Bezier Curve Arc Length: Polynomial Approximation
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.