Analytic geometry

Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of .

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 language), and its philosophical principles, provided the foundation for calculus in Europe.

=Important themes of analytical geometry=

vector space

definition of the plane (mathematics)

distance problems

the dot product, to get the angle of two vectors

the cross product, to get a perpendicular vector of two known vectors (and also their spatial volume)

intersection (set theory) problems

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

:Fleft(frac{a}{2},0 ight), Gleft(frac{a+b}{2},frac{e}{2} ight), Hleft(frac{b+c}{2},frac{e+f}{2} ight), Ileft(frac{c+d}{2},frac{f+g}{2} ight), Xleft(frac{a+b+c}{4},frac{e+f}{4} ight), and Yleft(frac{a+b+c+d}{4},frac{e+f+g}{4} ight).

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, :AEequiv 0pmod{4} (see modular arithmetic) so AE=4.

=Other uses=

Analytic geometry, for algebraic geometry, 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.