the ellipse E : x^{2}/a^{2} + y^{2}/b^{2} = 1, a > b > 0
Observe that, as well as the symmetries e, h
as for the parabola P, the ellipse E also has the
symmetries v, reflection in the yaxis, and r, the
halfturn about the origin. Again, these are clear
from the algebra, since if (x,y)εE, then we have
x^{2}/a^{2} + y^{2}/b^{2} = 1, so that we also have
(x,y), (x,y)
and (x,y) on E.
Again, we will assume that there are no other
symmetries, so that the symmetry group of E
is {e,h,v,r}.
These symmetries are very useful in sketching.
Once we know the shape of the part of E which
lies in the first quadrant  {(x,y) : x, y ≥ 0}, we
can infer the rest of the curve by reflecting, first
in the yaxis  to get the top half, and then in the
xaxis to get the rest.
Note that the points (a,0),(a,0),(0,b),(0,b) are on E.
Suppose (x,y)εE, so that 
x^{2}/a^{2} + y^{2}/b^{2} = 1 
i.e. 
x^{2}/a^{2} = 1  y^{2}/b^{2} 
then 
x^{2}/a^{2} ≤ 1, as y^{2}/b^{2} ≥ 0 
i.e. 
a ≤ x ≤ a 
Thus, E lies between the lines x = a and x = a.
Similarly, E lies between the lines y = b and y =b.
To get the shape of the portion of the curve in the
first quadrant, we could plot some points. The reader
familiar with differentiation will find a more formal
treatment for the ellipse and hyperbola here
Finally, we get the picture on the right. The ellipse E
is shown in blue. The green lines are the axes of
symmetry. The red lines form the "box" within which
E lies. They are tangents to E.
some further notation
The centre, O, of the halfturn is known as the
centre of the ellipse.
Each of the axes of symmetry cuts E in two points,
but the chord on the axis has length 2a, so is longer
than that on the yaxis, which has length 2b. Hence,
the xaxis is called the major axis of the ellipse, and
the yaxis is called the minor axis of the ellipse.

