Conic Section -- from Wolfram MathWorld

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 perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola (Hilbert and Cohn-Vossen 1999, p. 8). The curve produced by a plane intersecting both nappes is a hyperbola (Hilbert and Cohn-Vossen 1999, pp. 8-9).

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 that 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 that moves in the plane of a fixed point called the focus and a fixed line called the conic section directrix (with not on ) such that the ratio of the distance of from to its distance from is a constant called the eccentricity. If , the conic is a circle, if , the conic is an ellipse, if , the conic is a parabola, and if , it is a hyperbola.

A conic section with conic section directrix at , focus at , and eccentricity has Cartesian equation

(1)

(Yates 1952, p. 36), where is called the focal parameter. Plugging in gives

(2)

for an ellipse,

(3)

for a parabola, and

(4)

for a hyperbola.

The polar equation of a conic section with focal parameter is given by

(5)

The pedal curve of a conic section with pedal point at a focus is either a circle or a line. In particular the ellipse pedal curve and hyperbola pedal curve are both circles, while the parabola pedal curve is a line (Hilbert and Cohn-Vossen 1999, pp. 25-27).

Five points in a plane determine a conic (Coxeter and Greitzer 1967, p. 76; Le Lionnais 1983, p. 56; Wells 1991), as do five tangent lines in a plane (Wells 1991). This follows from the fact that a conic section is a quadratic curve, which has general form

(6)

so dividing through by to obtain

(7)

leaves five constants. Five points, for , ..., 5, therefore determine the constants uniquely. The geometric construction of a conic section from five points lying on it is called the Braikenridge-Maclaurin Construction. The explicit equation for this conic is given by the equation

|x^2 xy y^2 x y 1; x_1^2 x_1y_1 y_1^2 x_1 y_1 1; x_2^2 x_2y_2 y_2^2 x_2 y_2 1; x_3^2 x_3y_3 y_3^2 x_3 y_3 1; x_4^2 x_4y_4 y_4^2 x_4 y_4 1; x_5^2 x_5y_5 y_5^2 x_5 y_5 1|=0.

(8)

The general equation of a conic section in trilinear coordinates is

(9)

(Kimberling 1998, p. 234). For five points specified in trilinear coordinates , the conic section they determine is given by

|alpha^2 beta^2 gamma^2 betagamma gammaalpha alphabeta; alpha_1^2 beta_1^2 gamma_1^2 beta_1gamma_1 gamma_1alpha_1 alpha_1beta_1; alpha_2^2 beta_2^2 gamma_2^2 beta_2gamma_2 gamma_2alpha_2 alpha_2beta_2; alpha_3^2 beta_3^2 gamma_3^2 beta_3gamma_3 gamma_3alpha_3 alpha_3beta_3; alpha_4^2 beta_4^2 gamma_4^2 beta_4gamma_4 gamma_4alpha_4 alpha_4beta_4; alpha_5^2 beta_5^2 gamma_5^2 beta_5gamma_5 gamma_5alpha_5 alpha_5beta_5|=0.

(10)

(Kimberling 1998, p. 235).

Two conics that do not coincide or have an entire straight line in common cannot meet at more than four points (Hilbert and Cohn-Vossen 1999, pp. 24 and 160). There is an infinite family of conics touching four lines. However, of the eleven regions into which plane division cuts the plane, only five can contain a conic section which is tangent to all four lines. Parabolas can occur in one region only (which also contains ellipses and one branch of hyperbolas), and the only closed region contains only ellipses.

Let a polygon of sides be inscribed in a given conic, with the sides of the polygon being termed alternately "odd" and "even" according to some definite convention. Then the points where an odd side meet a nonadjacent even side lie on a curve of order (Evelyn et al. 1974, p. 30).

REFERENCES:

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.

Chasles, M. Traité des sections coniques. Paris, 1865.

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

Coxeter, H. S. M. "Conics" §8.4 in Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115-119, 1969.

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

Downs, J. W. Practical Conic Sections. Palo Alto, CA: Dale Seymour, 1993.

Evelyn, C. J. A.; Money-Coutts, G. B.; and Tyrrell, J. A. The Seven Circles Theorem and Other New Theorems. London: Stacey International, p. 30, 1974.

Hilbert, D. and Cohn-Vossen, S. "The Cylinder, the Cone, the Conic Sections, and Their Surfaces of Revolution." §2 in Geometry and the Imagination. New York: Chelsea, pp. 7-11, 1999.

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

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Klein, F. "Famous Problems of Elementary Geometry: The Duplication of the Cube, the Trisection of the Angle, and the Quadrature of the Circle." In Famous Problems and Other Monographs. New York: Chelsea, pp. 42-44, 1980.

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

Lebesgue, H. Les Coniques. Paris: Gauthier-Villars, 1955.

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, 1960.

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

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

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 238-240, 1999.

Weisstein, E. W. "Books about Conic Sections." http://www.ericweisstein.com/encyclopedias/books/ConicSections.html.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 175, 1991.

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

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Weisstein, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConicSection.html