Normal forms for singularities of pedal curves produced by non-singular dual curve germs in S n

摘要

For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are non-singular, 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 dual curve germs are non-singular. As an application of our list, we characterize C ∞ left equivalence classes of pedal curve germs (I, s 0) → S n produced by non-singular dual curve germ from the viewpoint of the relation between tangent space and tangent space.