pottmann

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

145 Scopus citations

Abstract

We present a novel framework to treat shapes in the setting of Riemannian geometry. Shapes - triangular meshes or more generally straight line graphs in Euclidean space - are treated as points in a shape space. We introduce useful Riemannian metrics in this space to aid the user in design and modeling tasks, especially to explore the space of (approximately) isometric deformations of a given shape. Much of the work relies on an efficient algorithm to compute geodesics in shape spaces; to this end, we present a multi-resolution framework to solve the interpolation problem - which amounts to solving a boundary value problem - as well as the extrapolation problem - an initial value problem - in shape space. Based on these two operations, several classical concepts like parallel transport and the exponential map can be used in shape space to solve various geometric modeling and geometry processing tasks. Applications include shape morphing, shape deformation, deformation transfer, and intuitive shape exploration.

Original languageEnglish (US)
Article number1276457
JournalACM Transactions on Graphics
Volume26
Issue number3
DOIs
StatePublished - Jul 29 2007

Keywords

  • Geodesic
  • Isometric deformation
  • Parallel transport
  • Riemannian geometry
  • Shape exploration
  • Shape space

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design

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