I wish to approximate the length of a Bézier curve in a 2D plane. I could do this iteratively – and may – but I want to experiment using Simpson's rule. I am not sure how to set this up and could use some advice. I am generating the Bézier curve from cubic parametric equations. Currently I am thinking
1) Divide Bézier curve using parameter
2) For each segment, derive quadratric fit equation using start, end and additional point on the curve, $ax^2+bx+c$
3) Here's where I need help – should I try to define the arc length function parametrically using $x(t)$ and $y(t)$, or as a function $f(x)$. Are the two cases equivalent in the 2D quadratic case – as $x(t)=t$ , and $y(t)=f(x)$?
Any advice or links appreciated!