The video explains properties of geodesic curves on surfaces and gives a glimpse of their application to numerical methods on surfaces. As an example, we apply geodesics to the computation and visualization of point waves on surfaces whose wave fronts evolve along geodesics. Here, remarkable phenomena like caustics of point waves are studied which are unkown in euclidean space. Further, the extension of the concept of straightest geodesic curves to piecewise linear surfaces is introduced which has turned out as a suitable concept when transfering standard numerical algorithms to arbitrary surfaces like the study of flows on surfaces.

The visualization of waves on arbitrary surfaces was a major task and needed new concepts. Such waves overlap regions of the surface a number of times and interfere with other parts of the wave. We used a special isometric texture map technique for arbitrary surfaces consisting of planar triangles and extended it to allow multiple coverings of a surface. Our technique allows surfaces to carry textures where each point of the surface has an associated stack of texel values and the height of the stack may vary over the surface. We call this a branched texture map, similiar to branched covering maps in mathematics. When the numerics are done and the evolution of the wave is computed, we blend the different layers at each point to simulate the interference. The resulting texel value is associated to the point on the surface.

The numerical simulation and computation of waves is done completely in the mathematical visualization system Oorange developed at our department. From the numerical data we create branched texture maps in Oorange and compute the interference in a blending process. This results in a final texture which can be locally isometric mapped onto the surface even with commercial animation systems like Softimage which we use for final rendering.