ANALYTIC GEOMETRY

also called Coordinate Geometry, the mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry. The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa; the methods of either subject can then be used to solve problems in the other. Many mathematicians of ancient times were aware that the geometry of figures was related to the algebra of numbers. Even the Greeks were limited, however, by the primitive state of algebraic symbolism and procedures and by their view of mathematics as being tied to and representative of the physical world. For example, the Greeks thought of ordinary numbers in terms of line segments, the product of two numbers in terms of areas, and the product of three numbers in terms of volumes. Since lengths, areas, and volumes are the only three types of geometric measurements inherent in the physical world, the Greeks were unable to consider the geometric equivalents of other types of algebraic relationships such as y = x4. It was only when algebra had become a more complete and useful subject in its own right, and mathematics had moved away from a complete dependence on and relationship to the physical world, that the possibility of establishing a fruitful correspondence between geometry and algebra became evident. Analytic geometry was established in France in the 17th century by Ren Descartes and Pierre de Fermat, who independently pointed out the correspondence between ordered pairs of real numbers and the distances of a point from two intersecting lines in the plane, called the axes, or coordinate axes. Once the axes are selected, every geometric point has a unique representation by an ordered pair of real numbers (x,y) and, conversely, every ordered pair of real numbers represents a unique geometric point. Modern analytic geometry employs axes that are perpendicular to each other; this type of coordinate system, and the coordinates (x,y) themselves, are called Cartesian, after Ren Descartes. The correspondence established between points in a plane and ordered pairs of real numbers can easily be extended to a correspondence between points in three-dimensional space and ordered triples of real numbers (x,y,z) by using a three-dimensional Cartesian coordinate system. It is perfectly reasonable to consider and work with ordered sets of as many real numbers as is desired. Therefore, the methods of analytic geometry allow mathematicians to study the theoretical properties of spaces with dimensions greater than three, even though such spaces cannot exist in the real world. The coordinate systems developed by Descartes and Fermat are not the only ones possible; one of the most useful alternatives is the polar coordinate system developed by Sir Isaac Newton. In this system any point A in the plane can be represented by its distance r from a reference point O and by the angle q between the half-line from O through A and a reference direction; the ordered pair (r,q) specifies the polar coordinates of the point. Certain geometric curves have much simpler algebraic equivalents when expressed in polar coordinates than when expressed in Cartesian coordinates: an example is the logarithmic spiral. In Cartesian coordinates the equation for this curve would be with a an arbitrary constant. In polar coordinates the equation is the much simpler r = aq. also called coordinate geometry mathematical subject in which algebraic symbolism and methods are used to represent and solve problems in geometry. The importance of analytic geometry is that it establishes a correspondence between geometric curves and algebraic equations. This correspondence makes it possible to reformulate problems in geometry as equivalent problems in algebra, and vice versa; the methods of either subject can then be used to solve problems in the other. After the work of David Hilbert, at the turn of the 20th century, the foundations of geometry were generalized and the classical concepts of space and objects in space, which derived from intuition, were replaced with abstract ideas. A step toward the generalization of classical geometry was taken when analytical geometry was created and first used in 1637 by Ren Descartes. Descartes applied algebra to geometry not just in the use of algebra to manipulate the dimensions of geometric figures but also in the representation of a point by a pair of numbers and the representation of lines and curves by equations. It was a powerful general method of solving certain geometric problems and one that could be applied to certain types of curves more readily than the geometry of the Greeks that was based on axioms. The basis of analytic geometry is the idea that a point in space can be specified by numbers giving its position. The notion that any point, for example, can be indicated by its latitude, longitude, and height above the Earth goes back to Archimedes and