Foci and Dandelin spheres.

Step 1 of 6

The gardener’s ellipse

A gardener who wants an oval bed drives in two pegs, ties a slack string to them, and pulls it taut with a stick that marks the line in the soil. The stick runs round an ellipse, and the pegs are its foci: at every point $P$⁠ of the ellipse, $PF_1 + PF_2$⁠ equals the string’s length.

In 1822 Germinal Pierre Dandelin proved this with the cone. Show the focus spheres shows two balls that fit snugly inside the cone, one above the cut and one below. Each ball touches the cut at one point, a focus, and the cone along a circle. From $P$⁠ you can measure to each ball in two ways: to the focus, or along the cone’s side to the circle, and those two tangents are equal. Work out the sum of the distances.

Step 2 of 6

a, b and c

The semi-major axis $a$⁠, the semi-minor axis $b$⁠ and the distance to a focus $c$⁠ are linked by a right triangle: from the end of the minor axis, the distance to each focus is $a$⁠, because the sum is $2a$⁠ and the two distances are equal. So $a^2 = b^2 + c^2$⁠. Work out $b$⁠.

Step 3 of 6

One focus

As the tilt approaches $60^\circ$⁠, the ellipse stretches out, and the larger ball is pushed up the cone. Predict what happens to it at the parabola, then increase Tilt of the cut.

Step 4 of 6

The directrix

Each ball touches the cone along a circle, and that circle’s plane cuts the plane of the cut in a straight line: the directrix. On every conic section, the ratio of a point’s distance to the focus, $PF$⁠, to its distance to the directrix, $PD$⁠, is the eccentricity. The point running along the curve measures both distances. This parabola has one focus and one directrix, and it is also the parabola of the quadratic function $y = ax^2$⁠.

Step 5 of 6

Hyperbola

On a hyperbola the balls are in the two halves of the cone, one above the tip and one below, and each focus is inside one of the branches. Move Distance from the tip, and watch both balls stay touching the cut.

Step 6 of 6

Vertex and focus

Back to the first ellipse. The vertices are the two points farthest apart, and the foci sit between them on the same axis.