A central conic is a nonempty, nondegenerate real conic section having a finite center of symmetry: a half-turn about this point maps the curve onto itself. The central conics are the ellipse, including the circle as a special case, and the hyperbola (Coxeter 1969, p. 117). A nondegenerate parabola has no finite center.
The following table compares standard forms centered at the origin. Here 
;
for the noncircular ellipse, 
; and for the hyperbola,
and 
. The last column gives the eccentricity 
. Each drawing is one representative of
its class. In the drawings, the curve is blue. Orange segments join the marked point 
to the center 
or the foci 
and 
, where present. The green lines
in the hyperbola drawing are its asymptotes.
In Cartesian coordinates, write the equation of such a conic section as
 |
(1)
|
Its center 
, when finite, satisfies
 |  |  |
(2)
|
 |  |  |
(3)
|
These conditions remove the linear terms under the translation 
, 
, so the translated equation
is unchanged by 
.
When 
,
the unique solution is 
and 
.
Thus the conic section is central iff 
. For 
it is an ellipse, while
for 
it is a hyperbola.
For an ellipse, its center is the midpoint of its two foci. For a hyperbola, it is both the midpoint of the foci and the intersection of the asymptotes.
