Apparent contours in Minkowski 3-space and first order ordinary differential equations

摘要

We consider projections of a smooth and regular surface M in the Minkowski 3-space ?~3 _1 along lightlike directions to a fixed transverse plane. The lightlike directions in ?~3 _1can be parametrized by a circle on the lightcone and the resulting 1-parameter family of projections can be considered as viewing M along a special 'camera motion'. The associated 1-parameter families of contour generators and apparent contours reveal some aspects of the extrinsic and intrinsic geometry of M. We characterize geometrically the generic A_e-codimension 1 singularities of a given projection and consider their bifurcations in the family of projections. We show that the families of contour generators and apparent contours are solutions of certain first order ordinary differential equations and obtain their generic local configurations.