Derivative of a Bernstien Polynomial
Derivative of a Bezier Curve and Hodograph
Derivative of a Bernstien Polynomial --
(*junk*)
Derivative of a Bezier Curve and Hodograph
Formula 1: Let (*junk*) be a Bezier curve. Check it!
Note: The derivative of a bezier curve (usually called the derivative curve) is another bezier curve since it can be written as a linear combination of Bernstein polynomials.
Definition: The above derivative curve is often called a Hodograph of the Bezier curve.
Note: This derivative curve does not live in anymore. Its coefficients are the differences of the points, i.e. vectors, which are elements of . To visualize the derivative curve and polygon in , we can construct a control polygon of the derivative curve in that consists of the points . Here a is an arbitrary point; we often choose a=0.
Higher Order Derivatives Formula 2: Let (*junk*) be a Bezier curve. The r-th derivative of the curve is given by (*junk*) Click here for an explanation of notation and the idea of the proof.
Derivatives of generalized Bernstein polynomials
Directional derivatives When we discussed derivatives for tensor product patches, we considered partials because they are easily computed for those surfaces. The situation is different for triangular patches; the appropriate derivative here is the directional derivative. Let u1 and u2 be two points in the domain. Their difference d = u2 - u1 defines a vector. The directional derivative of a surface at x(u) with respect to d is given by A geometric interpretation: in the domain, draw the straight line through u that is parallel to d. This straight line will be mapped to a curve on the patch. Its tangent vector at x(u) is the desired directional derivative. The partials of a Bezier patch are not hard to compute; we have, for the u-partial: Working out similar expressions for the v- and w-partials, we have found our directional derivative: A closer look at the terms in square brackets brings to mind the de Castlejau algorithm, and we may write the above equation as Thus a directional derivative is obtained by performing one step of the de Castlejau algorithm with respect to the direction vector d, and n-1 more with respect to the position u. Such configurations can be succintly expressed in blossom form:
We may continue taking derivatives, arriving at The r-th directional derivative at b(u) is therefore found by performing r steps of the de Castlejau algorithm with respect to d, and then by performing n-r more steps with respect to u. Of course it is irrelevant in which order we take these steps.
Mixed directional derivatives In the same way, we may compute mixed directional derivatives: if d1 and d2 are two vectors in the domain, then their mixed directional derivatives are This blossom result may also be expressed in terms of Bernstein polynomials. Taking n-r steps of the de Castlejau algorithm with respect to u, and then r more with respect to d, gives Or, we might have taken r steps with respect to d first, and then n-r ones with respect to u. This gives