Analytic geometry

• 1. Introduction

  ›

  Analytic geometry begins with a deceptively simple idea: every point in the plane can be named by two numbers. That pairing of algebra and shape is so fundamental that it now underlies algebraic, differential, discrete, and computational geometry alike. It also reaches far beyond mathematics into physics, engineering, aviation, rocketry, space science, economics, and the social sciences. But how did this fusion of number and form come to exist? The answer stretches from ancient Greece through medieval Persia to seventeenth-century Europe, and it hinges on a dispute about which comes first: the curve, or the equation that describes it.

• 2. Ancient Roots

  ›

  Menaechmus, the Greek mathematician, solved problems and proved theorems using a method that bore a strong resemblance to the use of coordinates, and scholars have sometimes credited him with introducing analytic geometry. Apollonius of Perga pushed further. In his work On Determinate Section, he tackled problems in what amounts to an analytic geometry of one dimension, finding points on a line held in a fixed ratio to others.

  In the Conics, Apollonius developed methods so close to modern analytic geometry that his work is sometimes said to have anticipated Descartes by roughly eighteen hundred years. He used a diameter and a tangent as reference lines, measuring distances along the diameter from the point of tangency as abscissas, and segments parallel to the tangent as ordinates. He even developed relations between abscissas and their corresponding ordinates that are equivalent to what we would call equations of curves, though he expressed them in words rather than symbols.

  Yet Apollonius fell short of the full invention. He never accounted for negative magnitudes, and in every case he superimposed a coordinate system onto a curve that was already given, rather than letting an equation determine the curve in advance. Equations followed curves; curves did not follow equations. That reversal was still centuries away.

• 3. Omar Khayyam's Bridge

  ›

  The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and worked to close the gap between them. His geometric solution of the general cubic equations moved the field in the right direction. Khayyam is credited with identifying the foundations of algebraic geometry.

  His book Treatise on Demonstrations of Problems of Algebra, written in 1070, laid down principles that would become part of analytic geometry. That text was part of the body of Persian mathematics eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam stands as a direct precursor to Descartes. The decisive step, however, came later.

• 4. Descartes and Fermat

  ›

  Analytic geometry was independently invented by Rene Descartes and Pierre de Fermat, though Descartes is sometimes given sole credit. The very name Cartesian geometry, the common alternative term for the field, is named after Descartes.

  Descartes set out his ideas in an essay titled La Geometrie, published in 1637 as one of three appendices accompanying his Discourse on Method. Written in his native French, the work provided a foundation for calculus in Europe. It was not well received at first. The many gaps in its arguments and its complicated equations put readers off. Only after van Schooten translated it into Latin in 1649, and added commentary, did the work receive the recognition it deserved.

  Fermat's contribution circulated in manuscript form in Paris in 1637, just before Descartes published. His Introduction to Plane and Solid Loci was clearly written and well received, and it too laid the groundwork for analytical geometry. The two men approached the subject from opposite ends. Fermat always started with an algebraic equation and then described the geometric curve that satisfied it. Descartes started with geometric curves and derived their equations. Because Descartes worked from curves to equations, he had to contend with more complicated polynomial equations of higher degree and had to develop the methods to handle them. It was Leonhard Euler who later first applied the coordinate method in a systematic study of space curves and surfaces.

• 5. The Language of Coordinates

  ›

  In the Cartesian coordinate system, every point in the plane is identified by an ordered pair of real numbers, one for its horizontal position and one for its vertical position. In three-dimensional space the same logic extends to an ordered triple. The Cartesian system is the most common, but it is not the only one.

  Polar coordinates represent each point by its distance r from the origin and its angle from the positive x-axis, measured counterclockwise. Cylindrical coordinates add a height z to that scheme, while spherical coordinates describe every point in space using a distance from the origin and two angles. The choice of system depends on the geometry of the problem at hand.

• 6. Curves, Equations, Sections

  ›

  In analytic geometry, any equation involving coordinates specifies a subset of the plane called a locus. The equation y equals x, for instance, describes the set of all points where the two coordinates are equal, which traces a straight line. Linear equations in x and y always specify lines; quadratic equations specify conic sections.

  The conic sections appear in the Cartesian system as the graphs of quadratic equations in two variables. Whether such an equation represents an ellipse, a parabola, a hyperbola, or a circle depends on the discriminant of that equation. A circle is a special case of an ellipse. In three dimensions, a single equation typically yields a surface rather than a curve, and the analogous objects are quadric surfaces: ellipsoids, paraboloids, hyperboloids, cylinders, cones, and planes.

  Distance and angle within this framework are defined by formulas that stay consistent with underlying Euclidean geometry. The distance between two points follows a version of the Pythagorean theorem. The angle a line makes with the horizontal is tied directly to its slope. In three dimensions the dot product of two vectors encodes the angle between them, defined precisely as the product of their magnitudes and the cosine of the angle between them.

• 7. Intersections and Tangents

  ›

  Finding where two geometric objects meet is one of the central tasks of analytic geometry. For two objects represented by relations, the intersection is simply the set of all points satisfying both relations simultaneously. Traditional methods for solving such systems include substitution and elimination. For conic sections, as many as four points might lie in a single intersection.

  The tangent line to a curve at a given point is the straight line that just touches the curve there, passing through the point and carrying the slope given by the derivative at that point. It is the best straight-line approximation to the curve at the point of tangency. The same idea extends to surfaces: the tangent plane to a surface at a point is the plane that just touches the surface there. The concept of a tangent is one of the most fundamental notions in differential geometry, and its generalisation leads to the idea of a tangent space, a construction that carries the coordinate geometry of Descartes and Fermat into the mathematics of curved spaces that underpin modern physics.