Hyperbola
Conic sectionsThe two-branched conic with a constant difference of focal distances, hugging its asymptotes forever.
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 Hyperbola?
The hyperbola is the conic with eccentricity greater than 1: the set of points whose focal distances differ by a constant 2a. Unlike the ellipse it splits into two branches, and unlike any other conic it comes with a pair of built-in guide lines — the asymptotes y = ±(b/a)x that the branches approach but never touch.
Constant difference of distances is exactly what a time difference of arrival measures, which made hyperbolas the mathematics of LORAN radio navigation before GPS. The same curve appears as y = 1/x in disguise (a rotated rectangular hyperbola), as the shadow of a sundial tip through a day, and as the profile of power-plant cooling towers.
Key properties
- Foci at (±c, 0) with c² = a² + b²; eccentricity e = c/a > 1.
- Difference of focal distances is constant: |r₁ − r₂| = 2a.
- Asymptotes y = ±(b/a)x; a = b gives the rectangular hyperbola (y = 1/x rotated 45°).
- Reflective property: rays aimed at one focus reflect toward the other.
- Conjugate hyperbola swaps the roles of the two axes.
Where you'll see it
Radio navigation (LORAN), cooling tower silhouettes, sundial shadow paths, gravitational slingshot trajectories, and y = 1/x.