Mathematics

Conic sections

Parabola, ellipse, hyperbola, eccentricity, foci, and an orbit sketch.

Basics

Slices and definitions

Slice a cone with a plane and you get a circle, ellipse, parabola, or hyperbola (parabola when the plane is parallel to a generator). Focus–directrix: distance to the focus equals e times distance to the directrix. e=1 parabola, e<1 ellipse, e>1 hyperbola, e=0 circle. In coordinates the discriminant of ax²+bxy+cy²+…=0 names the type. Rotate and translate to reach a standard form.

Ellipse

In the standard form x²/a² + y²/b² = 1 (a>b) the major axis is 2a, the minor 2b, the focal distance c=√(a²−b²), eccentricity e=c/a. The sum of distances to the two foci is the constant 2a. Kepler’s first law uses this ellipse. A circle a=b is the special case with foci collapsed to the centre. The parametrisation x=a cos t, y=b sin t is handy to draw.

Parabola and hyperbola

The parabola y²=4ax has focus (a,0) and directrix x=−a. Its reflection property is why headlamps and dish antennas use this curve. The hyperbola x²/a² − y²/b² = 1 has a constant difference 2a of distances to two foci. The rectangular hyperbola xy=c² is the inverse-proportion graph. Asymptotes are the skeleton at large |x|, |y|.

Polar form and orbits

A unified polar form is r = ed / (1 + e cosθ) (or 1 − e cosθ in the denominator). Under an inverse-square gravity, bound orbits have e<1 (ellipses), escape e=1 (parabola), and e>1 a hyperbola. Periapsis at θ=0 is a common convention. This is a sketch; perturbations and relativity (Mercury’s perihelion) are extra layers.

Formulas

Eccentricity

e = c / a (타원·쌍곡선)

Ellipse e<1, parabola 1, hyperbola >1.

Symbols

c centre-to-focus distance
a semi-major or transverse half-axis

Standard ellipse

x²/a² + y²/b² = 1, b² = a² − c²

If a>b the major axis is horizontal.

Symbols

b semi-minor axis

Parabola

y² = 4 a x

Focus (a,0), directrix x = −a.

Symbols

a focal length

Polar conic

r = (e d) / (1 + e cosθ)

Directrix to the right of the pole. d is pole-to-directrix distance.

Symbols

e eccentricity
θ angle (true anomaly analogue)

Key table

Discriminant	b²−4ac <0 ellipse (circle), =0 parabola, >0 hyperbola (general conic with xy term)
Earth orbit e	about 0.017 — a nearly circular ellipse
Reflection	ellipse: focus to focus. parabola: axis-parallel rays to the focus