Parametric continuity

Short description: Notion of smoothness for parametric curves

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.

Parametric continuity

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 $s:[0,1]\to {\mathbb{ℝ}}^n$ is said to be of class C^k if the derivatives of $s$ up to order $k$ exist and are continuous on $[0,1]$, where derivatives at the end-points $0$ and $1$ are taken to be one-sided derivatives (from the right at $0$ and from the left at $1$).

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

Two Bézier curve segments attached in a way that is only C⁰ continuous

Two Bézier curve segments attached in such a way that they are C¹ continuous

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

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

Geometric continuity

"Geometric continuity" redirects here. It is not to be confused with Geometrical continuity or Continuous geometry.

Curves with G¹-contact (circles,line)

$(1-{\epsilon }^2)x^2-2px+y^2=0,p>0,\epsilon \ge 0$
pencil of conic sections with G²-contact: p fix, $\epsilon$ variable
($\epsilon =0$: circle,$\epsilon =0.8$: ellipse, $\epsilon =1$: parabola, $\epsilon =1.2$: hyperbola)

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

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

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

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

$\begin{array}{rl}{g}^{(1)}(0)& ={\beta }_1{f}^{(1)}(1)\\ g^{(2)}(0)& ={\beta }_1^2{f}^{(2)}(1)+{\beta }_2{f}^{(1)}(1)\\ g^{(3)}(0)& ={\beta }_1^3{f}^{(3)}(1)+3{\beta }_1{\beta }_2{f}^{(2)}(1)+{\beta }_3{f}^{(1)}(1)\\ g^{(4)}(0)& ={\beta }_1^4{f}^{(4)}(1)+6{\beta }_1^2{\beta }_2{f}^{(3)}(1)+(4{\beta }_1{\beta }_3+3{\beta }_2^2)f^{(2)}(1)+{\beta }_4{f}^{(1)}(1)\\ \end{array}$

where ${\beta }_2$, ${\beta }_3$, and ${\beta }_4$ are arbitrary, but ${\beta }_1$ is constrained to be positive. In the case $n=1$, this reduces to $f^{\prime }(1)\ne 0$ and $f^{\prime }(1)=k{g}^{\prime }(0)$, for a scalar $k>0$ (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 $G^1$ 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 $G^2$ or higher continuity to ensure smooth reflections in a car body.

A rounded rectangle (with ninety-degree circular arcs at the four corners) has $G^1$ continuity, but does not have $G^2$ 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 $G^2$ continuity is required, then cubic splines are typically chosen; these curves are frequently used in industrial design.