### 2.49.6 Conic reduction : `conique_reduite`

`conique_reduite` takes two arguments : the equation of a conic and a vector of variable names.
`conique_reduite` returns a list whose elements are:

- the origin of the conic,
- the matrix of a basis in which the conic is reduced,
- 0 or 1 (0 if the conic is degenerated),
- the reduced equation of the conic
- a vector of it’s parametric equations.

Input :

`conique_reduite(2*x```^```2+2*x*y+2*y```^```2+5*x+3,[x,y])`

Output :

`[[-5/3,5/6],[[-1/(sqrt(2)),1/(sqrt(2))],[-1/(sqrt(2)), -1/(sqrt(2))]],1,3*x```^```2+y```^```2+-7/6,[[(-10+5*i)/6+ (1/(sqrt(2))+(i)/(sqrt(2)))*((sqrt(14)*cos(‘ t‘))/6+ ((i)*sqrt(42)*sin(‘ t‘))/6),‘ t‘,0,2*pi,(2*pi)/60]]]`

Which means that the conic is not degenerated, it’s reduced equation is

3x2+y2−7/6=0

origin is −5/3+5*i/6, axis are parallel to the vectors (−1,1) and (−1,−1). It’s parametric equation is

−10+5*i
6

+

(1+i)

√

2

*

(

√

14

*cos(t)+i*

√

42

*sin(t))

6

where the suggested parameters value for drawing are t from 0 to 2π with `tstep`= 2π/60.

Remark :
Note that if the conic is degenerated and is made of 1 or 2 line(s), the lines are not given by their parametric equation but by the list of two points of the line.
Input :

`conique_reduite(x```^```2-y```^```2+3*x+y+2)`

Output :

`[[(-3)/2,1/2],[[1,0],[0,1]],0,x```^;```2-y```^```2, [[(-1+2*i)/(1-i),(1+2*i)/(1-i)], [(-1+2*i)/(1-i),(-1)/(1-i)]]]`