Introduction

A Curve With a Long History

The ellipse is not just a mathematical curiosity — it is, famously, the shape traced by every planet's orbit around the sun, an idea with deep roots in ancient Indian astronomical thought: verses in the Rigveda describe the sun's yearly path as an unbroken, ever-repeating course, a picture later astronomers would formalise precisely as an elliptical orbit. Centuries afterward, the French mathematician Girard Desargues (c. 1591–1662) — an engineer and architect as well as a mathematician — made major original contributions to the study of conic sections, laying groundwork for what later became projective geometry.

What This Chapter Covers

Continuing from the previous chapter's focus–directrix definition of a conic, this chapter studies the case $0<e<1$: the ellipse. It derives the ellipse's standard equation, identifies its key features — the foci, directrices, eccentricity and latus rectum — and works out its parametric equations. It then studies lines in relation to the ellipse: the condition for a line to be a tangent, and the equations of the tangent and normal at a given point on the curve, both in Cartesian and parametric form.