Analytic geometry: Difference between revisions

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In [[mathematics]], '''analytic geometry''', also known as '''coordinate geometry''' or '''Cartesian geometry''', is the study of [[geometry]] using a [[coordinate system]]. This contrasts with [[synthetic geometry]].

Analytic geometry is used in [[physics]] and [[engineering]], and also in [[aviation]], [[Aerospace engineering|rocketry]], [[space science]], and [[~~spaceflight~~]]. It is the foundation of most modern fields of geometry, including [[Algebraic geometry|algebraic]], [[Differential geometry|differential]], [[Discrete geometry|discrete]] and [[computational geometry]].

Analytic geometry is used in [[physics]] and [[engineering]], and also in [[aviation]], [[Aerospace engineering|rocketry]], [[space science]], [[spaceflight]], [[statistics]], [[economics]], and the [[social science]]s. It is the foundation of most modern fields of geometry, including [[Algebraic geometry|algebraic]], [[Differential geometry|differential]], [[Discrete geometry|discrete]] and [[computational geometry]].

Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. 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]].

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===Polar coordinates (in a plane)===

In [[polar coordinates]], every point of the plane is represented by its distance ''r'' from the origin and its [[angle]] ''θ'', with ''θ'' normally measured counterclockwise from the positive ''x''-axis. Using this notation, points are typically written as an ordered pair (''r'', ''θ''). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: <math display="block">x = r\, \cos\theta,\, y = r\, \sin\theta; \, r = \sqrt{x^2+y^2},\, \theta = \arctan(y/x).</math> This system may be generalized to three-dimensional space through the use of [[Cylindrical coordinates|cylindrical]] or [[Spherical coordinates|spherical]] coordinates.

[[Image:Polära_koordinater.svg|thumb|Three [[Polar coordinate system#Vector calculus|vectors in the polar coordinate system]] all originating at the origin with separate angles <math>\theta</math> passing through the points <math>(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})</math>.]]

In [[polar coordinates]], every point of the plane is represented by its distance ''r'' from the origin and its [[angle]] ''θ'', with ''θ'' normally measured counterclockwise from the positive ''x''-axis. Using this notation, points are typically written as an ordered pair (''r'', ''θ''). One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae:<ref name=":0">{{Cite book |last=Wilson |first=Wallace Alvin |title=Analytic Geometry |last2=Tracey |first2=Joshua Irving |publisher=[[D.C. Heath and Company]] |year=1937 |edition=Alternate |location=[[Boston, New York]] |pages=153-155, 283-284 |language=en |lccn=37002786 |oclc=530055}}</ref> <math display="block">x = r\, \cos\theta,\, y = r\, \sin\theta; \, r = \sqrt{x^2+y^2},\, \theta = \arctan(y/x).</math> This system may be generalized to three-dimensional space through the use of [[Cylindrical coordinates|cylindrical]] or [[Spherical coordinates|spherical]] coordinates.<ref name=":0" />

===Cylindrical coordinates (in a space)===

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* {{citation|last=Cajori|first=Florian|title=A History of Mathematics|publisher=AMS|isbn=978-0821821022|year=1999}}

* [[John Casey (mathematician)|John Casey]] (1885) [https://archive.org/details/cu31924001520455 Analytic Geometry of the Point, Line, Circle, and Conic Sections], link from [[Internet Archive]].

* {{citation|last=Katz|first=Victor J.|title=A History of Mathematics: An Introduction (2nd Ed.)|publisher=Addison Wesley Longman |place=Reading|year=1998|isbn=0-321-01618-1|url=https://archive.org/details/historyofmathema00katz}}

* {{citation|last=Katz|first=Victor J.|author-link=Victor J. Katz|title=A History of Mathematics: An Introduction (2nd Ed.)|publisher=Addison Wesley Longman |place=Reading|year=1998|isbn=0-321-01618-1|url=https://archive.org/details/historyofmathema00katz}}

* [[Mikhail Postnikov]] (1982) [https://archive.org/details/postnikov-lectures-in-geometry-semester-i Lectures in Geometry Semester I Analytic Geometry] via Internet Archive

* {{citation|last=Struik|first=D. J.|title=A Source Book in Mathematics, 1200-1800|publisher=Harvard University Press |isbn=978-0674823556|year=1969}}

* {{citation|last=Struik|first=D. J.|author-link=Dirk Jan Struik|title=A Source Book in Mathematics, 1200-1800|publisher=Harvard University Press |isbn=978-0674823556|year=1969}}

