Eccentricity.

Step 1 of 5

One number for the shape

One number describes all four conic sections, the eccentricity $e$⁠. In a cone that opens at an angle $\alpha$⁠ from its axis, a cut at tilt $\theta$⁠ gives

$e = \frac{\sin\theta}{\cos\alpha}$$e =$$\frac{\sin\theta}{\cos\alpha}$⁠

Circle: $e = 0$⁠. Ellipse: $0 < e < 1$⁠. Parabola: $e = 1$⁠. Hyperbola: $e > 1$⁠. Here Tilt of the cut is $30^\circ$⁠ and $\alpha = 30^\circ$⁠. Work out $e$⁠, then choose On under Show the focus spheres: the sign beside the cone shows $e$⁠.

Step 2 of 5

Setting e

The more the cut tilts, the more the ellipse stretches and the bigger $e$⁠ grows. Find the tilt that gives a particular $e$⁠.

Step 3 of 5

Size

A cut further from the tip, at the same tilt, is an enlarged copy of the nearer one. Predict what will happen to $e$⁠ when Distance from the tip doubles.

Step 4 of 5

The foci

An ellipse has two foci, $F_1$⁠ and $F_2$⁠, on its major axis. The white points are where the focus spheres touch the cut. Write $a$⁠ for the semi-major axis, from the centre to a vertex, and $c$⁠ for the distance from the centre to each focus. Then the eccentricity is also

$e = \frac{c}{a}$⁠

In a circle both foci are at the centre, $c = 0$⁠. The longer the ellipse, the closer the foci are to its ends. Work out $c$⁠.

Step 5 of 5

When e is 1

The tilt at which $e = 1$⁠ is the border between ellipse and hyperbola: there the ellipse opens into a parabola. Find it.