Homogeneous coordinates

Why?

• Used in projective geometry
• Simpler formulas
• Points at infinity can be represented with finite coordinate
• Affine transformation can be represented with a single matrix

Definition

The representation x of a geometric object is homogeneous if x and $\lambda x$ represent the same object for $ \lambda x \neq 0 $, i.e. homogeneous coordinates are equivalent up to scale.

Transformations

Projective transformation: $x' = M x$

• Translation: 3 params

$$M = \lambda \begin{bmatrix} I & t \\ 0^T & 1 \end{bmatrix}$$

• Rotationn: 3 params

$$M = \lambda \begin{bmatrix} R & 0 \\ 0^T & 1 \end{bmatrix}$$

• Figid body transformation: 6 params

$$M = \lambda \begin{bmatrix} R & t \\ 0^T & 1 \end{bmatrix}$$

• Similarity transformation: 7 params 3 trans + 3 rot + 1 scale

$$M = \lambda \begin{bmatrix} mR & t \\ 0^T & 1 \end{bmatrix}$$

• Affine transoformation: 12 params 3 trans + 3 rot + 3 scale + 3 sheer

$$M = \lambda \begin{bmatrix} A & t \\ 0^T & 1 \end{bmatrix}$$

Properties

• Invert a transformation

$$x' = M x$$ $$x = M^{-1} x'$$

• Chaining transoformations via matrix products (not commutative)

$$M_1 M_2 x \neq M_2 M_1 x$$

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