We present a new algorithm to compute a geodesic path over a triangulated surface. Based on Sethian's
Fast Marching Method and Polthier's Straightest Geodesics theory, we are able to generate an iterative
process to obtain a good discrete geodesic approximation. It can handle both convex and non-convex
surfaces.
It is dened a new class of curves, called geodesic B´ezier curves, that are suitable for modeling on
manifold triangulations. As a natural generalization of B´ezier curves, the new curves are as smooth as
possible. We discuss the construction of C0 and C1 piecewise B´ezier splines. We also describe how
to perform editing operations, such as trimming, using these curves. Special care is taken to achieve
interactive rates for modeling tasks.
After giving an appropriated denition of convex sets on triangulations, we use it to study the convergence
of the geodesic algorithm, as well as the convex hull property of geodesic B´ezier curves. We prove
some results concerning convex sets.