Conic Section: Interactive Visualization

Conic sections in this visualization

A conic section (or conic) is the curve in which a plane meets a double cone: two cones joined at their tips, the apex, around one axis, running on for ever in both directions. The cone'shalf-angle $\alpha$ is the angle between the axis and its side lines; the plane makes an angle $\beta$ with the axis (from $0^\circ$ , parallel to the axis, to $90^\circ$ , square to it) and lies at a distance $d$ from the apex. Every section is a circle, an ellipse, a parabola or a hyperbola, or, when the plane passes through the apex, a point, one line or two lines.

One number decides the kind of curve

e = \frac{\cos\beta}{\cos\alpha} \qquad \beta > \alpha:\ e < 1 \qquad \beta = \alpha:\ e = 1 \qquad \beta < \alpha:\ e > 1

Turn the plane from $90^\circ$ down. The circle ( $e = 0$ ) becomes an ellipse that grows longer and longer; the moment the plane is parallel to a side line of the cone ( $\beta = \alpha$ ) the far end is gone to infinity and the curve is a parabola; turn a little further and the plane reaches the other cone too, where a second branch comes in from infinity: a hyperbola. The true shape window shows the section square on at a fixed scale, with its near end, the point closest to the apex, held in place, so the far end can be seen running out of the window. The kind of curve is decided from the angles exactly as typed: $\beta = 30.01^\circ$ against $\alpha = 30^\circ$ is an ellipse with $e \approx 0.9999$ , not a parabola.

Where the foci come from

Dandelin spheres sit inside the cone, each touching the cone along a circle and the plane at one point: those points are the foci $F_1$ and $F_2$ . From a point $P$ of the curve, every tangent line to the same sphere has the same length, so $PF_1$ equals the distance along the cone's side line through $P$ to the first touching circle, and $PF_2$ the distance to the second.
For an ellipse both spheres are in the same cone and $P$ lies between the circles, so $PF_1 + PF_2$ is the length of the side line between them, the same for every $P$ : $2a$ . For a hyperbola the spheres are in opposite cones and $|PF_1 - PF_2| = 2a$ . A parabola has room for only one sphere.
A directrix is the line where the plane meets the plane of a touching circle. The distance from $P$ to the focus and to the directrix keep the same ratio, $PF_1 / PD_1 = e$ ; for the parabola they are equal. The circle has no directrix: the two planes are parallel.

Things not to misread

Perspective makes a circle look like an ellipse in the 3D view; the true-shape window, or Face the section, shows the real shape. The window's scale never changes by itself: the numbers under it say where the far end or the other branch is.
The cone is drawn only near the apex and fades out at its ends; it goes on for ever. A long ellipse is still closed, however far its end, and a parabola is not half an ellipse: it has no far end at all.
The kind of curve and the relation between $\beta$ and $\alpha$ are exact; $e$ , the lengths and the distances are computed and marked ≈. The glass, its faint lines and the fading are only for display.

Controls

Turn $\beta$ with its knob or drag the blue handle on the plane; it catches at $\alpha$ and $90^\circ$ . The wheel moves the plane; $d = 0$ puts it through the apex. Angles can be typed in degrees or as multiples of π. With the spheres or the directrices on, drag $P$ along the curve. Presentation mode opens a full-screen view: ← → turn the plane, ↑ ↓ move it.

Further reading: Wikipedia: Conic section; Wikipedia: Dandelin spheres; OpenStax, Precalculus 2e, 10.5 Conic Sections in Polar Coordinates.

Conic sections in this visualization

Where the foci come from

Things not to misread

Controls

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