[[Image:Punktkoordinaten.PNG|thumb|Cartesian coordinates.]]

'''Analytic geometry''', also called '''coordinate geometry''' and earlier referred to as '''Cartesian geometry''' or '''analytical geometry''', is the study of [[geometry]] using the principles of [[algebra]]. That the algebra of the [[real numbers]] can be employed to yield results about the linear continuum of geometry relies on the [[Cantor-Dedekind axiom]]. Usually the [[Cartesian coordinate system]] is applied to manipulate [[equation]]s for [[Plane (mathematics)|plane]]s, [[line]]s, straight lines, and [[Square (geometry)|square]]s, 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 (spatial)|vector]] or a [[geometric shape|shape]]. Some consider that the introduction of analytic geometry was the beginning of modern [[mathematics]].

==History==
The Greek mathematician [[Menaechmus]] solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had analytic geometry.<ref>{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second Edition |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0471543977|chapter=The Age of Plato and Aristotle|pages=94-95|quote=Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a strong resemblance to the use of coordinates, as illustrated above, it has sometimes been maintained that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. It was shortcomings in algebraic notations that, more than anything else, operated against the Greek achievement of a full-fledged coordinate geometry.}}</ref> [[Apollonius of Perga]], in ''[[Apollonius of Perga#De Sectione Determinata|On Determinate Section]]'' dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others.<ref>{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second Edition |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0471543977|chapter=Apollonius of Perga |pages=142|quote=The Apollonian treatise ''On Determinate Section'' dealt with what might be called an analytic geometry of one dimension. It considered the following general problem, using the typical Greek algebraic analysis in geometric form: Given four points A, B, C, D on a straight line, determine a fifth point P on it such that the rectangle on AP and CP is in a given ratio to the rectangle on BP and DP. Here, too, the problem reduces easily to the solution of a quadratic; and, as in other cases, Apollonius treated the question exhaustively, including the limits of possibility and the number of solutions.}}</ref> Apollonius in the ''Conics'' further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve ''a posteriori'' instead of ''a priori''. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.<ref>{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second Edition |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0471543977|chapter=Apollonius of Perga |pages=156|quote=The method of Apollonius in the ''Conics'' in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not essentially different from the use fo a coordinate frame, whether rectangular or, more generally, oblique. Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed ''a posteriori'' upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down ''a priori'' for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed. Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions derived from a specific geometric situation; [...] That Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases.}}</ref>

The [[eleventh century]] [[Persian Empire|Persian]] mathematician [[Omar Khayyám]] saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra<ref name="Boyer Omar Khayyam positive roots"/> with his geometric solution of the general [[cubic equation]]s,<ref>Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", ''The Journal of the American Oriental Society'' '''123'''.</ref> but the decisive step came later with Descartes.<ref name="Boyer Omar Khayyam positive roots">{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=The Arabic Hegemony|pages=241-242|quote=Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an ''Algebra'' that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the sixteenth century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."}}</ref>

Analytic geometry has traditionally been attributed to [[René Descartes]]<ref name="Boyer Omar Khayyam positive roots"/><ref>{{cite book|first=John|last=Stillwell|authorlink=John Stillwell|title=Mathematics and its History |edition=Second Edition |publisher=Springer Science + Business Media Inc.|year=2004|chapter=Analytic Geometry|pages=105|isbn=0387953361|quote=the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments.}}</ref><ref>{{cite book|first=Roger |last=Cooke |authorlink=Roger Cooke |title=The History of Mathematics: A Brief Course|publisher=Wiley-Interscience |year=1997 |chapter=The Calculus |pages=326 |isbn=0471180823 |quote=The person who is popularly credited with being the discoverer of analytic geometry was the philosopher René Descartes (1596-1650), one of the most influential thinkers of the modern era.}}</ref> who made significant progress with the methods of analytic geometry when in [[1637]] in the appendix entitled ''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 [[French language|French]] tongue, and its philosophical principles, provided the foundation for [[calculus]] in Europe.

[[Abraham de Moivre]] also pioneered the development of analytic geometry. With the assumption of the [[Cantor-Dedekind axiom]], essentially that Euclidean geometry is [[interpretability|interpretable]] in the language of analytic geometry (that is, every theorem of one is a theorem of the other), [[Alfred Tarski|Alfred Tarski's]] proof of the [[decidability (logic)|decidability]] of the ordered real field could be seen as a proof that Euclidean geometry is consistent and [[decidability (logic)|decidable]].

==Themes==
Important themes of analytical geometry are
* [[vector space]]
* definition of the [[plane (mathematics)|plane]]
* [[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)|intersection]] problems

Many of these problems involve [[linear algebra]].

== Example ==

Here an example of a problem from the [[United States of America Mathematical Talent Search]] that can be solved via analytic geometry:

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

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

Using the [[midpoint]] formula, the points <math>F</math>, <math>G</math>, <math>H</math>, <math>I</math>, <math>X</math>, and <math>Y</math> are located at

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

Using the [[distance]] formula,

:<math>AE=\sqrt{d^2+g^2}</math>

and

:<math>XY=\sqrt{\frac{d^2}{16}+\frac{g^2}{16}}=\frac{\sqrt{d^2+g^2}}{4}.</math>

Since <math>XY</math> has to be an [[integer]],
:<math>AE\equiv 0\pmod{4}</math>
(see [[modular arithmetic]]) so <math>AE=4</math>.

==Other uses==

'''Analytic geometry''', for [[algebraic geometry|algebraic geometers]], is also the name for the theory of (real or) [[complex manifold]]s and the more general '''analytic spaces''' defined locally by the vanishing of [[analytic function]]s 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]]''.

==References==
{{reflist}}

[[Category:Geometry]]
[[Category:Algebraic geometry]]
[[Category:Mathematics education]]

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