: With geodesic paths we can parallel transport the difference between a smiling (upper left) and a neutral face (lower left) along a path (bottom row) towards a disgusted face (bottom right), resulting in a smile with a frown (upper right). We prove both in the smooth and discrete setting that the Hessian of an elastic deformation energy results in a proper Riemannian metric on the space of shells (modulo rigid body motions). Based on this foundation we develop a time-and space-discrete geodesic calculus. In particular we show how to shoot geodesics with prescribed initial data, and we give a construction for parallel transport in shell space. This enables, for example, natural extrapolation of paths in shell space and transfer of large nonlinear deformations from one shell to another with applications in animation, geometric, and physical modeling. Finally, we examine some aspects of curvature on shell space.
Besides, REF develop a time-and space-discrete geodesic calculus to shoot geodesics with prescribed initial data, they all need heavy computation.
Exploring the Geometry of the Space of Shells
Comput. Graph. Forum
Comput. Graph. Forum
Mathematics Computer Science