Curves on Surfaces

Differential geometric curves on surfaces such as geodesics, asymptote lines, line of curvature, shadow lines and many others.

Small segments of a geodesic line minimize the distance between its endpoints among all curves on the same surface. They generalize the concept of straightest lines to curved surfaces. Asymptote lines are integral curves of a field of asymptote directions which exists on negatively curved surfaces. Curvature lines have maximal normal curvature, and its tangent vectors are called principal curvature directions. At each point on a smooth surface exist exactly two orthogonal principal curvature directions.


  1. Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces, Prentice-Hall Englewood Cliffs, NJ (1976).
  2. Alfred Gray: Modern Differential Geometry of Curves and Surfaces, CRC Press (1994).
  3. A. Besse: Manifolds all of whose geodesics are closed, Springer Verlag, Berlin (1978).

Technical Note

Each curve should be accompanied by the underlying surface.

As a guide, polygons should be connected, no degenerate edges, no duplicate vertices. Surface meshes should have no holes, no degenerate triangles and elements, no duplicate vertices. Surfaces should have meshes with an adjacency relation.