# polar: Polar line of point with respect to a conic In RConics: Computations on Conics

## 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

 1 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.

## References

Richter-Gebert, J<c3><bc>rgen (2011). Perspectives on Projective Geometry - A Guided Tour Through Real and Complex Geometry, Springer, Berlin, ISBN: 978-3-642-17285-4

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42  # 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") 

RConics documentation built on May 30, 2017, 5:22 a.m.