Parametric Curves

Geometric and Parametric Continuity

Geometric Continuity

* G0: curves are joined
* G1: first derivatives are proportional at the join point
The curve tangents thus have the same direction, but not necessarily the same magnitude. i.e., C1'(1) = (a,b,c) and C2'(0) = (k*a, k*b, k*c).
* G2: first and second derivatives are proportional at join point

Parametric Continuity

* C0: curves are joined, There may be a sharp point where they meet.
* C1: first derivatives equal, The curves have identical tangents at the breakpoint. (The tangent is the slope at the breakpoint.) The curves join smoothly. C1 curves also have positional continuity.
* C2: first and second derivatives are equal,
The curves have identical curvature at the breakpoint. (Curvature is defined as the rate of change of the tangents.) Curvature continuity implies both tangential and positional continuity.


If t is taken to be time, this implies that the acceleration is continuous.
* Cn: nth derivatives are equal