Parametric continuity

In computer graphics, the terms parametric continuity (C k ) and geometric continuity (G n ) 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.

In computer graphics, the terms parametric continuity (C k ) and geometric continuity (G n ) 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 C 1 continuity when its velocity varies continuously, and C 2 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=== segments attached in a way that is only C 0 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== 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{align} 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{align} 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'(1)\neq0 and f'(1) = kg'(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 (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 , 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. ==References==

Text derived from https://en.wikipedia.org/w/index.php?oldid=1361111127 under CC BY-SA 4.0; modifications have been made.