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 Ck curve has k or more successive continuous derivative vectors. A G0 curve is just a C0 continuous curve. A Gl 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.