Reflection

In computer graphics, reflection is a transformation that flips an object across a line, plane, or point. There are different types of reflections, such as planar reflection, point reflection, and line reflection. 

Here, we will discuss planar reflection, which involves reflecting an object across a plane. To perform planar reflection, we need to define a plane in 3D space, which will act as the mirror plane for reflection. Then, we can apply the following steps:

1. Translate the object so that the plane of reflection passes through the origin. This can be done by applying a translation matrix `T1` that moves the origin to the plane of reflection.

```

T1 = [ 1  0  0  -dX ]

     [ 0  1  0  -dY ]

     [ 0  0  1  -dZ ]

     [ 0  0  0   1  ]

```

Here, `(dX, dY, dZ)` is the signed distance from the origin to the plane of reflection.

2. Reflect the object across the plane of reflection. This can be done by applying a reflection matrix `R` that flips the object across the plane of reflection.

```

R = [ 1-2*Nx*Nx   -2*Nx*Ny   -2*Nx*Nz   0 ]

    [ -2*Nx*Ny   1-2*Ny*Ny   -2*Ny*Nz   0 ]

    [ -2*Nx*Nz   -2*Ny*Nz   1-2*Nz*Nz   0 ]

    [ 0         0         0         1 ]

```

Here, `(Nx, Ny, Nz)` is the normal vector to the plane of reflection.

3. Translate the object back to its original position. This can be done by applying a translation matrix `T2` that moves the origin back to its original position.

```

T2 = [ 1  0  0  dX ]

     [ 0  1  0  dY ]

     [ 0  0  1  dZ ]

     [ 0  0  0  1  ]

```

4. Combine the above transformations to form a composite transformation matrix `M`.

```

M = T2 * R * T1

```

Note that the order of the transformations is important. We first translate the object so that the plane of reflection passes through the origin, then reflect it across the plane of reflection, and finally translate it back to its original position. If we change the order of the transformations, we may get a different result.