Approximating Circular Arcs with Cubic Splines

Some software and some machine tools have Bezier spline curves as primitives, but do not have circular arcs. If you want a circle, you need to piece an approximation together out of splines. Here we use mechanical Expressions to find the control points for a spline which approximates a quarter of the unit circle.

The diagram shows the Bezier Spline defined by a control polygon whose geometry is parametrized by length k. Measuring the distance of the mid point of the curve to the origin, we get an expression in terms of k. If we find a value of k such that this expression is equal to 1, then the mid point of the spline will lie on the arc.

A more extensive version of this example is available here.