Geometrically Continuous Cubic Splines

Abstract

Recall that we can construct a double tangent cubic spline which is tangent-vector geometrically continuous at its join points by choosing pairs of entry and exit tangent vectors at each point p _i , where each pair may have differing magnitudes, but the same direction. It is common to call a tangent vector geometrically continuous curve a G¹ curve, in the same way that a tangent vector algebraically continuous curve is commonly called a C¹ curve. A C⁰ curve is merely a continuous curve, and in general, a C^k curve has k or more successive continuous derivative vectors. A G⁰ curve is just a C⁰ continuous curve. A G^l curve has a continuous unit tangent vector curve, and a G² curve also has a continuous curvature function. A regularly parameterized C^k curve, whose tangent vector does not vanish, is necessarily also a G^k curve. We often wish to focus on a particular point x(t) of a space curve x and consider whether x is G^k continuous at that point.