# Conic Sections

**Type:** Concept  
**Domain:** Geometry  
**Codex URL:** /mathematics/geometry/conic-sections/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
Circle, ellipse, parabola, hyperbola — curves from slicing a cone. Studied by Apollonius c.200 BCE; later shown by Kepler/Newton to govern planetary orbits.

## Definitions
- Ellipse: sum of distances to two foci constant. x²/a²+y²/b²=1
- Parabola: equidistant from focus and directrix. y²=4px
- Hyperbola: difference of distances to two foci constant. x²/a²−y²/b²=1
- Eccentricity e: 0=circle, 0<e<1=ellipse, e=1=parabola, e>1=hyperbola

## Kepler and Newton
Kepler (1609): Mars's orbit is an ellipse, Sun at one focus (First Law) — broke 2000+ years of assumed circular orbits. Newton (1687, Principia): derived elliptical orbits rigorously from inverse-square gravitation law.

## Orbit shapes
Bound orbit=ellipse; escape-velocity trajectory=parabola; hyperbolic excess velocity=hyperbola (e.g. 'Oumuamua, 2017).

## Apollonius's Conics (c.200 BCE)
First to derive all 4 curves from one double cone; named the curves. Earlier (Menaechmus, c.350 BCE) used different cones for each curve separately.

## Projective geometry
Poncelet (early 19th c.): all conics are projectively equivalent — differences arise only from relation to the "line at infinity."

## Sources
### Tier 1
- Heath, T.L. (1961 reprint). *Apollonius of Perga: Treatise on Conic Sections*.
- Newton, I. (1687). *Principia Mathematica*.
### Tier 2
- Coxeter, H.S.M. (2003). *Projective Geometry*. 2nd ed.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*