第二章 Projective Geometry and Transformations of 2D

2.1 Planar geometry

Entities	point	line	conic
Algebraic	vector	vector	symmetric matrix

2.2 The 2D projective plane

2.2.1 Points and lines

Homogeneous representation of lines
A common line is represented by equation $ax+by+c=0$, thus, a line is represented by the vector $(a,b,c{)}^T$

The set of vectors in ${\mathbb{R}}^3-(0,0,0{)}^T$ forms the projective space ${\mathbb{P}}^2$.

Homogeneous representation of points
An arbitrary homogeneous vector $x=(x_1,x_2,x_3{)}^T$ represents the point $(\frac{x_1}{x_3},\frac{x_2}{x_3}{)}^T$ in ${\mathbb{R}}^2$.

Result 2.1 The point $\mathbf{x}$ lies on the line $\mathbf{l}$ if and only if $x^{T}l=0$.

Degrees of freedom (dof) Both point and line have 2 DoF.

Intersection of lines
Result 2.2 The intersection of two lines $\mathbf{l}$ and ${\mathbf{l}}^{\prime }$ is the point $x=l\times l^{\prime }$.

Line joining points
Result 2.4. The line through two points $\mathbf{x}$ and ${\mathbf{x}}^{\prime }$ is $l=x\times x^{\prime }$.

2.2.2 Ideal points and the line at infinity

Intersection of parallel lines.
Ideal points and the line at infinity.
The set of all ideal points can be written in $(x_1$