conic section(redirected from Conic Sections in Polar Coordinates)
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conic(kŏn`ĭk), curve formed by the intersection of a plane and a right circular conecone
or conical surface,
in mathematics, surface generated by a moving line (the generator) that passes through a given fixed point (the vertex) and continually intersects a given fixed curve (the directrix).
..... Click the link for more information. (conical surface). The ordinary conic sections are the circlecircle,
closed plane curve consisting of all points at a given distance from some fixed point, called the center. A circle is a conic section cut by a plane perpendicular to the axis of the cone.
..... Click the link for more information. , the ellipseellipse,
closed plane curve consisting of all points for which the sum of the distances between a point on the curve and two fixed points (foci) is the same. It is the conic section formed by a plane cutting all the elements of the cone in the same nappe.
..... Click the link for more information. , the parabolaparabola
, plane curve consisting of all points equidistant from a given fixed point (focus) and a given fixed line (directrix). It is the conic section cut by a plane parallel to one of the elements of the cone.
..... Click the link for more information. , and the hyperbolahyperbola
, plane curve consisting of all points such that the difference between the distances from any point on the curve to two fixed points (foci) is the same for all points.
..... Click the link for more information. . When the plane passes through the vertex of the cone, the result is a point, a straight line, or a pair of intersecting straight lines; these are called degenerate conic sections. There are many examples of the conic sections, both in nature and in technology. The orbits of planets and satellites are elliptical, and parallel reflectors (e.g., in telescopes) are parabolic in shape.
a curve obtained by cutting a right circular cone by planes that do not pass through its vertex.
There are three types of conic sections. (1) The cutting plane intersects all the generators of the cone at points of just one of its nappes (Figure l,a). The curve of intersection is a closed oval curve called an ellipse; a circle, a particular case of an ellipse, is obtained when the cutting plane is perpendicular to the axis of the cone. (2) The cutting plane is parallel to one of the tangent planes of the cone (Figure l,b). The curve of intersection is a nonclosed curve extending to infinity, called a parabola, lying entirely in one nappe. (3) The cutting plane intersects both nappes of the cone (Figure l,c). The curve of intersection, called a hyperbola, consists of two identical nonclosed parts (branches of the hyperbola) extending to infinity, one in each nappe of the cone.
From the standpoint of analytic geometry, conic sections are real irreducible curves of the second degree. When a conic section has a center of symmetry (center), that is, it is an ellipse or hyperbola, its equation can be reduced (by shifting the origin to the center) to the form
a11x2 + 2a12xy + a22Y2 = a33
Further investigation of these (central) conic sections shows that their equations can be reduced to a still simpler form:
(1) Ax2 + BY2 = C
by taking their principal axes (axes of symmetry) as the coordinate axes. If A and B have the same sign (coinciding with the sign of C), then equation (1) defines an ellipse; if A and B have different signs, then the equation defines a hyperbola.
The equation of a parabola cannot be reduced to the form (1). By means of a suitable selection of the coordinate axes (one axis being the unique axis of symmetry of the parabola, while the other axis is a line perpendicular to it and passing through the vertex of the parabola), its equation can be reduced to the form
Y2 = 2px
Ancient Greek mathematicians (such as Menaechmus, fourth century B.C.) already knew of the conic sections. Certain construction problems (doubling of a cube, for example) that could not be solved by means of the simplest drawing instruments, such as a compass and straightedge, were solved by means of these curves. In the earliest studies that have come down to us, Greek geometers had obtained conic sections as curves of intersection of a cone and a plane perpendicular to a generator of the cone. Depending on the vertex angle of the cone (that is, the largest angle between the generating rays of one nappe), the curve of intersection proved to be an ellipse if the angle was acute, a parabola if it was a right angle, and a hyperbola if it was obtuse. The most complete treatise devoted to these curves was Conic Sections by Apollonius of Perga (c. 200 B.C.). Further advances in the theory of conic sections are due to the creation of new geometric methods in the 17th century, that of projective geometry (the French mathematicians G. Desarques and B. Pascal) and, particularly, coordinate geometry (the French mathematicians R. Descartes and P. Fermat).
Relative to a suitable coordinate system, the equation of a conic section takes the form
Y2 = 2px + λx2
where p and λ are constants. If p λ 0, then this equation defines a parabola when λ = 0, an ellipse when λ < 0, and a hyperbola when λ > 0. The geometric characterization of the conic sections described by the last equation was already well known to ancient Greek geometers and was used by Apollonius of Perga as a basis for assigning the names that are still used for the different types of conic sections. Thus, the term “parabola” (from the Greek parabole) means application (since in Greek geometry the transformation of a rectangle of given area y2 into a rectangle of equal area and with given base 2p is called the application of this rectangle to this base), the term “ellipse” (from the Greek elleipsis) means defect (application with defect), and the term “hyperbola” (from the Greek hyperbole) means excess (application with excess).
In the transition to modern methods of investigation, the stereometric definition of the conic sections was replaced by planimetric definitions of these curves as geometric loci in the plane. For example, an ellipse is defined as the locus of points the sum of whose distances from two given points (foci) has a given value.
Another planimetric definition of conic sections that encompasses all the three types of these curves is possible: A conic section is the locus of points such that the ratio of the distances of any point from a given point (the focus) and a given line (the directrix) is equal to a given positive number (the eccentricity) e. If e < 1, the conic section is an ellipse; if e > 1, it is a hyperbola; and if e = 1, it is a parabola.
Interest in the conic sections has been sustained owing to their occurrence in different phenomena of nature and in human activity. In science, conic sections acquired special importance after the German astronomer J. Kepler, who relied on observations, and the British scientist I. Newton, who relied on theoretical deduction, discovered the laws of planetary motion. One of these laws asserts that the planets and comets of the solar system move along conic sections, at one of whose foci is the sun. The following examples pertain to the different types of conic sections: a projectile or stone thrown at an angle to the horizon traces a parabola (the true form of the curve is somewhat distorted by the resistance of the air); elliptically shaped gears are used in certain mechanisms; the hyperbola serves as the graph of the relation of inverse proportionality that is often observed in nature (for example, Boyle’s law [Marriotte’s law]).
REFERENCESAleksandrov, P. S. Lektsii po analiticheskoi geometrii. Moscow, 1968.
Van der Waerden, B. L. Probuzhdaiushchaiasia nauka. (Translated from Dutch.) Moscow, 1959.
V. I. BITIUTSKOV