Conic Sections Explorer

Conic sections are the curves you get when you slice a double-napped cone at different angles. The four types are the circle, ellipse, parabola, and hyperbola.

Every conic section can be written as a second-degree equation in $x$ and $y$.

The value of $B^2 - 4AC$ determines the type. When $B = 0$ (no rotation), the type depends on the relationship between $A$ and $C$.

A circle is the set of all points at a fixed distance (the radius) from a center point.

An ellipse is the set of points where the sum of distances to two foci is constant. Its eccentricity $e$ is between 0 and 1. A circle is a special ellipse with $e = 0$.

The semi-major axis $a$ is always the larger value. The foci lie along the major axis at distance $c = \sqrt{a^2 - b^2}$ from the center.

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Its eccentricity is always $e = 1$.

The value $p$ is the distance from the vertex to the focus. When $p > 0$ the parabola opens upward (or rightward for horizontal form). The directrix is the same distance from the vertex as the focus, but on the opposite side.

Parabolas appear in satellite dishes, car headlights, and the trajectory of projectiles (ignoring air resistance).

A hyperbola is the set of points where the absolute difference of distances to two foci is constant. It has two separate branches and its eccentricity is always $e > 1$.

The asymptotes have slopes $\pm \frac{b}{a}$ for a horizontal hyperbola. The foci are at distance $c = \sqrt{a^2 + b^2}$ from the center. Note: unlike an ellipse, here $c > a$.

Hyperbolas appear in GPS positioning (using time differences from satellites), the shape of sonic booms, and certain orbital trajectories.