Ellipse
Conic sectionsThe locus with a constant sum of distances to two foci — the true shape of planetary orbits.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Ellipse?
The ellipse is the set of points whose distances to two foci add to a constant 2a — the geometry behind the pins-and-string drawing method. Squash a circle by a uniform factor along one axis and you get an ellipse; that is exactly what the parametrization x = a·cos t, y = b·sin t says.
Kepler’s first law of 1609 made the ellipse the most important curve in the sky: every planet orbits the Sun along an ellipse with the Sun at one focus. Its reflective property — a signal leaving one focus arrives at the other — powers whispering galleries and the shock-wave lithotripters used to break kidney stones.
Key properties
- Foci at (±c, 0) with c² = a² − b²; eccentricity e = c/a < 1.
- Area is exactly πab; the perimeter has no elementary closed form (Ramanujan’s approximation is the standard workaround).
- Sum of focal distances is constant: r₁ + r₂ = 2a.
- Reflective property: one focus maps to the other.
- A circle is the special case a = b (e = 0).
Where you'll see it
Planetary and satellite orbits, whispering galleries, medical lithotripsy, and elliptical gears and cams.