Asymptotes of a conic

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Description

Find the slopes of the asymptotic directions of a conic.

Usage

1

Arguments

x

a 6-length vector or a symmetric 3x3 matrix

Details

The conicAsymptotes function calculates the slopes of the asymptotic directions of a conic specified by its coefficients or by its symmetric matrix.

If the equation of the conic is

1
    v_1 x_1^2 + v_2 x_1x_2 + v_3 x_2^2 + v_4 x_1 + v_5 x_2 + v_6 = 0

the slopes of the asymptotes are the roots of the equation at infinity of the conic:

1
    v_1 + v_2 t + v_3 t^2 = 0

where t=x_2/x_1.

Value

A vector containing the slopes: two values in the case of a hyperbola or of intersecting lines, one value in the case of a parabola or of parallel lines. In the case of an ellipse (which has no points at infinity), the function returns an empty vector.

Author(s)

Bernard Desgraupes
bernard.desgraupes@u-paris10.fr
University of Paris Ouest - Nanterre
Lab Modal'X (EA 3454)

See Also

conicAxes, conicCenter, conicMatrix, conicPlot

Examples

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# Hyperbola
# Equation: 2*x_1^2 + 2*x_1*x_2 - 2*x_2^2 - 20*x_1 + 20*x_2 + 10 = 0
v <- c(2,2,-2,-20,20,10)
conicAsymptotes(v)

# Ellipse
# Equation: 2*x_1^2 + 2*x_1*x_2 + 2*x_2^2 - 20*x_1 - 28*x_2 + 10 = 0
v <- c(2,2,2,-20,-28,10)
# Should return an empty vector (an ellipse has no asymptotes!):
conicAsymptotes(v)