math - How to best approximate a geometrical arc with a Bezier curve?

Question

When drawing an Arc in 2D, using a Bezier Curve approximation, how does one calculate the two control points given that you have a center point of a circle, a start and end angle and a radius?

Answer 1

This is an 8-year-old question, but one that I recently struggled with, so I thought I'd share what I came up with. I spent a lot of time trying to use solution (9) from this text and couldn't get any sensible numbers out of it until I did some Googling and learned that, apparently, there were some typos in the equations. Per the corrections listed in this blog post, given the start and end points of the arc ([x1, y1] and [x4, y4], respectively) and the the center of the circle ([xc, yc]), one can derive the control points for a cubic Bézier curve ([x2, y2] and [x3, y3]) as follows:

ax = x1 - xc
ay = y1 - yc
bx = x4 - xc
by = y4 - yc
q1 = ax * ax + ay * ay
q2 = q1 + ax * bx + ay * by
k2 = (4/3) * (sqrt(2 * q1 * q2) - q2) / (ax * by - ay * bx)

x2 = xc + ax - k2 * ay
y2 = yc + ay + k2 * ax
x3 = xc + bx + k2 * by                                 
y3 = yc + by - k2 * bx

Hope this helps someone other than me!