to Apollonius of Perga, who lived in the 3rd century BC. It was Descartes and another 17th-century Frenchman, Pierre de Fermat, however, who developed that notion systematically. The idea of negative distances is due to Sir Isaac Newton and Gottfried Wilhelm Leibniz, in the 17th century. Additional reading Studies of the topic include D.M.Y. Sommerville, Analytical Geometry of Three Dimensions (1934, reprinted 1959); A. Robson, An Introduction to Analytical Geometry (1940); N.V. Efimov, An Elementary Course in Analytical Geometry, 2 vol. (1966; originally published in Russian, 1950); and Marcel Berger, Geometry, 2 vol. (1987; originally published in French, 1977), a well-regarded and copiously illustrated treatment of many topics in geometry from an analytic viewpoint. H.F. Baker, Principles of Geometry, vol. 2, 2nd ed., (1930); Luther Pfahler Eisenhart, Coordinate Geometry (1939, reissued 1966); and Ross R. Middlemiss, John L. Marks, and James R. Smart, Analytic Geometry, 3rd ed. (1968), are three general references on conic sections. The history of the conic sections is treated in Julian Lowell Coolidge, A History of the Conic Sections and Quadric Surfaces (1945, reissued 1968). Information on special curves is featured in Julian Lowell Coolidge, A Treatise on Algebraic Plane Curves (1931, reprinted 1959); Robert C. Yates, A Handbook on Curves and Their Properties, rev. ed. (1952, reprinted 1974); J. Dennis Lawrence, A Catalog of Special Plane Curves (1972); and Egbert Brieskorn and Horst Knrrer, Plane Algebraic Curves (1986; originally published in German, 1981). A considerable amount of information on special curves may be found in books on analytic geometry, calculus, differential equations, the calculus of variations, and also mechanics. Raymond Clare Archibald Nathan Altshiller Court Ralph G. Sanger The Editors of the Encyclopdia Britannica Projective and solid analytic geometry Analytic projective geometry The 16th17th-century German astronomer Johannes Kepler conceived the idea of extending the Euclidean (or affine) plane by postulating a line at infinity the points of which lie on collections of parallel lines called pencils. The plane so extended is called the projective plane. In terms of barycentric coordinates, the line at infinity has the equation restricting the {tk} to equal 0 (see 108). General projective coordinates (x1, x2, x3) are derived by the substitution ti = kixi, in which the coefficients ki are three constants. The line at infinity is now expressed in terms of a linear combination of coordinates (see 109) and any other homogeneous linear equation represents an ordinary line. The transition to projective geometry is completed by waiving the distinction, so that a point at infinity is treated just like any other point. This vital step was taken by another 19th-century German mathematician, Karl Georg Christian von Staudt. It enables the points of the projective plane to be defined as the ordered triads of numbers (x1, x2, x3), not all zero, with the convention that (kx1, kx2, kx3) is the same point for all nonzero values of k (see below Projective geometry). Given two points (x1, x2, x3) and (y1, y2, y3), or more concisely, (x) and (y), an arbitrary point collinear with them may be expressed in the form ( x + ty) (see 110). In this notation, two triangles in perspective from (u) may be expressed in the concise notation (see 111). Corresponding sides of the triangles meet in three collinear points (see 112) in agreement with the two-triangle theorem of the 17th-century French mathematician Girard Desargues. The general homogeneous linear transformation xi = Scijxj (summed over j), in which det(cij) 0, transforms collinear points into collinear points; i.e., it is a collineation. For instance, the collineation that leaves unchanged the first two coordinates and modifies the third with multiplication by a constant (see 113) is a homology that leaves invariant every line through the point (0, 0, 1) and every point on the line x3 = 0. Solid analytic geometry Much of analytic geometry may be extended from two dimensions to three or more. In ordinary space a point has three Cartesian coordinates (x, y, z), a plane has a linear equation (see 114), and a line may be specified in various ways; for example, as the intersection of two planes or as the join of two points or as proceeding from a given point in a given direction. The last aspect yields parametric equations that are linear in the parameter and in the coordinates (see 115). Polar coordinates have two spatial counterparts: cylindrical coordinates (r, q, z), in which r and q are related to x and y in the usual way; and spherical polar coordinates, which consist of distance from the origin, latitude (or colatitude), and longitude. Barycentric (and also other projective) coordinates are referred to a tetrahedron P1P2P3P4.