Analytic geometry

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. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. 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. In Mathematics, the real numbers may be described informally in several different ways The phrase Cantor-Dedekind axiom has been used to describe the thesis that the Real numbers are order- Isomorphic to the linear continuum of Geometry Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, straight lines, and squares, often in two and sometimes in three dimensions of measurement. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Classification A square (regular Quadrilateral) is a special case of a Rectangle as it has four right angles and equal parallel sides 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. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined Some consider that the introduction of analytic geometry was the beginning of modern mathematics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

1 History
2 Themes
3 Example
4 Other uses
5 References
6 External links

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. There is also a Menaechmus in Plautus ' play The Menaechmi. Menaechmus (Μέναιχμος 380 – 320 BC was a Greek ^[1] 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. ^[2] 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. ^[3]

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^[4] with his geometric solution of the general cubic equations,^[5] but the decisive step came later with Descartes. The Persian Empire was a series of Iranian empires that ruled over the Iranian plateau, the original Persian homeland and beyond in Western Asia For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. ^[4]

Analytic geometry has traditionally been attributed to René Descartes^[4]^[6]^[7] 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. La Géométrie was published in 1637 as an appendix to Discours de la méthode ( Discourse on Method) written Organization How to think correctly The Method of Science Morals Maxims deduced from this Method Proof of God and the Soul Physics the heart This work, written in his native French tongue, and its philosophical principles, provided the foundation for calculus in Europe. French ( français,) is a Romance language spoken around the world by 118 million people as a native language and by about 180 to 260 million people Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Abraham de Moivre also pioneered the development of analytic geometry. "Moivre" redirects here for the French commune see Moivre Marne. 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. The phrase Cantor-Dedekind axiom has been used to describe the thesis that the Real numbers are order- Isomorphic to the linear continuum of Geometry The concept of interpretability is one in Mathematical logic. Alfred Tarski ( January 14, 1901, Warsaw, Russian ruled Poland – October 26, 1983, Berkeley California In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas In Logic, the term decidable refers to the existence of an Effective method for determining membership in a set of formulas

Themes

Important themes of analytical geometry are

vector space
definition of the 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 problems

Many of these problems involve linear algebra. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Distance is a numerical description of how far apart objects are In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently Linear algebra is the branch of Mathematics concerned with

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 $A B C D E$ , the sides have lengths $1$ , $2$ , $3$ , $4$ , and $5$ , though not necessarily in that order. The United States of America Mathematical Talent Search ( USAMTS) is a Mathematics competition open to all United States students in or below High school Let $F$ , $G$ , $H$ , and $I$ be the midpoints of the sides $A B$ , $B C$ , $C D$ , and $D E$ , respectively. Let $X$ be the midpoint of segment $F H$ , and $Y$ be the midpoint of segment $G I$ . The length of segment $X Y$ is an integer. Find all possible values for the length of side $A E$ .

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 $X Y$ has to be an integer,

$AE\equiv 0\pmod{4}$

(see modular arithmetic) so $A E = 4$ . The midpoint (also known as class mark in relation to Histogram) is the middle point of a Line segment. Distance is a numerical description of how far apart objects are The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers

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). Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, This article is about both real and complex analytic functions The theory of functions of several complex variables is the branch of Mathematics dealing with functions f ( z1 z2 It is closely linked to algebraic geometry, especially through the work of Jean-Pierre Serre in GAGA. In Mathematics, algebraic geometry and analytic geometry are two closely related subjects

References

^ Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "The Age of Plato and Aristotle", A History of Mathematics, Second Edition, John Wiley & Sons, Inc. , 94-95. ISBN 0471543977. “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. ”
^ Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Apollonius of Perga", A History of Mathematics, Second Edition, John Wiley & Sons, Inc. , 142. ISBN 0471543977. “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. ”
^ Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "Apollonius of Perga", A History of Mathematics, Second Edition, John Wiley & Sons, Inc. , 156. ISBN 0471543977. “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. ”
^ ^a ^b ^c Boyer (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "The Arabic Hegemony", , 241-242. “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. "”
^ Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", The Journal of the American Oriental Society 123.
^ Stillwell, John (2004). John Stillwell (born 1942 is an Australian Mathematician on the faculties of the University of San Francisco and Monash University. "Analytic Geometry", Mathematics and its History, Second Edition, Springer Science + Business Media Inc. , 105. ISBN 0387953361. “the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments. ”
^ Cooke, Roger (1997). "The Calculus", The History of Mathematics: A Brief Course. Wiley-Interscience, 326. ISBN 0471180823. “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. ”

External links

Build analytic geometry objects

Dictionary

-noun

(geometry) a branch of mathematics that investigates properties of figures through the coordinates of their points.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents