Parabola
Conic sectionsThe set of points equidistant from a focus and a directrix — the shape of projectile paths and satellite dishes.
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 Parabola?
The parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). It is also the conic section cut by a plane parallel to the cone’s slant side. Menaechmus studied it around 350 BC while attacking the cube-doubling problem; Apollonius gave it its name.
Two properties make it ubiquitous. Galileo showed that projectiles under uniform gravity follow parabolas. And its reflective property — rays from the focus emerge parallel to the axis, and incoming parallel rays converge to the focus — puts a parabola inside every satellite dish, car headlight, and radio telescope.
Key properties
- Focus at (0, p) and directrix y = −p for x² = 4py.
- All parabolas are geometrically similar — they differ only by scale.
- Reflective property: focal rays exit parallel to the axis.
- The latus rectum (chord through the focus, parallel to the directrix) has length 4p.
- Eccentricity is exactly 1 — the boundary between ellipses and hyperbolas.
Where you'll see it
Projectile motion, satellite dishes, headlight reflectors, suspension-free arch bridges, and quadratic functions everywhere.