15.6.9  Quadric reduction

The reduced_quadric command finds the reduced equation of a quadric.

• reduced_quadric takes two arguments:
  • eq, the equation of a quadric.
  • vars, a vector of variable names.
• reduced_quadric(eq,vars) 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 degenerate),
  • the reduced equation of the quadric
  • a vector with its parametric equations.
• Note that u,v will be used as parameters of the parametric equations; these variables should not be assigned (purge them before calling reduced_quadric).

Example

The 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:

  ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣        √ 6 3 √ 5 5 − √ 30 15 √ 6 6 0 √ 30 6 − √ 6 6 2 5 √ 5 √ 30 30 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

  .

• 1, hence the quadric is not degenerated.
• the reduced equation of the quadric:

  9 x2+3 y2+3 z2−

  602 27

  .

• The parametric equations (in the original frame):

  ⎡ ⎢ ⎢ ⎢ ⎣ ⎡ ⎢ ⎢ ⎣ 9  √ 6   √ 1806  sinu cosv 3· 243 + 9  √ 5   √ 602  sinu sinv 5· 81 − 9  √ 30   √ 602  cosu 15· 81 + 11 27 ,                        9  √ 6   √ 1806  sinu cosv 6· 243 + 9  √ 30   √ 602  cosu 6· 81 − 26 27 ,                       − 9  √ 6   √ 1806  sinu cosv 6· 243 + 9· 2  √ 5   √ 602  sinu sinv 5· 81 + 9  √ 30   √ 602  cosu 30· 81 − 29 54 ⎤ ⎥ ⎥ ⎦ ,                          u=0… π ,v=0… 2 π ,ustep= π 20 ,vstep= 2 20  π ⎤ ⎥ ⎥ ⎥ ⎦ .

Hence the quadric is an ellipsoid 9x²+3y²+3z²+(−602)/27=0. After the change of origin to [11/27,(−26)/27,(−29)/54], the matrix of basis change is:

Its parametric equation is:

Remark.

Note that if the quadric is degenerate and made of one or two plane(s), each plane is not given by its parametric equation but by the list of a point in the plane and a normal vector.

Example