Bezier Curve Arc Length: Polynomial Approximation
- Ryan Hirst
Example: The order 3 Bezier Curve Let g(t) be the length² of the tangent: And . The arc length of the order 3 Bezier curve is: . The integral will not bow to formal manipulation. But f(t) can be easily evaluated at a series of points. Using these points, we can approximate f(x) by polynomials which are easily integrated. Here is a function explorer for selecting and arranging the interpolating polynomials.
- The Bezier curve may have at most two cusps where the tangent reaches a local minimum. Here, the cusps can be given by the red control points. Since these points present the most difficulty for polynomial approximation, I have built the test function so that both cusps exist, and can be made arbitrarily sharp (drag either red point toward the x-axis).
- The Lagrange polynomial is the ordinary polynomial through n points (order n-1). The Hermitian polynomial matches f(x) and its first derivative at each control point. For n points, the curve is order 2n-1.
- If you think you have a good approximation, remember to drag the red points around. Do you have a plan for convergence?
- The second graphics view provides information about the error for the selected subinterval.
- GGB will agree that two numbers are equal if the relative difference is roughly of the order 10^(-8).