Analytic geometry turns shapes into equations on axes
Also called coordinate or Cartesian geometry, the field studies figures with a coordinate system instead of purely synthetic constructions. It underpins algebraic, differential, discrete, and computational geometry and serves physics, engineering, aviation, rocketry, statistics, and economics. Descartes and Fermat invented it independently in the seventeenth century.
Analytic geometry assigns coordinates so geometric questions become algebraic. School versions emphasize representing shapes numerically and reading numerical facts from those representations, usually with Cartesian axes for lines, planes, and circles in two or three dimensions. Greek foreshadowing appears in Menaechmus's coordinate-like methods and in Apollonius of Perga's Conics, whose diameters and tangents anticipate Descartes by nearly eighteen centuries. Omar Khayyam linked geometry to algebra in the eleventh century with geometric solutions of cubics and is credited with foundations of algebraic geometry, yet the decisive step waited for Descartes.
René Descartes and Pierre de Fermat invented the subject independently; Cartesian geometry takes Descartes's name. Descartes published methods in La Géométrie (1637), an appendix to the Discourse on Method. Written in French with gaps and heavy equations, it was poorly received until van Schooten's Latin translation and commentary of 1649. Fermat's Introduction to Plane and Solid Loci circulated in Paris in 1637 before Descartes's book; his viewpoint differed from Descartes's even while laying the same groundwork. The plane gets coordinates so each point is a real pair; space gets triples. Polar coordinates use radius and angle; cylindrical and spherical systems extend the idea to three dimensions.
An equation in the coordinates cuts out a locus—the solution set. Linear equations in x and y describe lines; quadratics describe conic sections; y = x is the line of equal coordinates; x squared + y squared = r squared is a circle centered at the origin. Trivial equations may fill the whole plane or a single point. In three dimensions one equation usually yields a surface; curves arise as intersections or parametric systems. Non-vertical lines often use slope-intercept form. Planes admit a point-normal equation: a point and a nonzero normal vector define the set of points whose connecting vector is orthogonal to the normal, written with a dot product or as ax + by + cz + d = 0.
Space lines usually need parametric equations with a direction vector parallel to the line rather than a single linear equation. Every quadratic Ax squared + Bxy + Cy squared + Dx + Ey + F = 0 with A, B, C not all zero graphs as a (possibly degenerate) conic, and every conic arises this way. Coordinate geometry thus became the numerical language that later fields of geometry still speak.
Source: Analytic geometry
