P-36 Exploring 2nd Natural Mates and Families of Primitive Curves

Buller Hall 207

The natural mate of a unit speed curve is generated by letting its principal normal vector be the tangent vector of the new curve. Expanding on Deshmukh et al., we demonstrate that given a curve, the primitive curve of which it is the natural mate is not uniquely determined, giving a family of curves with the same natural mate. We then explore the 2nd natural mate, demonstrating a simple relationship between its curvature and torsion, and proving several smaller theorems regarding how constraints on the primitive, 1st natural mate, and 2nd natural mate may constrain the other curves.