This paper develops a shape space framework for collision-aware geometric modeling, where basic geometric operations automatically avoid inter-penetration. Shape spaces are a powerful tool for surface modeling, shape analysis, nonrigid motion planning, and animation, but past formulations permit nonphysical intersections. Our framework augments an existing shape space using a repulsive energy such that collision avoidance becomes a first-class property, encoded in the Riemannian metric itself. In turn, tasks like intersection-free shape interpolation or motion extrapolation amount to simply computing geodesic paths via standard numerical algorithms. To make optimization practical, we develop an adaptive collision penalty that prevents mesh self-intersection, and converges to a meaningful limit energy under refinement. The final algorithms apply to any category of shape, and do not require a dataset of examples, training, rigging, nor any other prior information. For instance, to interpolate between two shapes we need only a single pair of meshes with the same connectivity. We evaluate our method on a variety of challenging examples from modeling and animation.

Abstract This paper introduces the construction of a low‐dimensional nonlinear space capturing the variability of a non‐rigid shape from a data set of example poses. The core of the approach is a Sparse Principal Geodesic Analysis (SPGA) on the Riemannian manifold of discrete shells, in which a pose of a non‐rigid shape is a point. The SPGA is invariant to rigid body motions of the poses and supports large deformation. Since the Riemannian metric measures the membrane and bending distortions of the shells, the sparsity term forces the modes to describe largely decoupled and localized deformations. This property facilitates the analysis of articulated shapes. The modes often represent characteristic articulations of the shape and usually come with a decomposing of the spanned subspace into low‐dimensional widely decoupled subspaces. For example, for human models, one expects distinct, localized modes for the bending of elbow or knee whereas some more modes are required to represent shoulder articulation. The decoupling property can be used to construct useful starting points for the computation of the nonlinear deformations via a superposition of shape submanifolds resulting from the decoupling. In a preprocessing stage, samples of the individual subspaces are computed, and, in an online phase, these are interpolated multilinearly. This accelerates the construction of nonlinear deformations and makes the method applicable for interactive applications. The method is compared to alternative approaches and the benefits are demonstrated on different kinds of input data.