Sunday, 17 August 2008

Ellipse? Circle? How Are They Related?

In maths study of shapes and geometry, we always come across the term "ellipse" and "circle".

How are they created? And are they related in any way?

Answer is they are related, and closely too!

Circle is no problem to anyone, but ... ellipse?

Ok, let me show you some interesting pictures.

We will use the torchlight to explain the concept of circle and ellipse generation.

Diagram 1: Circle generation from a vertically shining torchlight

A light output from a torchlight is a good tool to illustrate the concept of Circle. (Diagram 1).

When we shine the torchlight vertically onto a flat surface, we can expect to see a circle formed by the light from the torch. Many of us played with this ever since young.

What if we slant the torch slightly at an angle?

See diagram 2.

Diagram 2: Ellipse formation due to the slanted torch

In diagram 2, it is obvious the light created onto the surface is an elongated circle. This is the ellipse (Condition: only if the slanting angle is small).

(It is actually an "egg" shape if the slant is too much.)
The above 2 diagrams serve to illustrate that circle when stretched may evolved into one special shape of an ellipse having 2 radii (refer below for formula).

From the above observations, we can conclude that:
  • Ellipse covers more area than a circle.

  • Circle is part (or a subset) of the larger ellipse geometry

  • They are closely related (in fact of the same family)

  • Circle has constant radius

  • Ellipse has more than one radius (with one being the same as the circle)

The area and radius part deserve some mention.

Ellipse has the area given by AREA = (pi) x r1 x r2
where r1 and r2 is as shown in diagram 3.

Diagram 3: Two radii within

For circle, the AREA = (pi) x r12 = (pi) x r1 x r1
where r1 is a constant throughout the rotation.

From the AREA of both shapes, you discovered that the formula for both are actually the same. But since the radius "r1" is the same as "r2" for a circle, the latter formula is true.

That's all for ellipse and circle.

Enjoy your fun with the torchlight!


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