Best way to find a point on a circle closest to a

Question

Given a point (pX, pY) and a circle with a known center (cX,cY) and radius (r), what is the shortest amount of code you can come up with to find the point on the circle closest to (pX, pY) ?

I've got some code kind of working but it involves converting the circle to an equation of the form (x - cX)^2 + (y - cY)^2 = r^2 (where r is radius) and using the equation of the line from point (pX, pY) to (cX, cY) to create a quadratic equation to be solved.

Once I iron out the bugs it'll do, but it seems such an inelegant solution.

Answer 1

Treat the centre of the circular as your origin, convert the co-ordinates of (pX, pY) to polar co-ordinates, (theta, r') replace r' with the original circle's r and convert back to cartesian co-ordinates (and adjust for the origin).

Answer 2

Easy way to think about it in terms of a picture, and easy to turn into code: Take the vector (pX - cX, pY - cY) from the center to the point. Divide by its length sqrt(blah blah blah), multiply by radius. Add this to (cX, cY).