conics (version 0.3)

conicAsymptotes: Asymptotes of a conic

Description

Find the slopes of the asymptotic directions of a conic.

Usage

conicAsymptotes(x)

Arguments

x

a 6-length vector or a symmetric 3x3 matrix

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.

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

    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:

    v_1 + v_2 t + v_3 t^2 = 0

where t=x_2/x_1.

See Also

conicAxes, conicCenter, conicMatrix, conicPlot

Examples

# 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)