Analytic geometry

Article By:

Blumenthal, Leonard M. Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.

Last reviewed:October 2019

DOI:https://doi.org/10.1036/1097-8542.032000

A branch of mathematics in which algebra is applied to the study of geometry. Because algebraic methods were first systematically applied to geometry in 1637 by the French philosopher-mathematician René Descartes, the subject is also called Cartesian geometry. The basis for an algebraic treatment of geometry is provided by the existence of a one-to-one correspondence between the elements, “points” of a directed line g, and the elements, “numbers,” that form the set of all real numbers. Such a correspondence establishes a coordinate system on g, and the number corresponding to a point of g is called its coordinate. The point O of g with coordinate zero is the origin of the coordinate system. A coordinate system on g is Cartesian provided that for each point P of g, its coordinate is the directed distance $\overline{\mathrm{OP}}$ . Then all points of g on one side of O have positive coordinates (forming the positive half of g) and all points on the other side have negative coordinates. The point with coordinate 1 is called the unit point. Since the relation $\overline{\mathrm{OP}}$ + $\overline{\mathrm{PQ}}$ = $\overline{\mathrm{OQ}}$ is clearly valid for each two points P, Q, of directed line g, then $\overline{\mathrm{PQ}}$ = $\overline{\mathrm{OQ}}$ − $\overline{\mathrm{OP}}$ = q − p, where p and q are the coordinates of P and Q, respectively. Those points of g between P and Q, together with P, Q, form a line segment. In analytic geometry it is convenient to direct segments, writing PQ or QP accordingly as the segment is directed from P to Q or from Q to P, respectively. To find the coordinate of the point P that divides the segment P₁ P₂ in a given ratio r, put $\frac{\overline{P{}_{\text{1}}P}}{\overline{P{}_{\text{2}}P}}$ = r. Then (x − x₁)/(x − x₂) = r, where x₁, x₂, x are the coordinates of P₁, P₂, P, respectively, and solving for x gives x = (x₁ − rx₂)/(1 − r). Clearly r is negative for each point between P₁, P₂ and is positive for each point of g external to the segment. The midpoint of the segment divides it in the ratio −1, and hence its coordinate x = (x₁ + x₂)/2. See also: Mathematics

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