Projective geometry （投影几何）（一） 之 2D project plane (2D投影平面)

Row and column vector 通常情况向量$\mathbf{x}$代表着列向量， 行向量使用${\mathbf{x}}^{\mathbf{T}}$。

一个读者的感悟。
所有的知识点都是来源于书Multiple View of Geometry in Computer Vision 。读书的过程中觉得数学是一个很奇妙的东西。高中学习的知识在projective geometry中莫名被串了起来，形成了新的知识和新的系统。每一个定义和证明都非常巧妙有趣。受益匪浅。
PS 写博客过程中被导师提醒用英语写作可以锻炼research writing 的能力，所以博客后半段用英语写了。难以理解的地方尽量用中文注解。

Points and lines

在投影机和中点和线如何表示。通常情况下一条直线的表示方式为$ax+by+c=0$， 于是一条直线的向量我们可以表示为 $(a,b,c{)}^T$。然而这个表示并不是唯一的， 因为$(ka)x+(kb)y+(kc)=0$对于任意$k$, 表示的都是同一条直线， 所以向量可以表示为 $k(a,b,c{)}^T$。于是这里我们引用 Homogeneous representation of points.

Homogeneous representation of points

点$\mathbf{x}=(x,y{)}^T$ 在直线$\mathbf{l}=(a,b,c{)}^T$上的条件为$ax+by+c=0$. 可以表示为$(x,y,1)(a,b,c{)}^T=(x,y,1)\mathbf{l}=(kx,ky,k)\mathbf{l}=0$ 这里的$(kx,ky,k)$就是Homogeneous representation，所以投影空间的点可以表示为$(x_1,x_2,x_3)$，与之对应的二维空间的点应当表示为$(x_1/x_3, x_2/x_3).

Result2.1. The point x lies on the line l if and only if $x^{T}l=0$.
Note that $x^{T}l=l^{T}x=x.l$.

两条直线的交点 （intersection of two lines）

两条直线 $l=(a,b,c{)}^T$ and $l^{\prime }=(a^{\prime },b^{\prime },c^{\prime }{)}^T$. $x=l\times l^{\prime }$（叉乘）。因为$l.(l\times l^{\prime })=l^{\prime }.(l\times l^{\prime })$， 所以$l^{T}x=l^{\prime T}x=0$。因此，如果点$x$属于直线$l$与$l^{\prime }$， $x$就是$l$与$l^{\prime }$的焦点。

Result2.2. The intersection of two lines l and l’ is the point $x=l\times l^{\prime }$

例如：直线x = 1 可以被写成-1x + 1 = 0, $l=(-1,0,1{)}^T$同理直线y = 1可以被写作$l^{\prime }=(0,-1,1{)}^T$.
$$x=l\times l^{\prime }=\left|\begin{array}{ccc}i& j& k\\ -1& 0& 1\\ 0& -1& 1\end{array}\right|=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$$

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

Ideal points and the line at infinity

理想点和无穷远处的线的定义。

平行直线的交点

两条直线 $l=(a,b,c{)}^T$和 $l^{\prime }=(a,b,c^{\prime }{)}^T$。两条直线互相平行。交点 $l\times l^{\prime }=(c^{\prime }-c)(b,-a,0{)}^T$，由于
$(c^{\prime }-c)$是常数，所以交点为$(b,-a,0{)}^T$。当我们想要将这个点表示为2D的点的时候，我们发现这个点在无穷远处。所以在投影几何中，我们认为平行的直线相较于无穷远点。

例如：直线x = 1 可以被写成-1x + 1 = 0, $l=(-1,0,1{)}^T$同理直线x = 2可以被写作$l^{\prime }=(-1,0,2{)}^T$.
$$x=l\times l^{\prime }=\left|\begin{array}{ccc}i& j& k\\ -1& 0& 1\\ -1& 0& 2\end{array}\right|=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$$

Ideal points and the line at infinity

For the homogeneous points, 如果向量 $x=(x_1,x_2,x_3)$中， $x_3=0$， 那么这个点被称为ideal points or points at infinity。这一系列ideal points 是位于一条直线上的 叫做line at infinity $l_{\infty }=(0,0,1)$，因为$(0,0,1)(x_1,x_2,0{)}^T=0$.
Note that $l=(a,b,c)$ and $l^{\prime }=(a,b,c^{\prime })$与 $l_{\infty }$相交于同一点$(b,-a,0{)}^T$。在普通几何学中，$(b,-a)$是一个与直线相切的向量，所以与直线互相垂直的向量可以表示为$(a,b)$，可以表示直线的方向。Consequently, the direction of the lines varies as the coordinate of the ideal point $(b,-a,0{)}^T$ so the line at infinity can be regarded as a set of direction of the lines in the plane.

Model for the projective plane

Duality

The results above have duality. The equation $l^{T}x=0$ for the line and point is symmetric since $l^{T}x=0$ implies $x^{T}l=0$. Similarly, the results 2.2 and 2.4 have duality as well. So we can get the result of Duality principle.

Result 2.6. Duality principle To any theroem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theorem.

Conics and dual conics

Conic is a curve (这里的conic是指圆锥曲线) described by second-degree equation. Conics: hyperbola, ellipse and parabola (non-degenerate conics).

Note that:

The equation of conic can be written as:

$$a{x}^2+bxy+c{y}^2+dx+ey+f=0$$

We can rewrite it with homogeous coordinate by replacement: $x=x_1/x_3$, $y=x_2/x_3$, so that the equation is:
$$a{x}_1^2+b{x}_1{x}_2+c{x}_2^2+d{x}_1{x}_3+e{x}_2{x}_3+f{x}_3^2=0$$
or formula $x^{T}Cx=0$, where the conic coeffieient matrix $C$ is given by:
$$C=\left(\begin{array}{ccc}a& b/2& d/2\\ b/2& c& e/2\\ d/2& e/2& f\end{array}\right)$$

It is symmetric and has 5 degree of freedom (五个未知数).

Thus, 5 points can define a conic.

Tangent lines to conics

Result 2.7. The line $l$ tangent to $C$ at point $x$ on $C$ is given by $l=Cx$

Proof: （需添加proof）

Dual conics

The description in Reuslt 2.7 is defined by points. We can change it to lines which we denote as $C^{\ast }$. A line tangent to conic $C$ satisfied $l^T{C}^{\ast }l=0$. $C^{\ast }$ is the adjoint matrix of $C$. For non-singular symmetric matrix $C^{\ast }=C^{-1}$.

What’s more, from Reuslt 2.7, at the point x, the tangent line is $l=Cx$, which indicates that the points x at the line $l$ which intersects the C as $x=C^{-1}l$. Since $x^{T}Cx=0$, we obtain $(C^{-1}l{)}^{T}C(C^{-1}l)=l^T{C}^{\ast }l=0$ (because C is symmetric matrix, we can get $C^T=C^{-1}$).

Similarly, 5 lines can define a dual conic.

Degenerate conics

When the matrix C does not have full rank, then the conic is degenerate. Degenerate point conics include two lines (rank 2), and a repeated line (rank 1).

(需添加example)