math - Draw a quadratic Bézier curve through three given points

Question

I have three points in 2D and I want to draw a quadratic Bézier curve passing through them. How do I calculate the middle control point (x1 and y1 as in quadTo)? I know linear algebra from college but need some simple help on this.

How can I calculate the middle control point so that the curve passes through it as well?

Answer 1

Let P0, P1, P2 be the control points, and Pc be your fixed point you want the curve to pass through.

Then the Bezier curve is defined by

P(t) = P0*t^2 + P1*2*t*(1-t) + P2*(1-t)^2

...where t goes from zero to 1.

There are an infinite number of answers to your question, since it might pass through your point for any value of t... So just pick one, like t=0.5, and solve for P1:

Pc = P0*.25 + P1*2*.25 + P2*.25

P1 = (Pc - P0*.25 - P2*.25)/.5

   = 2*Pc - P0/2 - P2/2

There the "P" values are (x,y) pairs, so just apply the equation once for x and once for y:

x1 = 2*xc - x0/2 - x2/2
y1 = 2*yc - y0/2 - y2/2

...where (xc,yc) is the point you want it to pass through, (x0,y0) is the start point, and (x2,y2) is the end point. This will give you a Bezier that passes through (xc,yc) at t=0.5.