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2.49.8  Quadric reduction : quadrique_reduite

quadrique_reduite takes two arguments : the equation of a quadric and a vector of variable names.
quadrique_reduite returns a list whose elements are:

Warning ! u,v will be used as parameters of the parametric equations : these variables should not be assigned (purge them before calling quadrique_reduite).
Input :

quadrique_reduite(7*x^2+4*y^2+4*z^2+ 4*x*y-4*x*z-2*y*z-4*x+5*y+4*z-18)

Output is a list containing :

Hence the quadric is an ellipsoid and its reduced equation is :

9*x2+3*y2+3*z2+(−602)/27 

after the change of origin [11/27,(−26)/27,(−29)/54], the matrix of basis change P is :













 
6
3
5
5
30
15
 
6
6
0
 
30
6
 −
6
6
 
2
5
5
 
30
30












Its parametric equation is :



















x = 
6
602
243
sin(u)cos(v)
3
+
5
602
81
sin(u)sin(v)
5
30
602
81
cos(u)
15
+
11
27
y = 
6
602
243
sin(u)cos(v)
6
+
30
602
81
cos(u))
6
26
27
z = 
6
602
243
*sin(u)cos(v)
6
+
2
5
602
81
sin(u)sin(v)
5
+
30
602
81
cos(u)
30
29
54

Remark :
Note that if the quadric is degenerated and made of 1 or 2 plan(s), each plan is not given by its parametric equation but by the list of a point of the plan and of a normal vector to the plan.
Input :

quadrique_reduite(x^2-y^2+3*x+y+2)

Output :

[[(-3)/2,1/2,0],[[1,0,0],[0,1,0],[0,0,-1]],0,x^2-y^2, [hyperplan([1,1,0],[(-3)/2,1/2,0]), hyperplan([1,-1,0],[(-3)/2,1/2,0])]]

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