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
the 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