In computer graphics, the terms parametric continuity (Ck) and geometric continuity (Gn) were introduced by Brian Barsky, to show that the smoothness of a curve can be measured either with respect to a particular parametrization or after allowing changes in the speed with which the parameter traces out the curve.
Parametric continuity Parametric continuity (Ck) is a concept applied to parametric curves, which describes the smoothness of the curve as a function of its parameter. A (parametric) curve s : [ 0 , 1 ] → R n {\displaystyle s:[0,1]\to \mathbb {R} ^{n}} is said to be of class Ck if the derivatives of s {\displaystyle s} up to order k {\displaystyle k} exist and are continuous on [ 0 , 1 ] {\displaystyle [0,1]} , where derivatives at the end-points 0 {\displaystyle 0} and 1 {\displaystyle 1} are taken to be one-sided derivatives (from the right at 0 {\displaystyle 0} and from the left at 1 {\displaystyle 1} ). As a practical application of this concept, a curve describing the motion of an object with a parameter of time has C1 continuity when its velocity varies continuously, and C2 continuity when its acceleration varies continuously. For smoother motion, such as that of a camera's path while making a film, higher orders of parametric continuity may be required.
Order of parametric continuity
The various orders of parametric continuity can be described as follows:
C 0 {\displaystyle C^{0}} : zeroth derivative is continuous (curves are continuous)
C 1 {\displaystyle C^{1}} : zeroth and first derivatives are continuous
C 2 {\displaystyle C^{2}} : zeroth, first and second derivatives are continuous
C n {\displaystyle C^{n}} : 0-th through n {\displaystyle n} -th derivatives are continuous
Geometric continuity
A curve or surface can be described as having G n {\displaystyle G^{n}} continuity, with n {\displaystyle n} being an increasing measure of smoothness. Consider the segments on either side of a point on a curve:
G 0 {\displaystyle G^{0}} : The curves touch at the join point.
G 1 {\displaystyle G^{1}} : The curves also share a common tangent direction at the join point.
G 2 {\displaystyle G^{2}} : The curves also share a common center of curvature at the join point. In general, G n {\displaystyle G^{n}} continuity holds when the curves can be reparameterized so that they have C n {\displaystyle C^{n}} parametric continuity. A reparametrization of the curve is geometrically identical to the original; only the parameter is affected. Equivalently, two vector functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} such that f ( 1 ) = g ( 0 ) {\displaystyle f(1)=g(0)} have G n {\displaystyle G^{n}} continuity at the point where they meet if they satisfy equations known as Beta-constraints. For example, the Beta-constraints for G 4 {\displaystyle G^{4}} continuity are:
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