Ellipse — Introduction

Definition. An ellipse is the locus of a point in a plane that moves so that its distance from a fixed point (the focus $S$) bears a constant ratio $e$, with $0<e<1$, to its distance from a fixed line (the directrix $d$). Symbolically, if $M$ is the foot of the perpendicular from a point $P$ on the ellipse to the directrix, then $\dfrac{PS}{PM}=e$ and $PS=e\cdot PM$ — this is exactly the general focus–directrix definition of section 7.1.3, specialised to $0<e<1$.

Geometrically, an ellipse is also what results from slicing a double cone with a plane oblique to the axis (section 7.1.2, case 3) — a closed, oval cross-section, in contrast to the parabola's single open curve. …