Conic Section

The conic sections are the nondegenerate curves generated by the intersections of a Plane with one or two Nappes of a Cone. For a Plane parallel to a Cross-Section, a Circle is produced. The closed curve produced by the intersection of a single Nappe with an inclined Plane is an Ellipse or Parabola. The curve produced by a Plane intersecting both Nappes is a Hyperbola. The Ellipse and Hyperbola are known as Central Conics.

Because of this simple geometric interpretation, the conic sections were studied by the Greeks long before their application to inverse square law orbits was known. Apollonius wrote the classic ancient work on the subject entitled On Conics. Kepler was the first to notice that planetary orbits were Ellipses, and Newton was then able to derive the shape of orbits mathematically using Calculus, under the assumption that gravitational force goes as the inverse square of distance. Depending on the energy of the orbiting body, orbit shapes which are any of the four types of conic sections are possible.

A conic section may more formally be defined as the locus of a point $P$ that moves in the Plane of a fixed point $F$ called the Focus and a fixed line $d$ called the Directrix (with $F$ not on $d$) such that the ratio of the distance of $P$ from $F$ to its distance from $d$ is a constant $e$ called the Eccentricity. For a Focus $(0,0)$ and Directrix $x=-a$, the equation is

$$x^2+y^2=e^2(x+a)^2.$$

If $e=1$, the conic is a Parabola, if $e<1$, the conic is an Ellipse, and if $e>1$, it is a Hyperbola.

In standard form, a conic section is written

$$y^2 = 2Rx-(1-e^2)x^2,$$

where $R$ is the Radius of Curvature and $e$ is the Eccentricity. Five points in a plane determine a conic (Le Lionnais 1983, p. 56).

See also Brianchon's Theorem, Central Conic, Circle, Cone, Eccentricity, Ellipse, Fermat Conic, Hyperbola, Nappe, Parabola, Pascal's Theorem, Quadratic Curve, Seydewitz's Theorem, Skew Conic, Steiner's Theorem

References

Conic Sections

Besant, W. H. Conic Sections, Treated Geometrically, 8th ed. rev. Cambridge, England: Deighton, Bell, 1890.

Casey, J. ``Special Relations of Conic Sections'' and ``Invariant Theory of Conics.'' Chs. 9 and 15 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 307-332 and 462-545, 1893.

Coolidge, J. L. A History of the Conic Sections and Quadric Surfaces. New York: Dover, 1968.

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 138-141, 1967.

Downs, J. W. Conic Sections. Dale Seymour Pub., 1993.

Iyanaga, S. and Kawada, Y. (Eds.). ``Conic Sections.'' §80 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 271-276, 1980.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983.

Lee, X. ``Conic Sections.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html

Ogilvy, C. S. ``The Conic Sections.'' Ch. 6 in Excursions in Geometry. New York: Dover, pp. 73-85, 1990.

Pappas, T. ``Conic Sections.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 196-197, 1989.

Salmon, G. Conic Sections, 6th ed. New York: Chelsea, 1954.

Smith, C. Geometric Conics. London: MacMillan, 1894.

Sommerville, D. M. Y. Analytical Conics, 3rd ed. London: G. Bell and Sons, 1961.

Yates, R. C. ``Conics.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 36-56, 1952.