An oval is a curve resembling a squashed circle but, unlike the ellipse, without a precise mathematical definition. The word oval derived from the Latin word "ovus" for egg. Ovals can have a single axis of symmetry. A planar egg is an oval with one end more pointed than the other. An ellipse has two ends of the same shape, but this property does not characterize ellipses; nonelliptical Cassini ovals can also have matching ends.
The table compares ovals defined by distances, a circular arc construction, and algebraic egg curves.
| illustration | curve | construction |
| ellipse | constant sum of distances to two foci |
| Cartesian ovals | constant weighted sum of distances to two foci |
| Cassini ovals | constant product of distances to two foci |
| Hügelschäffer egg | two-circle construction giving a cubic curve |
| Moss's egg | four tangent circular arcs |
| Yamamoto egg curve | egg-shaped form of a single Kepler's folium |
For an ellipse of eccentricity 
, Burleigh's
oval is a smooth ovoid obtained as a negative
pedal curve with respect to a focus. The family also
includes self-crossing curves, such as the fish curve
at 
.
The particular variant illustrated above can be constructed with a compass by joining together arcs of different radii
such that the centers of the arcs
lie on a line passing through the join point (Dixon 1991).
Albrecht Dürer used this method to design a Roman letter font. If the separation
between left and right caps is 
and the radii are 
and 
, respectively, with 
and 
, then the center 
and radius 
of the joining circle
are
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is a signed Cartesian
coordinate, rather than a positive distance. The
cap centers are 
and 
, and 
.
This is the four-arc construction used for Type I of Thom's eggs (Thom 1967, pp. 29-30). In the figure, 
, 
, and 
, giving 
.
The two cap centers and the center
of the upper joining circular arc therefore form
a right triangle with side
lengths in the ratio 
.
Its hypotenuse passes through the join with the smaller
cap.
Call the three circles 
, 
, and 
. Let the upper point of intersection of 
and 
be 
, let the angle between
the vertical dashed line and the line
through 
be 
,
and let the angle between the horizontal radius
of 
and dashed line
through 
be 
.
Then
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
and half the area enclosed by the oval is the sum of the areas of the leftmost quarter-disk,
the circular sector of 
,
and the circular sector of 
minus the area of the triangular
portion of the circular sector of 
lying below the 
-axis, so
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  | )/(2(R-r)^2)tan^(-1)((2a(R-r))/(a^2-(R-r)^2))].](/images/equations/Oval/Inline60.svg) |
(9)
|
As expected, this formula reduces to the area of a circle
 |
(10)
|
for 
, and to the area of
a stadium
 |
(11)
|
for 
.
