Ellipse

In the chapter on loci we saw that the parabola arises by requiring a point to be equidistant from a focus and a line. The ellipse arises from an analogous but “doubled” condition: instead of a single focus we take two, and we require that the sum of the distances from them be constant. From this simple requirement the whole geometry of the ellipse follows: the canonical form $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, the semi-axes $a$ and $b$, the foci at distance $c=\sqrt{a^2-b^2}$ from the centre, the eccentricity $e=c/a$ which measures its “flattening”, the directrices, and finally the recognition of translated ellipses through completing the square. We shall also see that the circle is nothing but a degenerate ellipse, the one of maximum roundness.

Sections

• Definition and canonical form
• Elements of the ellipse
• The circle as a degenerate ellipse
• Translated ellipses and recognition

Further topics

• Construction with nails and string

Exercises

• All the exercises in the chapter