Point Reflected in a Plane

Date: 7 Jun 1995 17:41:42 -0400
From: DAVID
Subject: Help fast

Hi!
                      
I need to know how I can calculate the coordinates of a point p'.  I have  
given the point p and a plane E. The point p is supposed to be reflected  
(don't know the real name) by E.  How do I get p'?
          E
          E
   p------E------p'
          E
          E
          E

Thanks Dog!
## CrossPoint v3.02 ##

Date: 8 Jun 1995 09:57:52 -0400
From: Dr. Ken
Subject: Re: Help fast

Hello there!

I'll go through an example with you, and then I'll leave it to you to
generalize the process.

Let's say we have the point p = (1,2,3) and the plane 2x + 3y + 5z = 3.  
The first thing we're going to try to do is find the point in the plane that's
closest to p, and then we're going to be done with the hard part of the
problem.  

The shortest distance to the plane is going to be the perpendicular
distance, i.e. along a line perpendicular to the plane.  Since the vector
(2,3,5) is perpendicular to our plane (you can always get a perpendicular
vector from the coefficients in the plane equation), we can find our point
by finding a point on the line (1,2,3) + t(2,3,5) that's in the plane, i.e.
solve the following equation for t:

2(2t+1) + 3(3t+2) + 5(5t+3) = 3
4t + 2 + 9t + 6 + 25t + 15 = 3
38t = -20
  t = -10/19

So now the point in the plane that's closest to p is (1,2,3) + -10/19*(2,3,5),
i.e. (-1/19, 8/19, 7/19).  Now to get p', just find the difference between
(1,2,3) and (-1/19, 8/19, 7/19), which is (-20/19, -30/19, -50/19), and add
it to (1,2,3) twice:  we get (-21/19, -22/19, -34/19).

See how that works?  If there's anything that's not clear to you in this,
write us back and ask about it.

-K