SLIDE 2D Linear Algebra Notes

1. Vector Spaces and Affine Spaces
2. Transformations in 2D

   1. Translation
   2. Rotation
   3. Scale
   4. Viewport

3. Matrix Properties

Vector Spaces and Affine Spaces

A vector space is a set of vectors which have a vector addition (u+v) and a scalar multiplication (cv). The vector spaces we will use most commonly in graphics are R² and R³, the 2D and 3D real coordinate systems.

For our purposes, an affine space a vector space offset from the origin.

We can represent points and vectors in an affine space by adding an extra coordinate. For points, this coordinate will be 1, for vectors it will be 0. So, in 2D the point, p at (1,1) will be written p=(1,1,1), but the vector v at (1,1) will be written v=(1,1,0).

You can visualize the 2D affine space as the w=1 plane in a 3-dimensional (x,y,w) space.

You can subtract points to get vectors (p₁-p₂=u), and you can add vectors to points to get other points (p₁+u=p₂), but you cannot add two points together in an affine space. (p₁+p₂ is outside the affine plane),

Transformations in 2D

Translation

x' = x + tx y' = y + ty	Translation can be written as a shear in an affine space.

Rotation

In the figure on the right, the point p is rotated by an angle a to the point p'. You could think of the same transformation as rotating the x-axis to x' and the y-axis to y'. Then the point p is in the correct rotated place with respect to x' and y'.

Scale

Uniform scale: p'=sp	Non-uniform scale: x' = sxx y' = syy

View Port

M_VP,W = T(u_min, v_min) * S( (u_max - u_min)/(x_max - x_min), (v_max - v_min)/(y_max - y_min) ) * T(-x_min, -y_min)

Matrix Properties

Transpose

(QR)^T = R^TQ^T

Inverse

Q Q^-1 = I
(QR)^-1 = R^-1Q^-1

Symmetric

Q = Q^T

Orthogonal

(QQ^T) = I
Q^T = Q^-1
All of the columns of an orthogonal matrix are orthonormal. This means that

• Each column is a vector with length 1.
• Any two different columns are orthogonal (perpendicular) to each other.

In other words, for Q = q_{i j} with rows R_i = q_{i *} and columns C_j = q_{* j}, then for all i and j

• i == j: C_i^TC_j = 1
• i != j: C_i^TC_j = 0