Inflectional Convex Space Curves

Abstract

Let Φ be a regular closed C2 curve on a sphere S in Euclidean three-space. Let H(S)[H(Φ) ] denote the convex hull of S[Φ]. For any point p ∈ H(S), let O(p) be the set of points of Φ whose osculating plane at each of these points passes through p.1. THEOREM ([8]). If Φ has no multiple points and p ∈ H(Φ), then |0(p) | ≧ 3[4] when p is [is not] a vertex of Φ.2. THEOREM ( [9]). a) If the only self intersection point of Φ is a doublepoint and p ∈ H(Φ) is not a vertex of Φ, then |O(p)| ≧ 2.b) Let Φ possess exactly n vertices. Then(1)|O(p)| ≦ nforp ∈ H(S) and(2)if the osculating plane at each vertex of Φ meets Φ at exactly one point, |O(p)| = n if and only if p ∈ H(Φ) is not vertex.