Maths - Reflection using Matrix

Matrix representation

normal component

Norm = Va P = (Va × P) * P/|P|²

This is derived on this page.

parallel component

Proj = Va || P = Va • P * P/|P|²

This is derived on this page.

Reflection matrix

Given the inversion I'll add the terms instead of subtracting them to give the reflection result:

p → 1 /
(Px² + Py² + Pz²)*
-Px² + Pz²+ Py² - 2 * Px * Py - 2 * Px * Pz
- 2 * Py * Px -Py² + Px² + Pz² - 2 * Py * Pz
- 2 * Pz * Px -2 * Pz * Py -Pz² + Py² + Px²
[p]

Note that this matrix is symmetrical about the leading diagonal, unlike the rotation matrix, which is the sum of a symmetric and skew symmetric part.

Simple cases

In order to check the above lets take the simple cases where the point is reflected in the various axis:

Reflection in yz

-1 0 0
0 1 0
0 0 1

Reflection in xz

1 0 0
0 -1 0
0 0 1

Reflection in xy

1 0 0
0 1 0
0 0 -1

Determinant and eigen values

Another check is that the determinant of reflection matrix is -1

Example

refection in 30° line

As an example we want to reflect the point (1,0,0) in a plane at 30 degrees.

P2 =
-Px² + Pz² + Py² - 2 * Px * Py - 2 * Px * Pz
- 2 * Py * Px -Py² + Px² + Pz² - 2 * Py * Pz
- 2 * Pz * Px -2 * Pz * Py -Pz² + Py² + Px²
[P1]

where Px,Py,Pz is the normal to the mirror which is: (-0.5,0.866,0)

and P1 is the initial point which is (1,0,0)

substituting these values gives:

P2 =
-0.25 + 0 + 0.75 - 2 * -0.433 0
- 2 * -0.433 -0.75 + 0.25 + 0 0
0 0 0 + 0.75 + 0.25

 

1
0
0

 

P2 =
0.5 0.866 0
0.866 -0.5 0
0 0 1

 

1
0
0

 

multiplying to vector by the matrix gives:

P2 =

 

0.5
0.866
0

 

 


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Correspondence about this page

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