# Asymptotes of a conic

### 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 |

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

1 2 3 4 5 6 7 8 9 10 | ```
# 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)
``` |