Dot product:

                        a × b = a₁b₁  +   a₂b₂ +  a₃b₃ =   ||a|| ||b|| cos a

Cross product:

a x b = = (a₂b₃ – a₃b₂, a₃b₁ – a₁b₃, a₁b₂ – a₂b₁)

Magnitude is:              || a x b || = ||a|| ||b|| sin a

3-D Transformations

Note: in OpenGL a matrix is represented as a 4 * 4 matrix of single- or double-precision floating-point values stored in column-major order. That is, the matrix is stored as

1.   Translation by a vector  u = :

2.   Scaling by      along the  x-, y-, and z- axis, respectively:

3.   Reflection is a special case of scaling with negative coefficients:  or  or . This achieves reflection about the coordinates planes. Need to be combined with translations and rotations for the general plane. E.g. reflection about the xz – plane (bottom plane) is

4.   Projection:  a projection P renders an object to a lower dimension. They are always idempotent: P ² = P .

Simplest case: projection to the coordinate planes. This is scaling with a coefficient being 0, the rest 1. E.g. projection to the xz – plane (bottom plane):

A perspective projection is from a single point. Simplest case: shadow cast on the bottom xz-plane.

The matrix is

Here  x maps to x’ = ,  y  maps  to  0  and  z  maps  to  z’ =   by the similarity of the appropriate triangles.

General projection: first we need the equations of the line and the plane.

Equation of line:                                 Ax + By + C = 0

Equation of plane:                              Ax + By + Cz + D = 0

These are easily proved using normal vectors n = (A, B) and n = (A, B, C), respectively, setting the dot products to 0 on a difference vector (2 vectors), respectively.

General projection to the plane     Ax + By + Cz + D = 0:

Here the (x, y, z) is the vector between the light source and the center of the object.

5.   Rotation: The rotation R_q , u by the agle q  about the unit vector  u = (x, y, z)  is

where    s = sin q    and   c = cos q . Also || u || = || (x, y, z) || = 1.