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6.54.7  Conic reduction: reduced_conic

The reduced_conic command finds the reduced equation of a conic.


Example.
Input:

reduced_conic(2*x^2+2*x*y+2*y^2+5*x+3,[x,y])

Output:

     
⎡
⎢
⎢
⎢
⎣
⎡
⎢
⎢
⎣		−
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
⎤
⎥
⎥
⎦		⎤
⎥
⎥
⎥
⎦
         

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

 3x2+y2−7/6=0 

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

−10+5*i
6
+ 
(1+i)
√
2
*
(√
14
*cos(t)+i*√
42
*sin(t))
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.
Input:

reduced_conic(x^2-y^2+3*x+y+2)

Output:

⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣		⎡
⎢
⎢
⎣		−
3
2
,
1
2
⎤
⎥
⎥
⎦		,⎡
⎢
⎣
10
01
⎤
⎥
⎦		,0,x2−y2,⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−3+i
2
−1+3 i
2
−3+i
2
−1−i
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦		⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
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