Is a Circle an Ellipse

A circle is a special type of ellipse.

A circle and an ellipse are conic sections. When a plane cuts a cone, we get a conic section. The angle (angle of intersection) by which the plane cuts a cone determines whether it is a circle or an ellipse. Both can be perceived in coordinate axis if we compare the two shapes keeping them side by side.

A cone has four sections — circle, ellipse, parabola, and hyperbola. This article will precisely cover how and why a circle is considered a special case of an ellipse.

Before we discuss, under what circumstances, a circle is considered to be an ellipse, first we need  to see the basic differences between the two shapes.

How Would You Compare a Circle to an Ellipse

Let us now understand the differences the two using the figure below.

The differences between circle and ellipse is presented in the table below with respect to the figure above.

Circle	Ellipse
It has one fixed point, called the center O.	Besides having a center, it has 2 fixed points called the foci F1 and F2 (singular: focus).
All points on the circle (P or Q) are equidistant from the center. Thus, it has a constant radius.	All the points on the ellipse are not equidistant from the center. Thus, it has a variable semi-major axis ‘a’ and semi-minor axis ‘b’.
Its shape always remains the same even if its view is changed.	The shape varies from flat to round depending on the distance between the 2 foci.
Eccentricity ‘e’ is always zero.	Eccentricity ‘e’ lies between 0 and 1.

Despite the differences between circles and ellipses, an ellipse can satisfy the properties of a circle under certain conditions. We will elaborate on it in the next section.

Why is a Circle a Special Case of Ellipse

Let us first try to squeeze the ellipse in the figure above to make it a perfect circle and see the outcome in the figure below.

When the major and minor axes of an ellipse are equal in length, then an ellipse becomes a circle. The two foci (focus or focal points) combine to become one, i.e., the center.

Thus, the following conditions make an ellipse a circle:

• Semi-major axis and semi-minor axis becomes equal (a = b)
• distance between the foci becomes zero, so the 2 foci combines to become 1 focus (center) (i.e.  F₁, F₂, & O are same)
• eccentricity becomes zero (e = 0)

The conditions above clearly prove that all circles are ellipses. However, let us think the other way round. Are all Ellipse Circles?

No, because ellipses not necessarily will always have equal axes. The length of the axes can vary.

Thus, all ellipses may not be circles, but all circles are ellipses.

Let us now see whether the above conditions are valid when represented by circle and ellipse equations.

Circle and Ellipse Equations

As we know, the equation of an ellipse in the standard form is x²/a²+y²/b²=1

Implementing the above conditions, i.e., when a = b

We get, the equation of ellipse as

x²/b²+y²/b²=1

∴ x²+y²=b²

The above equation is the equation of a circle. Thus the above conditions also apply to the equations.