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 :

$$\left[\vphantom{ \begin{array}{ccc} \displaystyle \frac{\sqrt 6}{... ...frac{2\sqrt{5}}{5} & \displaystyle \frac{\sqrt{30}}{30}\\ \end{array}}\right.$$$$\begin{array}{ccc} \displaystyle \frac{\sqrt 6}{3} & \displaystyl... ...ystyle \frac{2\sqrt{5}}{5} & \displaystyle \frac{\sqrt{30}}{30}\\ \end{array}$$$$\left.\vphantom{ \begin{array}{ccc} \displaystyle \frac{\sqrt 6}{... ...frac{2\sqrt{5}}{5} & \displaystyle \frac{\sqrt{30}}{30}\\ \end{array}}\right]$$

Its parametric equation is :

$$\left\{\vphantom{ \begin{array}{l} x =\displaystyle \frac{\sqrt 6... ...frac{\sqrt{30}\sqrt\frac{602}{81}\cos(u)}{30}-\frac{29}{54} \end{array}}\right.$$$$\begin{array}{l} x =\displaystyle \frac{\sqrt 6\sqrt\frac{602}{24... ...v)}{5}+\frac{\sqrt{30}\sqrt\frac{602}{81}\cos(u)}{30}-\frac{29}{54} \end{array}$$

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])]]