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A branch of geometry. The fundamental concepts of analytic geometry are the simplest geometric elements (points, straight lines, planes, second-order curves and surfaces). The principal means of study in analytic geometry are the method of coordinates and the methods of elementary algebra. The genesis of the method of coordinates is closely linked with the intense development of astronomy, mechanics and technology in the 17th century. In his Géometrie, R. Descartes (1637) gave a clear and exhaustive account of this method and of the foundations of analytic geometry. P. Fermat, a contemporary of Descartes, was also familiar with the principles of this method. Subsequent development of analytic geometry is due to the studies of G. Leibniz, I. Newton and, particularly, L. Euler. The tools of analytic geometry were utilized by J.L. Lagrange in his construction of analytic mechanics and by G. Monge in differential geometry. In our own days, analytic geometry has no significance as an independent branch of science, but its methods are extensively employed in various fields of mathematics, mechanics, physics and other sciences.
 
A branch of geometry. The fundamental concepts of analytic geometry are the simplest geometric elements (points, straight lines, planes, second-order curves and surfaces). The principal means of study in analytic geometry are the method of coordinates and the methods of elementary algebra. The genesis of the method of coordinates is closely linked with the intense development of astronomy, mechanics and technology in the 17th century. In his Géometrie, R. Descartes (1637) gave a clear and exhaustive account of this method and of the foundations of analytic geometry. P. Fermat, a contemporary of Descartes, was also familiar with the principles of this method. Subsequent development of analytic geometry is due to the studies of G. Leibniz, I. Newton and, particularly, L. Euler. The tools of analytic geometry were utilized by J.L. Lagrange in his construction of analytic mechanics and by G. Monge in differential geometry. In our own days, analytic geometry has no significance as an independent branch of science, but its methods are extensively employed in various fields of mathematics, mechanics, physics and other sciences.
  
The principle of the method of coordinates is as follows. Consider, for example, two mutually-perpendicular straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122702.png" /> in a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122703.png" />. These lines, including their direction, the coordinate origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122704.png" /> and the selected scale unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122705.png" />, form a so-called Cartesian orthogonal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122706.png" /> of coordinates on the plane. The straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122708.png" /> are called the abscissa (axis) and the ordinate (axis) respectively. The location of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a0122709.png" /> in the plane relative to this system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227010.png" /> can be determined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227012.png" /> be the projections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227013.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227015.png" />, and let the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227017.png" /> be the magnitudes of the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227019.png" /> (the magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227020.png" /> of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227021.png" />, for example, is equal to the length of this segment, taken with the plus sign if the direction from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227022.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227023.png" /> is the same as the direction of the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227024.png" />, and taken with the minus sign in the opposite case). The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227025.png" /> (the abscissa) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227026.png" /> (the ordinate) are said to be the Cartesian orthogonal coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227027.png" /> in the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227028.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227029.png" /> with abscissa <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227030.png" /> and ordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227031.png" /> is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227032.png" />. The coordinates of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227033.png" /> clearly determine its location with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227034.png" />.
+
The principle of the method of coordinates is as follows. Consider, for example, two mutually-perpendicular straight lines $  Ox $
 +
and $  Oy $
 +
in a plane $  \pi $.  
 +
These lines, including their direction, the coordinate origin $  O $
 +
and the selected scale unit $  e $,  
 +
form a so-called Cartesian orthogonal system $  Oxy $
 +
of coordinates on the plane. The straight lines $  Ox $
 +
and $  Oy $
 +
are called the abscissa (axis) and the ordinate (axis) respectively. The location of any point $  M $
 +
in the plane relative to this system $  Oxy $
 +
can be determined as follows. Let $  M _ {x} $
 +
and $  M _ {y} $
 +
be the projections of $  M $
 +
onto $  Ox $
 +
and $  Oy $,  
 +
and let the numbers $  x $
 +
and $  y $
 +
be the magnitudes of the segments $  O M _ {x} $
 +
and $  O M _ {y} $(
 +
the magnitude $  x $
 +
of the segment $  O M _ {x} $,  
 +
for example, is equal to the length of this segment, taken with the plus sign if the direction from $  O $
 +
to $  M _ {x} $
 +
is the same as the direction of the straight line $  Ox $,  
 +
and taken with the minus sign in the opposite case). The numbers $  x $(
 +
the abscissa) and $  y $(
 +
the ordinate) are said to be the Cartesian orthogonal coordinates of the point $  M $
 +
in the system $  Oxy $.  
 +
A point $  M $
 +
with abscissa $  x $
 +
and ordinate $  y $
 +
is denoted by the symbol $  M (x, y) $.  
 +
The coordinates of a point $  M $
 +
clearly determine its location with respect to the system $  Oxy $.
  
