RConics (version 1.0)

polar: Polar line of point with respect to a conic

Description

Return the polar line \(l\) of a point \(p\) with respect to a conic with matrix representation \(C\). The polar line \(l\) is defined by \(l = Cp\).

Usage

polar(p, C)

Arguments

p

a \((3 \times 1)\) vector of the homogeneous coordinates of a point.

C

a \((3 \times 3)\) matrix representation of the conic.

Value

A \((3 \times 1)\) vector of the homogeneous representation of the polar line.

Details

The polar line of a point \(p\) on a conic is tangent to the conic on \(p\).

References

Richter-Gebert, J<U+00FC>rgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Real and Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4

Examples

# NOT RUN {
  # Ellipse with semi-axes a=5, b=2, centered in (1,-2), with orientation angle = pi/5
  C <- ellipseToConicMatrix(c(5,2),c(1,-2),pi/5)
  
  # line
  l <- c(0.25,0.85,-1)
  
  # intersection conic C with line l:
  p_Cl <- intersectConicLine(C,l)
  
  # if p is on the conic, the polar line is tangent to the conic
  l_p <- polar(p_Cl[,1],C)
  
  # point outside the conic
  p1 <- c(5,-3,1)
  l_p1 <- polar(p1,C)
  
  # point inside the conic
  p2 <- c(-1,-4,1)
  l_p2 <- polar(p2,C)
  
  # plot
  plot(ellipse(c(5,2),c(1,-2),pi/5),type="l",asp=1, ylim=c(-10,2))
  # addLine(l,col="red")
  points(t(p_Cl[,1]), pch=20,col="red")
  addLine(l_p,col="red")
  points(t(p1), pch=20,col="blue")
  addLine(l_p1,col="blue")
  points(t(p2), pch=20,col="green")
  addLine(l_p2,col="green")
  
  # DUAL CONICS
  saxes <- c(5,2)
  theta <- pi/7
  E <- ellipse(saxes,theta=theta, n=50)
  C <-  ellipseToConicMatrix(saxes,c(0,0),theta)
  plot(E,type="n",xlab="x", ylab="y", asp=1)
  points(E,pch=20)
  E <- rbind(t(E),rep(1,nrow(E)))
  
  All_tangant <- polar(E,C)
  apply(All_tangant, 2, addLine, col="blue")
  
# }