Parametric continuity

In computer graphics, the terms parametric continuity (C^k) 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.¹²³

Parametric continuity (C^k) is a concept applied to parametric curves, which describes the smoothness of the curve as a function of its parameter. A (parametric) curve ${\displaystyle s:[0,1]\to \mathbb {R} ^{n}}$ is said to be of class C^k if the derivatives of ${\displaystyle s}$ up to order ${\displaystyle k}$ exist and are continuous on ${\displaystyle [0,1]}$, where derivatives at the end-points ${\displaystyle 0}$ and ${\displaystyle 1}$ are taken to be one-sided derivatives (from the right at ${\displaystyle 0}$ and from the left at ${\displaystyle 1}$).

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.

The various orders of parametric continuity can be described as follows:⁴

• ${\displaystyle C^{0}}$: zeroth derivative is continuous (curves are continuous)
• ${\displaystyle C^{1}}$: zeroth and first derivatives are continuous
• ${\displaystyle C^{2}}$: zeroth, first and second derivatives are continuous
• ${\displaystyle C^{n}}$: 0-th through ${\displaystyle n}$-th derivatives are continuous

A curve or surface can be described as having ${\displaystyle G^{n}}$ continuity, with ${\displaystyle n}$ being an increasing measure of smoothness. Consider the segments on either side of a point on a curve:

• ${\displaystyle G^{0}}$: The curves touch at the join point.
• ${\displaystyle G^{1}}$: The curves also share a common tangent direction at the join point.
• ${\displaystyle G^{2}}$: The curves also share a common center of curvature at the join point.

In general, ${\displaystyle G^{n}}$ continuity holds when the curves can be reparameterized so that they have ${\displaystyle C^{n}}$ parametric continuity.⁵⁶ A reparametrization of the curve is geometrically identical to the original; only the parameter is affected.

Equivalently, two vector functions ${\displaystyle f(t)}$ and ${\displaystyle g(t)}$ such that ${\displaystyle f(1)=g(0)}$ have ${\displaystyle G^{n}}$ continuity at the point where they meet if they satisfy equations known as Beta-constraints. For example, the Beta-constraints for ${\displaystyle G^{4}}$ continuity are:

${\displaystyle {\begin{aligned}g^{(1)}(0)&=\beta _{1}f^{(1)}(1)\\g^{(2)}(0)&=\beta _{1}^{2}f^{(2)}(1)+\beta _{2}f^{(1)}(1)\\g^{(3)}(0)&=\beta _{1}^{3}f^{(3)}(1)+3\beta _{1}\beta _{2}f^{(2)}(1)+\beta _{3}f^{(1)}(1)\\g^{(4)}(0)&=\beta _{1}^{4}f^{(4)}(1)+6\beta _{1}^{2}\beta _{2}f^{(3)}(1)+(4\beta _{1}\beta _{3}+3\beta _{2}^{2})f^{(2)}(1)+\beta _{4}f^{(1)}(1)\\\end{aligned}}}$

where ${\displaystyle \beta _{2}}$, ${\displaystyle \beta _{3}}$, and ${\displaystyle \beta _{4}}$ are arbitrary, but ${\displaystyle \beta _{1}}$ is constrained to be positive.^5: 65 In the case ${\displaystyle n=1}$, this reduces to ${\displaystyle f'(1)\neq 0}$ and ${\displaystyle f'(1)=kg'(0)}$, for a scalar ${\displaystyle k>0}$ (i.e., the direction, but not necessarily the magnitude, of the two vectors is equal).

While it may be obvious that a curve would require ${\displaystyle G^{1}}$ continuity to appear smooth, for good aesthetics, such as those aspired to in architecture and sports car design, higher levels of geometric continuity are required. For example, class A surface requires ${\displaystyle G^{2}}$ or higher continuity to ensure smooth reflections in a car body.

A rounded rectangle (with ninety-degree circular arcs at the four corners) has ${\displaystyle G^{1}}$ continuity, but does not have ${\displaystyle G^{2}}$ continuity. The same is true for a rounded cube, with octants of a sphere at its corners and quarter-cylinders along its edges. If an editable curve with ${\displaystyle G^{2}}$ continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design.