 
 
 
15.6.7  Conic reduction
The reduced_conic
command finds the reduced equation of a conic.
- 
reduced_conic takes two arguments:
- 
eq, the equation of a conic.
- vars, a list of the variable names.
 
- reduced_conic(eq,vars)
returns a list whose elements are:
- 
the origin of the conic,
- the matrix of a basis in which the conic is reduced,
- 0 or 1 (0 if the conic is degenerate),
- the reduced equation of the conic
- a vector of its parametric equations.
 
Example
| reduced_conic(2*x^2+2*x*y+2*y^2+5*x+3,[x,y]) | 
|  | |  | | ⎡ ⎢
 ⎢
 ⎣
 | − |  | , |  | ⎤ ⎥
 ⎥
 ⎦
 | , | | ⎡ ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎣
 |  | ⎤ ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎦
 | 
 | ,1,3 x2+y2− |  | , | ⎡ ⎢
 ⎢
 ⎣
 |  | + | ⎛ ⎜
 ⎜
 ⎝
 |  | + |  | i | √ |  | ⎞ ⎟
 ⎟
 ⎠
 | ⎛ ⎜
 ⎜
 ⎝
 |  |  | √ |  | cost+ |  | i | √ |  | sint | ⎞ ⎟
 ⎟
 ⎠
 | , | 
 |  |  |  |  |  |  |  |  |  |  |  | | t, 0, 2 π , |  | π , 2 x2+2 x y+2 y2+5 x+3, |  | + | | | ⎛ ⎜
 ⎜
 ⎝
 |  | + |  | i | √ |  | ⎞ ⎟
 ⎟
 ⎠
 | ⎛ ⎜
 ⎜
 ⎝
 |  |  | √ |  |  | ⎛ ⎝
 | 1−t2 | ⎞ ⎠
 | + |  | i | √ |  | t | ⎞ ⎟
 ⎟
 ⎠
 | 
 |  |  |  | 1+t2 | 
 | ⎤ ⎥
 ⎥
 ⎦
 | ⎤ ⎥
 ⎥
 ⎥
 ⎦
 | 
 |  |  |  |  |  |  |  |  |  | 
 | 
This means that the conic is not degenerate, its reduced equation is
its origin is −5/3+5i/6, its axes are
parallel to the vectors (−1,1) and (−1,−1), and its parametric equation is
where the suggested parameter values for drawing are
t from 0 to 2π with tstep=2π/60.
Remark.
Note that if the conic is degenerate and is made of
1 or 2 line(s), the lines are not given by their parametric equation
but by the list of two points of the line.
Example
| reduced_conic(x^2-y^2+3*x+y+2) | 
|  | | |  | ⎡ ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎣
 | ⎡ ⎢
 ⎢
 ⎣
 | − |  | , |  | ⎤ ⎥
 ⎥
 ⎦
 | , |  | ,0,x2−y2, | | ⎡ ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎢
 ⎣
 |  | ⎤ ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎦
 | 
 | ⎤ ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎥
 ⎦
 | 
 |  |  |  |  |  |  |  |  |  |  | 
 | 
 
 
