**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.