### 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)]]]