### 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])

| ⎡ ⎢ ⎢ ⎢ ⎣

⎡
⎢
⎢
⎣

−

5
3

,

5
6

⎤
⎥
⎥
⎦

,

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣

√

2

2

−

√

2

2

√

2

2

√

2

2

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

,1,3x2+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 , 2x2+2x y+2y2+5x+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

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦

⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