Question

I am having some issues calculating the nearest point on a quadratic curve to the mouse position. I have tried a handful of APIs, but have not had any luck finding a function for this that works. I have found an implementation that works for 5th degree cubic bezier curves, but I do not have the math skills to convert it to a quadratic curve. I have found some methods that will help me solve the problem if I have a t value, but I have no idea how to begin finding t. If someone could point me to an algorithm for finding t, or some example code for finding the nearest point on a quadratic curve to an arbitrary point, I’d be very grateful.

Thanks

Answer 1

I can get you started on the math.
I’m not sure how quadratic Bezier is defined, but it must be equivalent
to:

(x(t), y(t)) = (a_x + b_x t + c_x t^2, a_y + b_y t + c_y t^2),

where 0 < t < 1. The a, b, c’s are the 6 constants that define the curve.

You want the distance to (X, Y):

sqrt( (X - x(t))^2 + (Y - y(t))^2  )

Since you want to find t that minimizes the above quantity, you take its
first derivative relative to t and set that equal to 0.
This gives you (dropping the sqrt and a factor of 2):

0 = (a_x - X + b_x t + c_x t^2) (b_x + 2 c-x t) + (a_y - Y + b_y t + c_y t^2) ( b_y + 2 c_y t) 

which is a cubic equation in t. The analytical solution is known and you can find it on the web; you will probably need to do a bit of algebra to get the coefficients of the powers of t together (i.e. 0 = a + b t + c t^2 + d t^3). You could also solve this equation numerically instead, using for example Newton-Raphson.

Note however that if none of the 3 solutions might be in your range 0 < t < 1. In that case just compute the values of the distance to (X, Y) (the first equation) at t = 0 and t = 1 and take the smallest distance of the two.

EDIT:
actually, when you solve the first derivative = 0, the solutions you get can be maximum distances as well as minimum. So you should compute the distance (first equation) for the solutions you get (at most 3 values of t), as well as the distances at t=0 and t=1 and pick the actual minimum of all those values.