How can I reflect a point with respect to the plane?

Question

How can I reflect a point with respect to a plane?

Example:

I have three points (0,0,2), (4,0,0) and (0,8,0). And I have a point (x,y,z). From these, I want derive a composite transformation matrix. How can I do that?

Answer 1

Given the following:

• A as the 4x4 augmented translation matrix to move any of the point of the plane into the origin
• I as the 3x3 identity matrix
• N as the 3-dimensional unit normal vector for the plane (calculable by creating a cross product from any two non-parallel lines on the plane, then normalising them)

The calculation steps for the augmented reflection matrix are as follows:

1) Calculate the 3x3 reflection matrix R₀ for a plane with the same normal vector, but which lies at the origin

R_o = I - 2 NN^T

2) Augment the reflection matrix to create the augmented reflection matrix R_A

3) Calculate the final augmented reflection matrix by first applying the translation A, mirroring the point by R_A, then applying the reverse of the translation A^-1

R = AR_AA^-1

Answer 2

Three points, A, B and C define a plane. The lines AB and AC are both on the plane, so their normalised cross product, perpendicular to both lines, is the plane's normal n:

n = normalise( (B - A) ⨯ (C - A) )

The equation for the plane is given by: n · x + d = 0, where -d is the displacement between the plane and the origin in the normal direction, i.e.: d = -n · A.

An arbitrary point x can be described as the sum of two vectors, u pointing from plane to point in the normal direction, v being the other component:

u = (x · n + d) n

v = x - u

The mirror image of x can then be calculated by flipping the u-component: x' = -u + v. Now that x' is known, capturing the calculation in a transformation matrix is trivial: solve T x = x' for T.