3D transformation

Procedure is similar as in the case of two-dimensional geometric transformations.

Translation

Translation can best be described as linear change in position. This change can be represented by a delta vector [dx, dy, dz], where dx represents the change in the object's x position, dy represents the change in its y position, and dz its z position.

Translation is defined by a delta vector v  = (Tx, Ty, Tz) and by a transformation matrix

     | 1  0  0  0 |
     | 0  1  0  0 |
Ap = | 0  0  1  0 |
     |Tx Ty Tz  1 |

Example :
Point with coordinates (x, y, z) will be shifted by vector v =  (Tx, Ty, Tz).

             |  1  0  0  0 |
| x y z 1 |* |  0  1  0  0 | = | x+Tx y+Ty z+Tz 1 |
             |  0  0  1  0 |
             | Tx Ty Tz  1 |

Rotation

Rotation of an object in space is done by means of rotation around coordinate axes.

Transformation matrix of rotation about the x axis by angle u :

     | 1   0     0    0 |
     | 0  cos u sin u 0 |
Ax = | 0 -sin u cos u 0 |
     | 0   0     0    1 |


$$$APPLET

Applet Rotation of an object about the x axis

Transformation matrix of rotation about the y axis by angle u :

     | cos u  0 sin u  0 |
     |  0    1   0     0 |
Ay = | -sin u 0 cos u  0 |
     |  0    0   0     1 |

$$$APPLET

Applet Rotation about the y axis

Transformation matrix of rotation about the z axis by angle u :
 

    | cos u sin u  0  0 |
    |-sin u cos u  0  0 |
Az =|  0      0    1  0 |
    |  0      0    0  1 |


$$$APPLET

Applet Rotation of an object about the z axis

Example of rotation:

Let's take point (X, Y, Z) to rotate about the point (0, 0, 0)

1. The 2D rotation about the x axis by angle u

 X' = X
 Y' = Y * cos(u) - Z * sin(u)
 Z' = Y * sin(u)  + Z *cos(u)

2. rotate about the y axis by angle u

 X' = X * cos(u) + Z * sin(u)
 Y' = Y
 Z' = Z *cos(u) - X * sin(u)

3. rotate about the z axis by angle u

 X' = X * cos(u) - Y * sin(u)
 Y' = Y *cos(u)+-X * sin(u)
 Z' = Z

Finally, result coordinate of point are (X', Y', Z').

Scaling

Scale change - scaling in space is represented by the following transformation matrix:

     | Sx  1 0  0 |
     |  1 Sy 0  0 |
As = |  0  0 Sz 0 | ,
     |  0  0  0 1 |
where coefficients Sx, Sy and Sz determine scale point of the coordinate axis.

Example : Vector (x, y, z, 1) is to be magnified to (x*Sx, y*Sy, z*Sz, 1)

The vector will be multiplied by the transformation matrix :

| sx 0  0  0 |
|  x y  z  1 | * | 0 sy 0 0 | = |x*sx y*sy z*sz 1|
|  0 0 sz  0 |
|  0 0  0  1 |

Symetry

Like in surface, in space symetry is a special case of scale change, where values of Sx, Sy and Sz are either 1 or -1.

 
Sx
Sy
Sz
  symetry by x axis
<1
-1
-1
  symetry by y axis
-1
1
-1
  symetry by z axis
-1
-1
1
  symetry by xy plane
1
1
-1
  symetry by plane xy
1
-1
1
  symetry by plane xy
-1
1
1
  symetry by center
-1
-1
-1

Shear transformations

In space, we divide shear transformation according to the direction of the surfaces xy,xz and yz. Values of Sx,Sy and Sz determine shear transformation sizes for all the directions.

A shear transformation about the xy plane

      |  1  0 0 0 |
      |  0  1 0 0 |
Axy = | Sx Sy 0 0 |.
      |  0  0 0 1 |

A shear matrix about the xz plane

      |  1  0  0  0 |
      | Sx  1 Sz  0 |
Axz = |  0  1  1  0 |.
      |  0  0  0  1 |

A shear matrix about the yz plane

      | 1 Sy Sz 0 |
      | 0  1  0 0 |
Ayz = | 0  0  1 0 |.
      | 0  0  0 1 |