In a discussion with friends we talked about: Is it possible that an ellipse can completely cover an oval in shape (circumference and/or area) or vice versa that the oval covers the ellipse?

Can you explain it to us in a very simple way?

Thanks in advance.

note:

I think that an ‘ellipse’ consists of a closed curve starting from the foci and that an ‘oval’ consists of four rays equal two by two and that the major rays can also fall outside the figure.

## Answer

Dear Andre,

The shortest answer is that every ellipse is an oval, but not the other way around. An ellipse is a special case of an oval.

Let me go into a little more detail. Roughly speaking, an oval is any closed curve that somewhat resembles an ellipse, but that is of course not a strictly mathematical definition.

I believe an oval is well defined by saying:

An oval is any curve that satisfies the following properties:

a. The curve defines a single coherent region.

b. The curve defines a convex region.

c. The curve is differentiable.

What does all this mean?

A simple connected area is an area with no gaps in it. In the first figure you see a non-single connected area, if you think away the holes it becomes simple connected.

The curve defines a convex area. If you take two points inside the area, then the line segment connecting the two points is also inside that area. In the second figure on the left you see a convex area and on the right you see a non-convex area.

The term differentiable is more difficult to explain, but it actually means that there are no angles.

It is easy to see that an ellipse certainly fits that definition, but when exactly does an oval become an ellipse?

An ellipse has the additional property that an ellipse contains two points (called foci) such that the sum of the distances to a point on the ellipse and the two foci is constant (see Figure 3).

The example you cite that an oval has two rays two equals two by two is, in my opinion, actually an athletic field, that is certainly an oval, but an oval can be much more.

Hopefully this answers your question.

Regards.

## Answered by

#### prof. Dr. Kurt Barbara

Mathematics, statistics, probability, scientific calculation

Pleinlaan 2 1050 Ixelles

http://www.vub.ac.be/

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