get closest point to a line

Question

I\'d like to have a straight forward C# function to get a closest point (from a point P) to a line-segment, AB. An abstract function may look like this. I\'ve search through

Answer 1

This answer is based on ideas from projective geometry.

Compute the cross product (Ax,Ay,1)×(Bx,By,1)=(u,v,w). The resulting vector describes the line connecting A and B: it has the equation ux+vy+w=0. But you can also interpret (u,v,0) as a point infinitely far away in a direction perpendicular to that line. Doing another cross product you get the line joining hat point to P: (u,v,0)×(Px,Py,1). And to intersect that line with the line AB, you do another cross product: ((u,v,0)×(Px,Py,1))×(u,v,w). The result will be a homogenous coordinate vector (x,y,z) from which you can read the coordinates of this closest point as (x/z,y/z).

Take everything together and you get the following formula:

Using a computer algebra system, you can find the resulting coordinates to be the following:

x = ((Ax - Bx)*Px + (Ay - By)*Py)*(Ax - Bx) + (Ay*Bx - Ax*By)*(Ay - By)
y = -(Ay*Bx - Ax*By)*(Ax - Bx) + ((Ax - Bx)*Px + (Ay - By)*Py)*(Ay - By)
z = (Ax - Bx)^2 + (Ay - By)^2

As you notice, there are a lot of recurring terms. Inventing (pretty much arbitrary) names for these, you can get the following final result, written in pseudocode:

dx = A.x - B.x
dy = A.y - B.y
det = A.y*B.x - A.x*B.y
dot = dx*P.x + dy*P.y
x = dot*dx + det*dy
y = dot*dy - det*dx
z = dx*dx + dy*dy
zinv = 1/z
return new Point(x*zinv, y*zinv)

Benefits of this approach:

• No case distinctions
• No square roots
• Only a single division