Curves and Surfaces for CAGD, 5th Edition

11

Geometric Continuity

11.1 Motivation

Before we explain in detail the concept of geometric continuity, we will give an example of a curve that is curvature continuous yet not twice differentiable. Such curves (and, later, surfaces) are the objects that we will label geometrically continuous.

Figure 11.1 shows three parabolas with junction points at the midpoints of an equilateral triangle. According to (10.10), where we have to set all w_i equal to 1, all three parabolas have the same curvature at the junction points. We thus have a closed, curvature continuous curve. It is C¹ over a uniform knot sequence. But it is not C² as is easily seen by sketching the second derivative vectors at the junction points.

Figure 11.1 G² continuity: a closed ...