GEOMETRIC CONTINUITY CONDITIONS

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An alternate method for joining two successive curve sections is to specify conditions for geometric continuity. In This case, we only require parametric derivatives of the two sections to be proportional to each other at their common boundary instead of equal to each other.

Zero- order geometric continuity, described as G⁰ continuity, is the same as zero- order parametric continuity. That is, the two curves sections must have the same coordinate position at the boundary point. First order geometric continuity, or G¹ continuity, means that the parametric first derivatives are proportional at the intersection on two successive sections. If we denote the parametric position on the curve as P(u), the direction of the tangent vector P'(u), but not necessarily its magnitude, will be the same for two successive curve sections at their joining point under G¹ continuity. Second-order geometric continuity, or G² continuity, means that both the first and second parametric derivatives of the two curve sections are proportional at their boundary. Under G² continuity, curvatures of two curve sections will match at the joining position.

A curve generated with geometric continuity conditions is similar to one generated with parametric continuity, but with slight differences in curve shape. Figure below provides a comparison of geometric and parametric continuity. With geometric continuity, the curve is pulled toward the section with the greater tangent vector.

Figure 1: Curves with  G¹ continuity

Figure 1: Curves with  C¹ continuity