Singularities and dualities of pedal curves in pseudo-hyperbolic and de Sitter space

For the spherical unit speed nonlightlike curve in pseudo-hyperbolic space and de Sitter space Formula: see text and a given point P, we can define naturally the pedal curve of Formula: see text relative to the pedal point P. When the pseudo-sphere dual curve germs are nonsingular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the pseudo-sphere dual curve germs are nonsingular. Furthermore, we obtain the extension results in dualities, which has wide influence on the open and closed string field theory and string dynamics in physics, and can be used to better solve the dynamics of trajectory particle condensation process.