Enveloids and involutoids of spherical Legendre curves

Abstract

We define ϕ-enveloids for one-parameter families of spherical Legendre curves. The ϕ-enveloids are not only the generalizations of isogonal trajectories of regular spherical curves but also the generalizations of envelopes of spherical curves. We also consider the relationships between enveloids of spherical Legendre curves and plane Legendre curves. Furthermore, we generalize the notions of involutes and evolutes to the notions of involutoids and evolutoids of spherical curves from the view point of ϕ-enveloids. Then we give the duality theorem for involutoids and evolutoids. Finally, we give the relationships among involutes, evolutes, 0-enveloids (envelopes) and π/2-enveloids (normal envelopes).