math - Perpendicular on a line segment from a given point

Question

I want to calculate a point on a given line that is perpendicular from a given point.

I have a line segment AB and have a point C outside line segment. I want to calculate a point D on AB such that CD is perpendicular to AB.

I have to find point D.

It quite similar to this, but I want to consider to Z coordinate also as it does not show up correctly in 3D space.

Answer 1

Proof: Point D is on a line CD perpendicular to AB, and of course D belongs to AB. Write down the Dot product of the two vectors CD.AB = 0, and express the fact D belongs to AB as D=A+t(B-A).

We end up with 3 equations:

 Dx=Ax+t(Bx-Ax)
 Dy=Ay+t(By-Ay)
(Dx-Cx)(Bx-Ax)+(Dy-Cy)(By-Ay)=0

Subtitute the first two equations in the third one gives:

(Ax+t(Bx-Ax)-Cx)(Bx-Ax)+(Ay+t(By-Ay)-Cy)(By-Ay)=0

Distributing to solve for t gives:

(Ax-Cx)(Bx-Ax)+t(Bx-Ax)(Bx-Ax)+(Ay-Cy)(By-Ay)+t(By-Ay)(By-Ay)=0

which gives:

t= -[(Ax-Cx)(Bx-Ax)+(Ay-Cy)(By-Ay)]/[(Bx-Ax)^2+(By-Ay)^2]

getting rid of the negative signs:

t=[(Cx-Ax)(Bx-Ax)+(Cy-Ay)(By-Ay)]/[(Bx-Ax)^2+(By-Ay)^2]

Once you have t, you can figure out the coordinates for D from the first two equations.

 Dx=Ax+t(Bx-Ax)
 Dy=Ay+t(By-Ay)