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🍦 Conic Sections

Interactive conic sections simulator. Slice a double cone with an adjustable plane and watch the circle, ellipse, parabola or hyperbola form in real time, driven by the true eccentricity relationship.

Geometry2DEasy60 FPS

🍦 What It Demonstrates

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.

How to Use

Drag the Plane angle θ slider from 0° (flat, circular cut) up to 90° (vertical, hyperbolic cut).
Adjust the Cone half-angle α to make the cone narrower or wider — this shifts exactly which θ values produce each curve type.
Watch the side view of the cone and cutting plane update on the left, and the resulting 2D conic curve draw itself on the right from its polar equation.
Read the live eccentricity and conic type label as you adjust either slider.

Did You Know?

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.

About this simulation

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.

🔬 What it shows

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.

🎮 How to use

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.

💡 Did you know?

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.

Frequently asked questions

What determines whether a slice is a circle, ellipse, parabola or hyperbola?

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.

Why does a plane parallel to the cone's edge produce a parabola specifically?

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.

Why does a hyperbola have two branches?

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.

Is a circle really just a special ellipse?

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.

Where do conic sections appear outside pure geometry?

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.

⚙ Under the hood

Canvas 2DConic SectionsEccentricityPolar EquationEllipse

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install