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
| √ | | ⎞
⎟
⎟
⎠ | ⎛
⎜
⎜
⎝ | | | √ | | cos t+ | |
i | √ | | sin t | ⎞
⎟
⎟
⎠ | , |
| | | | | | | | | |
| | | 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, | ⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣ | | ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
| ⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦ |
| | | | | | | | | | |
|