Affine Transformations
======================
I am somewhat reorganizing the presentation of this material as compared to
what the slides do.


I. Motivate transformations
---------------------------
How do we represent an object?
  - a bunch of points
  - polygons with vertices

We'll work in 2D for a while...
Draw a square.
How do we position this "canonical square" on the plane?

We want:
  Rotation
  Translation

Plus maybe:
  Scaling
  Other distortions?

II. Basic Math - Linear xforms
------------------------------
How do we express a rotation
  - first, consider a point at 'old' = (1,0) on the unit circle...
     x = cos(theta)
     y = sin(theta)
  - then, consider a point at 'old' = (0,1) on the unit circle...
     x = -sin(theta)
     y = cos(theta)
  - for arbitrary point:
     xnew = cos(theta) * xold - sin(theta) * yold
     ynew = sin(theta) * xold + cos(theta) * yold

Is there a more convenient way to represent this?

  point = [x]
          [y]

  [xnew] =  [cos(theta)  -sin(theta)][xold]
  [ynew]    [sin(theta)  cos(theta)] [yold]

   R = rotation matrix

   pnew = R * pold;

What else can we do with a matrix like this?

Identity:
   [1 0]
   [0 1]

Uniform scaling:
   [a  0]
   [0  a]

Non-uniform scaling:
   [a  0]
   [0  b]

   takes square into rectangle...

Mirroring (negative scaling):
   [-1  0
   [ 0  0]

   etc.

Shear:
   [1  b]
   [0  1]

  x' = x + by
  y  = y 

   show how this affects a square.

III. Basic Math -- Translations
===============================
How do we translate a point?

1D:  x' = x + shift_amount

2D:  x' = x + shift_amount_x
     y' = y + shift_amount_y


Combining all of this (rotation + translation):

  p' = R*p + T, where T = [shift_amount_x]
                          [shift_amount_y]

This is referred to as an *affine transformation*.

This is sort of awkward - have to carry around both R and T.
So use a trick:

define p = [x]   (homogeneous coordinates)
           [y]
           [1]

Now,

 [x'] = [Rxx Ryx Tx][x]       x' = Rxx * x + Ryx * y + Tx;
 [y']   [Rxy Ryy Ty][y]       y' = Rxy * x + Ryy * y + Ty;
 [?]    [0    0   1][1]       1  = 0*x     + 0*y     + 1*1;

Show a translation...

Rotation about arbitrary points
-------------------------------
- Identify rotation point
- Translate rotation point to origin
- Perform rotation
- Translate back

Change of coordinate system
===========================
Can also view rotations and translations as representing a
*change of coordinate systems* rather than as *movement*.

Show two pairs of coordinate axes...

For rotation, project onto new axes as represented in old coordinate system:

x' = [ux uy]xold
     [vx vy]

THIS ONLY WORKS FOR ORTHOGONAL COORDINATE SYSTEMS

For translation, add translation vector to take new axes to old origin.

Change of coordinate systems for non-orthogonal coordordinate systems
=====================================================================

     -->      -->
P = a u  +  b  v

question: what is 'a', what is 'b'?

For orthogonal coordinate systems, dot the equation above by u, and then by
v.  For non-orthogonal coordinate systems, must explicitly solve the
system of equations.

REST OF LECTURE
===============
Take from Curless slides.
* Points and vectors (w=1 vs. w=0)
* Barycentric coordinates -- affine frame -- non-orthogonal axes
* Cross products -- |a||b|sin(theta) = area of parallelogram
                    right hand rule.