Analytic geometry

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File:Punktkoordinaten.PNG
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 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.

Contents

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>{{

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}}.{{ #if: 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. |  “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 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>{{

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}}.{{ #if: 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. |  “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>{{

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}}.{{ #if: 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. |  “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 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 equations,<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">{{

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}}.{{ #if: 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." |  “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>{{

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}}.{{ #if: the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments. |  “the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments.” }} </ref><ref>{{

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}}.{{ #if: 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. |  “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 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 interpretable in the language of analytic geometry (that is, every theorem of one is a theorem of the other), Alfred Tarski's proof of the decidability of the ordered real field could be seen as a proof that Euclidean geometry is consistent and decidable.

Important themes of analytical geometry

Many of these problems involve linear algebra

Example

Here 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.

References

<references />
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