Conic Sections Calculator

What Are Conic Sections? Conic sections are the curves formed by intersecting a cone with a plane at different angles. They were first systematically studied by Apollonius of Perga, a Greek mathematician, around 200 BC in Alexandria, Egypt. The four types: circle (plane perpendicular to axis), ellipse (tilted cut), parabola (parallel to a slant), hyperbola (steeper than slant). Conics appear everywhere in physics: planetary orbits (ellipses), projectiles (parabolas), and spacecraft escape trajectories (hyperbolas).

Ellipse Standard form: x²/a² + y²/b² = 1 (a > b > 0, major axis along x-axis) Or: x²/b² + y²/a² = 1 (a > b > 0, major axis along y-axis) Semi-major axis: a. Semi-minor axis: b. Foci: c = √(a² - b²) apart from center. Eccentricity: e = c/a (0 < e < 1). Area = π × a × b. There is no elementary formula for the perimeter, so this calculator uses Ramanujan’s second approximation, with h = ((a-b)/(a+b))²: Perimeter ≈ π(a+b) × [1 + 3h / (10 + √(4 - 3h))]

Ramanujan also published a simpler first formula, π[3(a+b) - √((3a+b)(a+3b))], and you will see it quoted more often. On an ordinary ellipse the second one is far better: at a=5, b=3 the true perimeter is 25.526999, the second formula is right to about one part in 10¹¹, and the first is out by about five parts in 10⁷.
Both degrade on a very flat ellipse, and it is worth knowing by how much. At a=100, b=1 the true perimeter is 400.1098. The first formula gives 398.74, low by 0.34%. The second gives 400.01, low by 0.024%, so it is roughly fourteen times closer but nowhere near exact. If you need a flat ellipse to full precision, neither approximation will do it and you want a numerical elliptic integral.

A circle is a special ellipse with a = b and e = 0.

Parabola Standard form: y = x²/(4p) or x = y²/(4p) p = distance from vertex to focus (= distance from vertex to directrix). Focus at (0, p). Directrix: y = -p (for upward parabola). Eccentricity: e = 1 (always). Parabolas focus parallel rays to a single point, which is why they turn up in telescope mirrors, satellite dishes and headlight reflectors.

Hyperbola Standard form: x²/a² - y²/b² = 1 (transverse axis along x) Foci: c = √(a² + b²). Eccentricity: e = c/a > 1. Asymptotes: y = ±(b/a)x. The difference of distances from any point to the two foci is constant = 2a. GPS and the older LORAN system both use hyperbolic positioning: a time difference between two stations puts you on one hyperbola, and a second pair narrows it to a point.

Eccentricity Summary e = 0: circle. 0 < e < 1: ellipse. e = 1: parabola. e > 1: hyperbola. Planetary orbit eccentricities: Earth = 0.017, Mars = 0.093, Mercury = 0.206, Pluto = 0.248, Halley’s Comet = 0.967. Earth’s orbit is so close to circular that a scale drawing of it looks like a circle. The Sun sitting off-center is what you would notice, not the shape.

General Equation Any conic: Ax² + Bxy + Cy² + Dx + Ey + F = 0. Discriminant B² - 4AC: < 0 → ellipse/circle; = 0 → parabola; > 0 → hyperbola.

Kepler’s Laws and Conics Kepler’s first law (1609): planets move in ellipses with the Sun at one focus. Open trajectories (escape or hyperbolic flyby): comets and spacecraft often travel on hyperbolic or parabolic paths relative to the Sun. The vis-viva equation gives orbital speed at any point: v² = GM(2/r - 1/a).