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 :
^
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 :
^
2+3*y^
2+3*z^
2+(-602)/27,Hence the quadric is an ellipsoid and its reduced equation is :
9*x2+3*y2+3*z2+(−602)/27 = 0 |
after the change of origin [11/27,(−26)/27,(−29)/54], the matrix of basis change P is :
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ |
Its parametric equation is :
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ |
|
Remark :
Note that if the quadric is degenerate and made of 1 or 2 plane(s),
each plane is not given by
its parametric equation but by the list of a point of the plane
and of a normal vector to the plane.
Input :
^
2-y^
2+3*x+y+2)Output :
^
2-y^
2, [hyperplan([1,1,0],[(-3)/2,1/2,0]), hyperplan([1,-1,0],[(-3)/2,1/2,0])]]