### 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:

- the origin,
- the matrix of a basis where the quadric is reduced,
- 0 or 1 (0 if the quadric is degenerated),
- the reduced equation of the quadric
- a vector with its parametric equations.

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 :

- The origin (center of symmetry) of the quadric
  `[11/27,(-26)/27,(-29)/54],`
- The matrix of the basis change:
  `[[(sqrt(6))/3,(sqrt(5))/5,(-(sqrt(30)))/15], [(sqrt(6))/6,0,(sqrt(30))/6], [(-(sqrt(6)))/6,(2*sqrt(5))/5,(sqrt(30))/30]],`
- 1 hence the quadric is not degenerated
- the reduced equation of the quadric :
  `0,9*x```^```2+3*y```^```2+3*z```^```2+(-602)/27,`
- The parametric equations (in the original frame) are :
  `[[(sqrt(6)*sqrt(602/243)*sin(u)*cos(v))/3+ (sqrt(5)*sqrt(602/81)*sin(u)*sin(v))/5+ ((-(sqrt(30)))*sqrt(602/81)*cos(u))/15+11/27, (sqrt(6)*sqrt(602/243)*sin(u)*cos(v))/6+ (sqrt(30)*sqrt(602/81)*cos(u))/6+(-26)/27, ((-(sqrt(6)))*sqrt(602/243)*sin(u)*cos(v))/6+ (2*sqrt(5)*sqrt(602/81)*sin(u)*sin(v))/5+ (sqrt(30)*sqrt(602/81)*cos(u))/30+(-29)/54], u=(0 .. pi),v=(0.. (2*pi)),ustep=(pi/20), vstep=((2*pi)/20)]]`

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])]]`