 Conical Sections

In General
* 3rd century BC: Eight monumental volumes by Apollonius of Perga, who studied conical sections only for their mathematical beauty, not for applications.
* Eccentricity: For an ellipse, e = (distance between foci)/(twice the semimajor axis), and 0 < e < 1; It can also be covariantly defined in terms of angular velocity at periastron and apastron for the Kepler motion.
* Polar coordinates: If l = semi-latus rectum, a = semi-major axis; e = eccentricity, φ0 = angle of closest approach,

l/R = a(1−e2)/R = 1 + e cos(φφ0)      (with the origin at one focus) .

* Cartesian coordinates: If u is the eccentric anomaly, satisfying ue sin u = n (tt0),

x = a (cos ue) ,      y = a (1−e2)1/2 sin u ;      or      x2/a2 + y2/b2 = 1 .

@ Variations: in Cariñena et al JMP(05)mp [on spaces of constant curvature].
@ History: Fried 12 [Edmond Halley's reconstruction of the lost book of Apollonius' Conics].

Ellipse > s.a. Elliptic Functions; Precession.
\$ Def: A set S of points in E2 is an ellipse if there is a pair of points P1 and P2 in E2 such that for all PS, |P1|2/a2 + |P1|2/b2 = 1, for some two real constants a and b.
* Elliptical coordinates: Starting from Cartesian coordinates x and y in the plane, and given a constant F, one can introduce coordinates u and v such that

x + iy = F cosh(u + iv) ,    or   x = F cosh u cos v    and   y = F sinh u sin v ;

Then the lines u = u0 are ellipses (x/a)2 + (y/b)2 = 1 with a = F cosh u0 and b = F sinh u0, and the lines v = v0 are hyperbolas (x/A)2 − (y/B)2 = 1 with A = F cos v0 and B = F sin v0; > s.a. MathWorld page, Wikipedia page.
* Curious fact: The elliptical shadow made on a plane by a sphere illuminated by a point source (at infinity or not) always has one of the foci at the point where the sphere touches the plane; The proof is simple for the case of the source at infinity (using the relationship between the position of the focus and the length of the axes of an ellipse, the fact that the semiminor axis is equal to the radius of the sphere, and some triangles); For the general case it can be shown using second-order algebraic equations, but the proof is not very transparent; If the focus of an ellipse has a projective meaning, then one might be able to use the first proof in this case also.