Example: The order 3 Bezier Curve[br]Let g(t) be the length² of the tangent: [math] g(t) = {\small ({\bf \dot x \cdot \dot x})}, [/math][br]And [math]{\small f(t) = \sqrt{g(t)}} [/math].[br]The arc length of the order 3 Bezier curve is: [math] S(t) = {\small \;\;\; \int f(x) dx} [/math].[br][br]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.[br][br]Here is a function explorer for selecting and arranging the interpolating polynomials.