points-and-lines-at-infinity

Reference

• Stanford CS231A

무한점, 무한선 현실에서는 평행한 두 선은 영원히 만나지 않지만, 기하학적으로는 무한대에 있는 한 점에서 만난다고 정의할 수 있다. 이 개념은 2D 이미지 한 장으로 잃어버린 3D 정보를 복구할 수 있는 거의 유일한 단서이다.

homogeneous vector $l=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$, slope of the line $-\frac{a}{b}$ and y-intercept $-\frac{c}{b}$ $p=\left[\begin{array}{c}x\\ y\end{array}\right]\left[\begin{array}{ccc}a& b& c\end{array}\right]\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=ax+by+c=0$

Quote

In general, two lines $l$ and $l^{\prime }$ will intersect at a point $x$. This point is defined as the cross product between $l$ and $l^{\prime }$.

Proof

The intersection point $x$ should lie on both $l$ and $l^{\prime }$. Therefore, $x=\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]l=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]l^{\prime }=\left[\begin{array}{c}{a}^{\prime }\\ b^{\prime }\\ c^{\prime }\end{array}\right]$ $x^{T}l=\left[\begin{array}{ccc}x& y& 1\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=ax+by+c=0x^T{l}^{\prime }=\left[\begin{array}{ccc}x& y& 1\end{array}\right]\left[\begin{array}{c}{a}^{\prime }\\ b^{\prime }\\ c^{\prime }\end{array}\right]=a^{\prime }x+b^{\prime }y+c^{\prime }=0$ Dot product (내적) 이 0이라는 것은 두 vector 가 orthogonal (직교) 한다는 것이다. $x$ 는 $l$ 과 $l^{\prime }$ 과 동시에 수직이어야 한다. 그러므로, $x=l\times l^{\prime }$

Example

$x=1y=1\Rightarrow -x+1=0-y+1=0$ $x=l\times l^{\prime }=\left|\begin{array}{ccc}i& j& k\\ -1& 0& 1\\ 0& -1& 1\end{array}\right|=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$ $\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)$ 는 Intersection point in homogeneous coordinate $\to \left(\begin{array}{c}1\\ 1\end{array}\right)$ in inhomogeneous coordinate

What about Parallel lines?

$l=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]l^{\prime }=\left[\begin{array}{c}a\\ b\\ c^{\prime }\end{array}\right]$ $x=l\times l^{\prime }=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]\times \left[\begin{array}{c}a\\ b\\ c^{\prime }\end{array}\right]=\left[\begin{array}{c}b{c}^{\prime }-cb\\ ca-a{c}^{\prime }\\ ab-ba\end{array}\right]=(c^{\prime }-c)\left[\begin{array}{c}b\\ -a\\ 0\end{array}\right]$

Quote

Everybody knowledge expects these lines to never intersect. However, this definition could be rewritten to say that these lines intersect at infinity.

Points at infinity

동차좌표계 (homogeneous-coordinate) 에서는 무한에 위치하는 점을 표현하고 싶을 때, 마지막 차원을 0으로 한다. 예를 들어, $\text{homogeneous coordinate}\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\to \text{inhomogeneous coordinate}(1,1)$ $\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]\to \left(\frac{1}{0},\frac{1}{0}\right)\to (\infty ,\infty )$ 위의 계산에 따르면, $l\times l^{\prime }\propto \left[\begin{array}{c}b\\ -a\\ 0\end{array}\right]=x_{\infty }$ . 그리고 이 점을 The point at infinity 혹은 ideal point라고 부른다.

Quote

One interesting property of a point at infinity is that all parallel lines with the same slope $-\frac{a}{b}$ pass through the ideal point as shown below: $l^T{x}_{\infty }=\left[\begin{array}{ccc}a& b& c\end{array}\right]\left[\begin{array}{c}b\\ -a\\ 0\end{array}\right]=ab-ba+0=0$

Example

two lines $x=1,x=2\Rightarrow -x+1=0,-x+2=0$ $l=\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right)l^{\prime }=\left(\begin{array}{c}-1\\ 0\\ 2\end{array}\right)$ $x=l\times l^{\prime }=\left|\begin{array}{ccc}i& j& k\\ -1& 0& 1\\ -1& 0& 2\end{array}\right|=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$

Lines at infinity

The concept of points of infinity can be extended to define lines at infinity.

Quote

Consider two or more pairs of parallel lines. Each pair of parallel lines intersect at a point at infinity. The line $l_{\infty }$ that passes through all these points at infinity must satisfy $l_{\infty }^T{x}_{\infty }=0$. This means that $l_{\infty }=\left[\begin{array}{c}0\\ 0\\ c\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$. Since $c$ is an arbitrary value (just simplified version).

Projective transformation $H$

Apply projective transformation $H$ to a point at infinity $p_{\infty }$ $p^{\prime }=H{p}_{\infty }=\left[\begin{array}{cc}A& t\\ v& b\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}{p}_x^{\prime }\\ p_y^{\prime }\\ p_z^{\prime }\end{array}\right]$ The last element of $p^{\prime }$ may become non-zero. A projective transformation generally maps points at infinity to points not infinity.

Let’s apply a projective transformation $H$ to a line $l$ to get a new line $l^{\prime }$.

&x^Tl = 0 \quad x'^Tl' = 0 & \\ &x' = Hx \quad H^{-1}x' = x & \\ \\ &x^TIl = x^TH^TH^{-T}l = 0 & \\ &(H^{-1}x')^TIl = x'^TH^{-T}H^TH^{-T}l = x'^TH^{-T}l = 0 = x'^Tl' & \\ &l' = H^{-T}l & \end{flalign}$$ Similar to points at infinity, the projective transformation of a line at infinity does not map to another line at infinity. #### Affine transformation $$p' = Hp_\infty = \begin{bmatrix} A & t \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} p_x' \\ p_y' \\ 0 \end{bmatrix}$$ [[transformations-in-2d#affine-transformations|Affine transformation]] maps points at infinity to points at infinity. (lines same)