math - Reflection matrix for an arbitrary plane

Question

I have seen the HouseHolder equation which creates an matrix that reflects an point about an plane but the equation assumes the plane only has a normal vector v.

My plane has 3 components

The normal unit vector              V
A point that lies on the plane      P
Distance of the plane from origin   D

All stored in seperate variables.

How would I extend the equation to take the point and distance into its calculation or do I need a different approach?

question from:https://stackoverflow.com/questions/65919995/reflection-matrix-for-an-arbitrary-plane

Answer 1

(I found out the solution anyway so here it is)

The aforementioned householder equation also suppose that your plan contains the origin. So we cannot directly apply to your solution.

However let consider P as the new origin, x coordinates in this system are now x - P and the plan goes by the origin, its normal remains the same so you can compute S the symetrie of x in this system thanks to the HouseHolder equation:

S' = (x-p) - 2v(v^h(x-p)).

and you can get its coordinate in the original system :

S(x) = (x-p) - 2v(v^h(x-p)) + p