2. Geometric transformations
Geometric transformations will map points in one
space to points in another: (x',y',z') = f(x, y, z).
These transformations can be very simple, such as
scaling each coordinate, or complex, such as nonlinear
twists and bends.
1
3. Linear Transformation
A 2 x 2 linear transformation matrix allows:
Scaling
Rotation
Reflection
Shearing
2
4. Affine Transformation 3
Definition:
P(Px, Py) is transformed into Q(Qx , Qy ) as follows:
Qx = aPx + cPy + Tx
Qy = bPx + dPy + Ty
5. Affine Properties 4
Preserves parallelism of lines, but not lengths and
angles.
Lines are preserved.
Proportional distances are preserved (Midpoints map
to midpoints).
9. Projective Properties 4
With projective geometry, two lines always meet in a
single point, and two points always lie on a single line.
Mapping from points in plane to points in plane
3 aligned points are mapped to 3 aligned points
Cross Ratio