How to get t for average distance of Bézier curve

Question

I am calculating the interpolation position of Bézier curve by using the formula:

pow(1 - t, 2) * start + 2.0 * (1 - t) * t * control + t * t * end

The problem is that if I linear step the t by for example 0.1 per segment, the length of segment on the Bézier curve is not average.

Is there any way to get the corresponding array of t for getting average or approximately average length of the segment on the curve.

Answer 1

It seems you want an approximate parametrization by arc length.

For the quadratic case, there is a closed-form expression for the arc length of a Bézier curve, but it is complicated and you still need table lookup. These papers discuss general techniques:

• Approximate Arc Length Parametrization, in SIBGRAPI 1996.

• Adaptive sampling of parametric curves, in Graphics Gems V, 1995.

• Computing the arc length of parametric curves, in IEEE Computer Graphics and Applications, 1990.

• Point-based methods for estimating the length of a parametric curve, in Journal of Computational and Applied Mathematics, 2006.