Conical
Sections| Conical
Sections |
In General
* 3rd cy 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 ito angular velocity at periastron and apastron for the Kepler motion.
* Polar coordinates:
If l = semi-latus rectum, a =
semi-major axis; e = eccentricity, 
0
= angle of closest approach,
l/R = a(1–e2)/R
= 1 + e cos(–0) (with
origin at one focus) .
* Cartesian coordinates: If u is the eccentric anomaly, satisfying u – e sin u = n (t–t0),
x = a (cos u – e) , y = a (1–e2)1/2 sin u ; or x2/a2 + y2/b2 = 1 .
@ Variations: in Cariñena et al mp/05 [on spaces of constant curvature].
Ellipse
$ 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 P 
S,
|P1–P|2/a2 +
|P1–P|2/b2 =
1, for some two real constants a and b.
* 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.
Hyperbola
$ Def: A set S of
points in E2 is a hyperbola if there is
a pair of points P1 and P2 in
E2 such that for all P S,
|P1–P|2/a2 –
|P1–P|2/b2 =
1, for some two real constants a and b.
Parabola
$ Def: The set of points which are equidistant from a point P = (x0,
y0) and a line 
in
the Euclidean plane.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
24 jan 2007