polar {RConics}

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.

Details

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

Value

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

Source

Richter-Gebert, Jürgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Real and Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4

Examples

# 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")