Find a line intersecting a known line at right angle, given a point

Question

This is basic graphics geometry and/or trig, and I feel dumb for asking it, but I can\'t remember how this goes. So:

1. I have a line defined by two points (x1, y1

Answer 1

A useful rule of thumb in this kind of computational geometry is that you should work with vectors as long as you can, switching to Cartesian coordinates only as a last resort. So let's solve this using vector algebra. Suppose your line goes from p to p + r, and the other point is q.

Now, any point on the line, including the point you are trying to find (call it s), can be expressed as s = p + λ r for a scalar parameter λ.

Now the vector from q to s must be perpendicular to r. Therefore

(q − (p + λ r)) · r = 0

Where · is the dot product operator. Expand the product:

(q − p) · r = λ (r · r)

And divide:

λ = (q − p) · r / r · r

When you come to implement it, you need to check whether r · r = 0, to avoid division by zero.