next up previous contents index
suivant: Multivariate calculus monter: Quadratic forms précédent: Graph of a quadric   Table des matières   Index


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 :

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

$\displaystyle \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.$$\displaystyle \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])]]


next up previous contents index
suivant: Multivariate calculus monter: Quadratic forms précédent: Graph of a quadric   Table des matières   Index
giac documentation written by Renée De Graeve and Bernard Parisse