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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:

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)]]]

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