# Intersections with conics

### Description

Point(s) of intersection between a conic and a line, and between two conics in homogeneous coordinates.

### Usage

 1 2 intersectConicLine(C, l) intersectConicConic(C1,C2) 

### Arguments

 C, C1, C2 (3 \times 3) matrix representation of conics. l a (3 \times 3) vector of the homogeneous representation of a line.

### Value

The homogeneous coordinates of the intersection points. If there are two points of intersection, it returns a (3 \times 2) matrix whose columns correspond to the homogeneous coordinates of the intersection points. If there is only one point, a (3 \times 1) vector of the homogeneous coordinates of the intersection point is returned. If there is no intersection, NULL is returned.

### References

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

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  # Ellipse with semi-axes a=8, b=2, centered in (0,0), with orientation angle = -pi/3 C <- ellipseToConicMatrix(c(8,2),c(0,0),-pi/3) # Ellipse with semi-axes a=5, b=2, centered in (1,-2), with orientation angle = pi/5 C2 <- ellipseToConicMatrix(c(5,2),c(1,-2),pi/5) # line l <- c(0.25,0.85,-3) # intersection conic C with line l: p_Cl <- intersectConicLine(C,l) # intersection conic C with conic C2 p_CC2 <- intersectConicConic(C,C2) # plot plot(ellipse(c(8,2),c(0,0),-pi/3),type="l",asp=1) lines(ellipse(c(5,2),c(1,-2),pi/5), col="blue") addLine(l,col="red") points(t(p_Cl), pch=20,col="red") points(t(p_CC2), pch=20,col="blue") 

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