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

You can find that point by considering first a generic point (x, y) along the line from (x1, y1) to (x2, y2):

x = x1 + t*(x2 - x1)
y = y1 + t*(y2 - y1)

and the computing the (squared) distance from this point from (xp, yp)

E = (x - xp)**2 + (y - yp)**2

that substituting the definition of x and y gives

E = (x1 + t*(x2 - x1) - xp)**2 +
    (y1 + t*(y2 - y1) - yp)**2

then to find the minimum of this distance varying t we derive E with respect to t

dE/dt = 2*(x1 + t*(x2 - x1) - xp)*(x2 - x1) +
        2*(y1 + t*(y2 - y1) - yp)*(y2 - y1)

that after some computation gives

dE/dt = 2*((x1 - xp)*(x2 - x1) + (y1 - yp)*(y2 - y1) +
           t*((x2 - x1)**2 + (y1 - y2)**2))

looking for when this derivative is zero we get an explicit equation for t

t = ((xp - x1)*(x2 - x1) + (yp - y1)*(y2 - y1)) /
    ((x2 - x1)**2 + (y2 - y1)**2)

so the final point can be computed using that value for t in the definition of (x, y).

Using vector notation this is exactly the same formula suggested by Gareth...

t = 
 / 

where the notation represents the dot product operation ax*bx + ay*by.

Note also that the very same formula works in an n-dimensional space.