The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm, thereby obtaining an exact solution even more quickly.

The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact "single source, all destination" algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lower-bound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm, thereby obtaining an exact solution even more quickly.

We present a method for naturally and continuously morphing two simple planar polygons with corresponding vertices in a manner that guarantees that the intermediate polygons are also simple. This contrasts with all existing polygon morphing schemes who cannot guarantee the non-self-intersection property on a global scale, due to the heuristics they employ. Our method achieves this property by reducing the polygon morphing problem to the problem of morphing compatible planar triangulations of corresponding point sets, which is performed by interpolating vertex barycentric coordinates instead of vertex locations. The reduction involves compatibly triangulating simple polygons and polygons with a single hole. We show how to achieve this using only a small number of extra (Steiner) vertices.

Two planar triangulations with a correspondence between the pair of vertex sets are compatible (isomorphic) if they are topologically equivalent. This work describes methods for morphing compatible planar triangulations with identical convex boundaries in a manner that guarantees compatibility throughout the morph. These methods are based on a fundamental representation of a planar triangulation as a matrix that unambiguously describes the triangulation. Morphing the triangulations corresponds to interpolations between these matrices.We show that this basic approach can be extended to obtain better control over the morph, resulting in valid morphs with various natural properties. Two schemes, which generate the linear trajectory morph if it is valid, or a morph with trajectories close to linear otherwise, are presented. An efficient method for verification of validity of the linear trajectory morph between two triangulations is proposed. We also demonstrate how to obtain a morph with a natural evolution of triangle areas and how to find a smooth morph through a given intermediate triangulation.

We show how to continuously map a texture onto a 3D triangle mesh when some of the mesh vertices are constrained to have given (u, v) coordinates. This problem arises frequently in interactive texture mapping applications and, to the best of our knowledge, a complete and efficient solution is not available. Our techniques always guarantee a solution by introducing extra (Steiner) vertices in the triangulation if needed. We show how to apply our methods to texture mapping in multi‐resolution scenarios and image warping and morphing.