6.54.9  Quadric reduction: reduced_quadric

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.

Warning ! u,v will be used as parameters of the parametric equations: these variables should not be assigned (purge them before calling reduced_quadric).

Example.
Input:

Output:

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 and its reduced equation is:

after the change of origin [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 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.

Example.
Input:

Output: