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ANALYTIC GEOMETRY

Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining geometrical shapes in a numerical way, and extracting numerical information from that representation. The numerical output, however, might also be a vector or a shape. Some consider that the introduction of analytic geometry was the beginning of modern mathematics.

René Descartes is popularly regarded as having introduced the foundation for the methods of analytic geometry in 1637 in the appendix titled Geometry of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native language (French), and its philosophical principles, provided the foundation for calculus in Europe.

The relationship between algebra and geometry is nontrivial. There exist proofs of this relationship based on the theory of geometry and notions of geometric length.

Important themes of analytical geometry

Many of these problems involve linear algebra

Example

Here is an example of a problem from the USAMTS that can be solved via analytic geometry:

Problem: In a convex pentagon ABCDE, the sides have lengths 1, 2, 3, 4, and 5, though not necessarily in that order. Let F, G, H, and I be the midpoints of the sides AB, BC, CD, and DE, respectively. Let X be the midpoint of segment FH, and Y be the midpoint of segment GI. The length of segment XY is an integer. Find all possible values for the length of side AE.

Solution: Let A, B, C, D, and E be located at A(0,0), B(a,0), C(b,e), D(c,f), and E(d,g).

Using the midpoint formula, the points F, G, H, I, X, and Y are located at

F\left(\frac{a}{2},0\right), G\left(\frac{a+b}{2},\frac{e}{2}\right), H\left(\frac{b+c}{2},\frac{e+f}{2}\right), I\left(\frac{c+d}{2},\frac{f+g}{2}\right), X\left(\frac{a+b+c}{4},\frac{e+f}{4}\right), and Y\left(\frac{a+b+c+d}{4},\frac{e+f+g}{4}\right).

Using the distance formula,

AE=\sqrt{d^2+g^2}

and

XY=\sqrt{\frac{d^2}{16}+\frac{g^2}{16}}=\frac{\sqrt{d^2+g^2}}{4}.

Since XY has to be an integer,

AE\equiv 0\pmod{4}

(see modular arithmetic) so AE = 4.

Other uses

Analytic geometry, for algebraic geometers, is also the name for the theory of (real or) complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables (or sometimes real ones). It is closely linked to algebraic geometry, especially through the work of Jean-Pierre Serre in GAGA. It is strictly a larger area than algebraic geometry, but studied by similar methods.