Conic reduction

### 14.6.7 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])

| ⎡ ⎢ ⎢ ⎢ ⎣
⎡ ⎢ ⎢ ⎣ | − |
5
3
| , |
5
6
| ⎤ ⎥ ⎥ ⎦ | , |
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
√ |
2
2
− |
√ |
2
2
√ |
2
2
√ |
2
2
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
| ,1,3 x2+y2− |
7
6
| ,
⎡ ⎢ ⎢ ⎣ |
−10+5i
6
| + | ⎛ ⎜ ⎜ ⎝ |
√ |
2
2
| + |
1
2
| i | √ |
2
| ⎞ ⎟ ⎟ ⎠ | ⎛ ⎜ ⎜ ⎝ |
3
18
|  | √ |
14
| cost+ |
1
6
| i | √ |
42
| sint | ⎞ ⎟ ⎟ ⎠ | ,
t, 0, 2 π , |
2
60
| π , 2 x2+2 x y+2 y2+5 x+3,
−10+5 i
6
| + |
⎛ ⎜ ⎜ ⎝ |
√ |
2
2
| + |
1
2
| i | √ |
2
| ⎞ ⎟ ⎟ ⎠ | ⎛ ⎜ ⎜ ⎝ |
3
18
|  | √ |
14
|  | ⎛ ⎝ | 1−t2 | ⎞ ⎠ | + |
2
6
| i | √ |
42
| t | ⎞ ⎟ ⎟ ⎠
1+t2
| ⎤ ⎥ ⎥ ⎦ | ⎤ ⎥ ⎥ ⎥ ⎦

This means that the conic is not degenerate, its reduced equation is

3x2+y2− |
7
6
| =0

its origin is −5/3+5i/6, its axes are parallel to the vectors (−1,1) and (−1,−1), and its parametric equation is

−10+5i
6
| + |
(1+i)
√ |
2
| · |
⎛ ⎜ ⎝ | √ |
14
| cost+i | √ |
42
| sint | ⎞ ⎟ ⎠
6

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)

| ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ | ⎡ ⎢ ⎢ ⎣ | − |
3
2
| , |
1
2
| ⎤ ⎥ ⎥ ⎦ | , |
⎡ ⎢ ⎣ |
1 | 0
0 | 1
| ⎤ ⎥ ⎦
| ,0,x2−y2, |
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ |
−3+i
2
−1+3 i
2
−3+i
2
−1−i
2
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
| ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