Note:
RConics# install "devtools" if not already done
if(!require("devtools")) install.packages("devtools")
devtools::install_github("emanuelhuber/RConics")
library(RConics) # load RGPR in the current R session
Many of the functions of the R-package RConics are based on projective geometry. In projective geometry parallel lines meet at an infinite point and all infinite points are incident to a line at infinity. Points and lines of a projective plane are represented by homogeneous coordinates, that means by 3D vectors:
point representation: $(x, y, z)$ - line representation: $(a, b, c)$ such that $ax + by + c = 0$ In projective geometry:
Euclidean points correspond to $(x, y, 1)$ - infinite points correspond to $(x, y, 0)$ - Euclidean lines correspond to $(a, b, c)$ with $a\neq 0$ or $b\neq 0$ - the line at infinity correspond to $(0, 0, 1)$.
The vector $(x, y, z)$ and $(\lambda x,\lambda y,\lambda z)$ define the same geometric point!
In projective geometry, points and lines have the same algebraic construction. A point can be viewed as a line and a line can be viewed as a point:
\[\text{points}\leftrightarrow\text{line}\]The interested reader is referred to the excellent book of Jürgen Richter-Gebert on projective geometry (Richter-Gebert, 2011). We recommend to use the R-package conics from Bernard Desgraupes to plot conics.
# two points in homogeneous coordinates
p1 <- c(3, 1, 1)
p2 <- c(0, 2, 1)
plot(rbind(p1, p2), xlab = "", ylab = "")
grid()
A point $(x, y, z)$ is contained in the line $(a, b, c)$ if $a\cdot x + b\cdot y + c\cdot z = 0$. That means that the vectors $(x, y, z)$ and $(a, b, c)$ are orthogonal.
Now assume that points $p = (p_1, p_2, p_3)$ and $q = (q_1, q_2, q_3)$ are on the same line $l = (l_1, l_2, l_3)$. Then, $l$ is orthogonal to $p$ and $q$ and is given by
\[p\times q =\begin{pmatrix} p_1\\ p_2\\ p_3\end{pmatrix}\times\begin{pmatrix} q_1\\ q_2\\ q_3\end{pmatrix} =\begin{pmatrix} +p_2 q_3 - p_3 q_2\\ -p_1 q_3 + p_3 q_1\\ +p_1 q_2 - p_2 q_1\end{pmatrix}\]It can be easily shown that this vector (i.e., the cross-product $p\times q$) is orthogonal to $p$ and $q$. For example, we obtain for $p$ :
\[p_1\cdot\left( p_2 q_3 - p_3 q_2\right) + p_2\cdot\left( -p_1 q_3 + p_3 q_1\right) + p_2\cdot\left( p_1 q_2 - p_2 q_1\right) = 0\]The line joining two points, $p$ and $q$ , is defined by the join operation:
\[\mathbf{meet}\left(p, q\right):= p\times q\]In RConics this is done with the join() function:
# homogeneous line joining p1 and p2
l_12 <- join(p1,p2)
l_12
plot(rbind(p1, p2), xlab = "", ylab = "")
addLine(l_12)
grid()
Exactly in the same way, the intersection between two lines, $l = (l_1, l_2, l_3)$ and $m = (m_1, m_2, m_3)$, is the point that is orthogonal to both $l$ and $m$. The intersection point is defined by the vector resulting from the cross-product of $l$ and $m$ . This operation is called meet and defined as follows:
\[\mathbf{meet}\left(l, m\right):= l\times m\]