===Articles===

@@ Line 4: / Line 4: @@
 In [[mathematics]], '''analytic geometry''', also known as '''coordinate geometry''' or '''Cartesian geometry''', is the study of [[geometry]] using a [[coordinate system]]. This contrasts with [[synthetic geometry]].
-Analytic geometry is used in [[physics]] and [[engineering]], and also in [[aviation]], [[Aerospace engineering|rocketry]], [[space science]], and [[spaceflight]]. It is the foundation of most modern fields of geometry, including [[Algebraic geometry|algebraic]], [[Differential geometry|differential]], [[Discrete geometry|discrete]] and [[computational geometry]].
+Analytic geometry is used in [[physics]] and [[engineering]], and also in [[aviation]], [[Aerospace engineering|rocketry]], [[space science]], [[spaceflight]], [[statistics]], [[economics]], and the [[social science]]s. It is the foundation of most modern fields of geometry, including [[Algebraic geometry|algebraic]], [[Differential geometry|differential]], [[Discrete geometry|discrete]] and [[computational geometry]].
 Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. 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]].
@@ Line 49: / Line 49: @@
 ===Polar coordinates (in a plane)===
 {{main|Polar coordinate system}}
-In [[polar coordinates]], every point of the plane is represented by its distance ''r'' from the origin and its [[angle]] ''θ'', with ''θ'' normally measured counterclockwise from the positive ''x''-axis. Using this notation, points are typically written as an ordered pair (''r'', ''θ'').  One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae: <math display="block">x = r\, \cos\theta,\, y = r\, \sin\theta; \, r = \sqrt{x^2+y^2},\, \theta = \arctan(y/x).</math> This system may be generalized to three-dimensional space through the use of [[Cylindrical coordinates|cylindrical]] or [[Spherical coordinates|spherical]] coordinates.
+[[Image:Polära_koordinater.svg|thumb|Three [[Polar coordinate system#Vector calculus|vectors in the polar coordinate system]] all originating at the origin with separate angles <math>\theta</math> passing through the points <math>(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})</math>.]]
+In [[polar coordinates]], every point of the plane is represented by its distance ''r'' from the origin and its [[angle]] ''θ'', with ''θ'' normally measured counterclockwise from the positive ''x''-axis. Using this notation, points are typically written as an ordered pair (''r'', ''θ'').  One may transform back and forth between two-dimensional Cartesian and polar coordinates by using these formulae:<ref name=":0">{{Cite book |last=Wilson |first=Wallace Alvin |title=Analytic Geometry |last2=Tracey |first2=Joshua Irving |publisher=[[D.C. Heath and Company]] |year=1937 |edition=Alternate |location=[[Boston, New York]] |pages=153-155, 283-284 |language=en |lccn=37002786 |oclc=530055}}</ref> <math display="block">x = r\, \cos\theta,\, y = r\, \sin\theta; \, r = \sqrt{x^2+y^2},\, \theta = \arctan(y/x).</math> This system may be generalized to three-dimensional space through the use of [[Cylindrical coordinates|cylindrical]] or [[Spherical coordinates|spherical]] coordinates.<ref name=":0" />
 ===Cylindrical coordinates (in a space)===
@@ Line 270: / Line 275: @@
 * {{citation|last=Cajori|first=Florian|title=A History of Mathematics|publisher=AMS|isbn=978-0821821022|year=1999}}
 * [[John Casey (mathematician)|John Casey]] (1885) [https://archive.org/details/cu31924001520455 Analytic Geometry of the Point, Line, Circle, and Conic Sections], link from [[Internet Archive]].
-* {{citation|last=Katz|first=Victor J.|title=A History of Mathematics: An Introduction (2nd Ed.)|publisher=Addison Wesley Longman |place=Reading|year=1998|isbn=0-321-01618-1|url=https://archive.org/details/historyofmathema00katz}}
+* {{citation|last=Katz|first=Victor J.|author-link=Victor J. Katz|title=A History of Mathematics: An Introduction (2nd Ed.)|publisher=Addison Wesley Longman |place=Reading|year=1998|isbn=0-321-01618-1|url=https://archive.org/details/historyofmathema00katz}}
 * [[Mikhail Postnikov]] (1982) [https://archive.org/details/postnikov-lectures-in-geometry-semester-i Lectures in Geometry Semester I Analytic Geometry] via Internet Archive
-* {{citation|last=Struik|first=D. J.|title=A Source Book in Mathematics, 1200-1800|publisher=Harvard University Press |isbn=978-0674823556|year=1969}}
+* {{citation|last=Struik|first=D. J.|author-link=Dirk Jan Struik|title=A Source Book in Mathematics, 1200-1800|publisher=Harvard University Press |isbn=978-0674823556|year=1969}}
 ===Articles===

Analytic geometry: Difference between revisions

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