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, *φ*_{0}
= angle of closest approach,

*l*/*R* = *a*(1–*e*^{2})/*R*
= 1 + *e* cos(*φ*–*φ*_{0}) (with
origin at one focus) .

* __Cartesian coordinates__: If *u* is the eccentric anomaly, satisfying
*u* – *e* sin *u* = *n* (*t*–*t*_{0}),

*x* = *a* (cos *u* – *e*) , *y* = *a* (1–*e*^{2})^{1/2} sin
*u* ; or *x*^{2}/*a*^{2}
+ *y*^{2}/*b*^{2} = 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 E^{2} is an ellipse if there
is a pair of points *P*_{1} and *P*_{2} in
E^{2} such that for all *P* ∈ *S*,
|*P*_{1}–*P *|^{2}/*a*^{2} +
|*P*_{1}–*P *|^{2}/*b*^{2} =
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* + i*y* = *F* cosh(*u* + i*v*) ,
or *x* = *F* cosh *u* cos *v*
and *y* = *F* sinh *u* sin *v* ;

Then the lines *u* = *u*_{0} are ellipses (*x*/*a*)^{2} + (*y*/*b*)^{2} = 1 with *a* = *F* cosh *u*_{0} and *b* = *F* sinh *u*_{0}, and the lines *v* = *v*_{0} are hyperbolas (*x*/*A*)^{2} – (*y*/*B*)^{2} = 1 with *A* = *F* cos *v*_{0} and *B* = *F* sin *v*_{0}; > 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.

> __Online resources__:
see MathWorld page; Wikipedia page; Malin Christersson's 3-ellipse (and Cassini oval) page.

**Hyperbola**

$ __Def__: A set *S* of
points in E^{2} is a hyperbola if there is
a pair of points *P*_{1} and *P*_{2} in
E^{2} such that for all *P* ∈ *S*,
|*P*_{1}–*P* |^{2}/*a*^{2} –
|*P*_{1}–*P *|^{2}/*b*^{2} =
1, for some two real constants *a* and *b*.

**Parabola**

$ __Def__: The set of points which are equidistant from a point *P* = (*x*_{0},
*y*_{0}) and a line *l* in the Euclidean plane.

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send feedback and suggestions to bombelli at olemiss.edu – modified
26 jul 2016