## 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.

Any convex quadrilateral can be mapped to any other convex quadrilateral.