A hyperbola is two mirror-image curves that open away from each other. Slice a double cone with a steep plane and you get one. Comets that won't return, navigation by LORAN, and the shadow of a lampshade — all hyperbolic.
Slide the eccentricity
hyperbola
e = 1.40
e = 0 → circle · 0<e<1 → ellipse · e = 1 → parabola · e > 1 → hyperbola
Equation
Note the minus sign — that's what makes it a hyperbola not an ellipse.
Quick anatomy
- Two branches — opening left and right (or up and down).
- Two foci — one for each branch.
- Asymptotes — straight lines the curves hug at infinity.
- |PF₁ − PF₂| = 2a — the constant *difference* of distances (vs. the *sum* for an ellipse).