What is reflection matrix 3x3?

Reflection Matrix (3x3)

A 3x3 reflection matrix is a linear transformation that reflects points in 3D space across a plane that passes through the origin. This plane is defined by its normal vector.

Here's a breakdown:

• General Form: While there's no single "general form" that is always applicable (it depends on the plane of reflection), any reflection matrix can be derived from a reflection%20transformation. The idea is to perform a rotation (if necessary) to align the reflection plane with one of the coordinate planes (xy, xz, or yz), perform a simple reflection across that coordinate plane, and then reverse the original rotation.

• Reflection across Coordinate Planes: Reflections across the principal coordinate planes have simple, easily memorized matrices:

  • Reflection across the XY-plane (z=0):

    [ 1  0  0 ]
    [ 0  1  0 ]
    [ 0  0 -1 ]
    

  • Reflection across the XZ-plane (y=0):

    [ 1  0  0 ]
    [ 0 -1  0 ]
    [ 0  0  1 ]
    

  • Reflection across the YZ-plane (x=0):

    [ -1  0  0 ]
    [  0  1  0 ]
    [  0  0  1 ]
    

• Reflection across an Arbitrary Plane (through the origin): If n = (nx, ny, nz) is the unit normal vector to the plane, the reflection matrix R can be computed as follows:

  R = I - 2 * n * n<sup>T</sup>
  

  Where:

  • I is the 3x3 identity matrix.
  • n is the column vector representation of the normal vector.
  • n^T is the transpose of n (a row vector).
  • n * n<sup>T</sup> is the outer product of n with itself, resulting in a 3x3 matrix.
  • 2 * n * n<sup>T</sup> is a 3x3 matrix where each element (i, j) is 2 * ni * nj.

  In component form:

  [ 1 - 2nx²  -2nxny   -2nxnz  ]
  [ -2nxny   1 - 2ny²  -2nynz  ]
  [ -2nxnz   -2nynz   1 - 2nz² ]
  

• Properties:

  • A reflection matrix is an orthogonal%20matrix, meaning its transpose is equal to its inverse (R^T = R^-1).
  • The determinant of a reflection matrix is -1. This indicates that it reverses the orientation of space.
  • Applying the same reflection twice returns the original vector (R * R = I).
  • Reflection matrices are used extensively in computer graphics, linear algebra, and physics to model reflections of objects and waves.

• Application: To reflect a 3D point v (represented as a column vector) across the plane, simply multiply the reflection matrix R by the vector v: v' = R * v.

• Important Note: If the plane of reflection does not pass through the origin, you will need to use a more general affine%20transformation which combines a linear transformation (the reflection) with a translation. This often involves using homogeneous coordinates.