Curved Figures

Until now, we have studied plane figures whose sides are straight lines. Now let us study figures which are bounded, fully or partly, by curved line.

The figures which are studied in geometry are circle, ellipse and parabola. One reason could be that all these could be drawn easily or observed in motions of bodies.

Let us do the following activity. Take a long piece of thread and knot both ends so that it forms a loop.

Drive a nail or a stick on the ground, and loop one end around it. Hold the loop tightly and using another stick inserted at the other end of the loop, draw a figure by moving it in any direction that it can move. The figure that you would draw would be a circle. The fixed stick would be its centre. The length of the loop would be its radius.

Now near the centre, put another stick and now ensure that the loop goes over both the sticks. Stretching the loop will now form a triangle! If the same process is followed as in the circle, now you will get a circle which has been slightly distorted. It would look as if on one side the circle has been stretched and in the perpendicular direction it has been compressed. This figure is called the Ellipse. The 2 points around which the loop was wound are the 2 foci (singular- focus) of the ellipse.

An interesting fact would be that both the original circle and the new ellipse would touch along the side where the ellipse seems to have been stretched.

Space the 2ndstick a little further away from the centre (of the circle) and drawn again. You will get another ellipse, which is even more stretched and compressed. This ellipse will also touch the circle and the first ellipse at the same point!

In the same way, you can draw a series of ellipses, whose foci will become farther apart and their shapes will get more compressed. All the ellipses would be contained inside the circle.

We see circles all around us – the Sun, the Moon, eyes etc. Many objects around us closely resemble a circle or part of it. The circle is one of the most studied figures. We will see more about circles in the subsequent chapters.

A chicken egg resembles an ellipse. The paths of the planets around the Sun are ellipses.

Another figure we see is a parabola. It is a path taken by a ball which we throw high into the air and away from us.

Conic sections

One of the earliest discoveries of Greek mathematicians was that all these shapes could be got by slicing a cone at different angles. The name of Apollonius is associated with the discovery of Conic Sections.

A cut perpendicular to the axis of the cone results in a circle. As the cut proceeds from the vertex to the base, the size of this circle keeps on increasing.

A cut which is at an angle less than a right angle, results in an ellipse. Varying the angle of the cut results in ellipses of different sizes.

A cut which is perpendicular to the base results in a parabola. As the cut moves closer to the axis, the shape of the parabola also changes.