3-D Coordinate Vector Transformation

Date: 8/27/96 at 1:37:56
From: Anonymous
Subject: 3-D Coordinate Vector Transformation

What is the approach to take for transforming one coordinate system to 
another where it isn't just a translation transformation but also 
rotations and even reflections? The general case is where one would 
have two coordinate systems, each defined by its origin, and three 
orthogonal vectors defining x,y,z axes. 

Date: 8/27/96 at 18:52:42
From: Doctor Tom
Subject: Re: 3-D Coordinate Vector Transformation

For details, get any book on 3D computer graphics, but here's
the general idea:

Add a fake "w" coordinate to make a 4D vector that looks like
(x, y, z, w), and in every case, just make the w coordinate
be 1.  So the point (3, 5, 7) will be represented by (3, 5, 7, 1).

Every one of the transformations you mention (rotation, translation,
scaling, mirroring, (and even shearing, perspective transformations,
and some other stuff) can be represented as a multiplication by
the correct 4x4 matrix.

Here are some examples:

To rotate about the z axis by angle t, multiply by:

|  cos t   sin t   0    0  |
| -sin t   cos t   0    0  |
|    0       0     1    0  |
|    0       0     0    1  |

To mirror about the x-axis, multiply by:

|   -1  0  0  0  |
|    0  1  0  0  |
|    0  0  1  0  |
|    0  0  0  1  |

To translate by 3 in x, 4 in y and 5 in z, multiply by:

|  1  0  0  0  |
|  0  1  0  0  |
|  0  0  1  0  |
|  3  4  5  1  |

To do combinations, multiply the appropriate combinations of matrices 
together.

This is related to projective geometry in some very interesting ways, 
so after you've fiddled with the stuff above a bit, you might look 
into projective geometry to see what's "really" going on!

Good luck.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/