Circle · Ellipse · Parabola · Hyperbola — driven by eccentricity e = sin(θ)/sin(α)
A conic section is the curve formed where a plane intersects a double cone (two cones joined tip to tip). Depending on the angle of the cutting plane relative to the cone's axis, four distinct curves appear: a circle, an ellipse, a parabola, or a hyperbola. This was studied exhaustively by the ancient Greek mathematician Apollonius of Perga around 200 BCE in his treatise Conics, which named these curves and derived most of their core properties without any algebra.
The determining quantity is the eccentricity, e = sin(θ)/sin(α), where θ is the angle the cutting plane makes with the cone's base (0° = a horizontal cut) and α is the cone's half-angle. When e = 0 the section is a circle; when 0 < e < 1 it is an ellipse; at exactly e = 1 the plane runs parallel to the cone's slant edge, producing a parabola; and when e > 1 the plane is steep enough to cut through both nappes' worth of curvature, producing a hyperbola.
Conic sections are far from a mathional curiosity: planetary and satellite orbits are ellipses (Kepler's first law), a parabola is the exact shape traced by a thrown projectile under uniform gravity (ignoring air resistance) and the shape used in satellite dishes and car headlight reflectors to focus parallel rays to a point, and hyperbolas describe the paths of objects on an unbound gravitational flyby, such as some comets and interstellar objects passing through the solar system only once.
This simulator draws a side-on 2D view of a double cone being sliced by a plane at an adjustable angle, and simultaneously plots the resulting conic section — circle, ellipse, parabola or hyperbola — using its true polar equation r = l/(1 + e·cos θ), where the eccentricity e is computed directly from the cutting-plane angle and the cone's half-angle.
The left panel renders the cone's outline and a line representing the cutting plane at the chosen angle θ. The right panel plots the actual conic curve using the eccentricity-driven polar equation, so the curve's shape genuinely changes from a closed circle through ellipses, to an open parabola, to a two-branch hyperbola as the eccentricity crosses 0, then 1.
Drag the Plane angle θ slider to tilt the cutting plane from flat (0°, circular cross-section) to vertical (90°, hyperbola). Adjust the cone's Half-angle α to make the cone sharper or wider, which shifts the θ value at which the curve transitions from ellipse to parabola to hyperbola. The Readout panel shows the live eccentricity and curve name.
All conic sections can be described by one single equation, Ax² + Bxy + Cy² + Dx + Ey + F = 0, with the specific curve type determined entirely by the sign of the discriminant B² − 4AC: negative gives an ellipse (or circle), zero gives a parabola, and positive gives a hyperbola — matching exactly the e = 0, e = 1 and e > 1 boundaries used in this simulation.
The eccentricity e = sin(θ)/sin(α), where θ is the cutting-plane's angle from horizontal and α is the cone's half-angle. A value of e = 0 (a flat, horizontal cut) gives a circle, 0 < e < 1 gives an ellipse, e = 1 (the plane exactly parallel to the cone's slant side) gives a parabola, and e > 1 gives a hyperbola.
When the cutting plane is tilted to exactly match the slope of the cone's slant surface, the resulting curve extends to infinity along one direction without ever closing up into an ellipse or splitting into two separate branches, which is the defining geometric property of a parabola.
Once the cutting plane is steep enough (e > 1), it intersects both the upper and lower nappe of the double cone, producing two separate, mirror-image curves — the two branches of the hyperbola — rather than a single closed or open curve.
Yes. A circle is the limiting case of an ellipse where the eccentricity is exactly zero, meaning both foci coincide at the centre. This happens geometrically when the cutting plane is perfectly perpendicular to the cone's axis.
Planetary and satellite orbits are ellipses, thrown projectiles under uniform gravity follow parabolic paths, parabolic mirrors and antennas focus parallel rays to a single point, and hyperbolic trajectories describe unbound flybys such as some comets or interstellar objects passing through the solar system only once.