## Geometric transformations

### Translation

x' = x + A
y' = y + B
Position is changed. Orientation and distance are fixed.
Image invariants: distance, orientation
A unit square aligned with the axes can be mapped to any other unit square aligned with the axes.

### Rotation

x' = x cos(w) - y sin(w) + A
y' = x sin(w) + y cos(w) + B
Position and orientation in plane perpendicular to line of sight are fixed.
Image invariants: Distances, angles.
A unit square can be mapped to any other unit square.

### Scale

x' = Ax
y' = Ay
Orientation is fixed. Distance varies.
Image invariants: Orientation, ratios between distance.
Any square can be mapped to another square of same alignment.

### Orthogonal transformations: scale + rotation

(no picture)
x' = Ax - By + C
y' = Bx + Ay + D
Surface plane remains normal to line of sight.
Invariants: Ratios between distances, angles
Any square can be mapped to any other square.

### Affine

x' = Ax + By + C
y' = Dx + Ey + F
(AE - BD > 0).

3-D motion. Distance from eye to surface is large, so distance to all parts of surface is essentially the same.
Invariants: Parallel lines. Ratios between distances on parallel lines.
Any parallelogram can be mapped to any other parallelogram.

### Projective (perspective)

x' = (Ax +By +C) / (Gx + Hy + I)
y' = (Dx +Ey +F) / (Gx + Hy + I)
not(G=H=I=0) and AE-BD != 0.
3-D motion. Surface is close to eye, so some parts are much closer than others.
Invariants: Collinearity of points; coincidence of lines.