Let a line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227035.png" /> be given on a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227036.png" /> with a given Cartesian orthogonal coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227037.png" />. Using the concept of coordinates of a point it is possible to introduce the concept of the equation of the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227038.png" /> with respect to system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227039.png" /> as an equation of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227040.png" />, which will be satisfied by the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227042.png" /> of any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227043.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227044.png" />, but not by those of any point which does not lie on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227045.png" />.
+
Let a line $  L $
 +
be given on a plane $  \pi $
 +
with a given Cartesian orthogonal coordinate system $  Oxy $.  
 +
Using the concept of coordinates of a point it is possible to introduce the concept of the equation of the line $  L $
 +
with respect to system $  Oxy $
 +
as an equation of the type $  F (x, y) = 0 $,  
 +
which will be satisfied by the coordinates $  x $
 +
and $  y $
 +
of any point $  M $
 +
on $  L $,  
 +
but not by those of any point which does not lie on $  L $.
  
The basic idea of the method of coordinates on a plane is that the geometric properties of the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227046.png" /> can be clarified by studying its equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227047.png" /> by analytic and algebraic tools. Thus, for instance, the problem of the number of intersection points between a straight line and a circle is reduced to the analytic problem of the number of solutions of the set of equations of the line and of the circle.
+
The basic idea of the method of coordinates on a plane is that the geometric properties of the line $  L $
 +
can be clarified by studying its equation $  F(x, y) = 0 $
 +
by analytic and algebraic tools. Thus, for instance, the problem of the number of intersection points between a straight line and a circle is reduced to the analytic problem of the number of solutions of the set of equations of the line and of the circle.
  
 
This was in fact the method of study of the properties of an [[Ellipse|ellipse]], a [[Hyperbola|hyperbola]] and a [[Parabola|parabola]], which are the lines of intersection between a circular cone and planes which do not pass through its vertex (cf. [[Conic sections|Conic sections]]).
 
This was in fact the method of study of the properties of an [[Ellipse|ellipse]], a [[Hyperbola|hyperbola]] and a [[Parabola|parabola]], which are the lines of intersection between a circular cone and planes which do not pass through its vertex (cf. [[Conic sections|Conic sections]]).
  
The so-called algebraic curves of the first and second orders are a subject of systematic study in plane analytic geometry; in Cartesian orthogonal coordinates these are described by algebraic equations of the first and second degree, respectively. The curves of the first order are rectilinear and, conversely, any straight line is described by an algebraic equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227048.png" /> of the first degree. The curves of the second order are described by equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227049.png" />. The principal technique in the study and classification of such curves is to select a Cartesian orthogonal coordinate system in which the equation assumes its simplest form, after which the simple equation is studied. Cf. [[Second-order curve|Second-order curve]].
+
The so-called algebraic curves of the first and second orders are a subject of systematic study in plane analytic geometry; in Cartesian orthogonal coordinates these are described by algebraic equations of the first and second degree, respectively. The curves of the first order are rectilinear and, conversely, any straight line is described by an algebraic equation $  Ax + By + C = 0 $
 +
of the first degree. The curves of the second order are described by equations of the form $  A x  ^ {2} + Bxy + C y  ^ {2} + Dx + Ey + F = 0 $.  
 +
The principal technique in the study and classification of such curves is to select a Cartesian orthogonal coordinate system in which the equation assumes its simplest form, after which the simple equation is studied. Cf. [[Second-order curve|Second-order curve]].
  
In the analytic geometry of space, the Cartesian orthogonal coordinates are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227051.png" /> (the abscissa, the ordinate and the applicate), and points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227052.png" /> are described exactly as in plane analytic geometry. Any given surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227053.png" /> in space has its own equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227054.png" /> in the coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227055.png" />, and its geometric properties are studied by studying its equation by analytic and algebraic means. A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227056.png" /> in space is given as the intersection of two surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227058.png" />. If the respective equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227060.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227062.png" />, these equations, considered together, are the equation of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227063.png" />. Thus, a straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227064.png" /> in space may be considered as the intersection of two planes. In analytic geometry of space a systematic study is made of the so-called algebraic surfaces of the first and second orders. It was found that only planes are algebraic surfaces of the first order. Surfaces of the second order have equations of the type
+
In the analytic geometry of space, the Cartesian orthogonal coordinates are $  x, y $
 +
and $  z $(
 +
the abscissa, the ordinate and the applicate), and points $  M $
 +
are described exactly as in plane analytic geometry. Any given surface $  S $
 +
in space has its own equation $  F(x, y, z) = 0 $
 +
in the coordinate system $  Oxyz $,  
 +
and its geometric properties are studied by studying its equation by analytic and algebraic means. A curve $  L $
 +
in space is given as the intersection of two surfaces $  S _ {1} $
 +
and $  S _ {2} $.  
 +
If the respective equations of $  S _ {1} $
 +
and $  S _ {2} $
 +
are $  F _ {1} (x, y, z) = 0 $
 +
and $  F _ {2} (x, y, z) = 0 $,  
 +
these equations, considered together, are the equation of the curve $  L $.  
 +
Thus, a straight line $  L $
 +
in space may be considered as the intersection of two planes. In analytic geometry of space a systematic study is made of the so-called algebraic surfaces of the first and second orders. It was found that only planes are algebraic surfaces of the first order. Surfaces of the second order have equations of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227065.png" /></td> </tr></table>
+
$$
 +
A x  ^ {2} + B y  ^ {2} + C z  ^ {2} + D x y + E y z + F x z +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012270/a01227066.png" /></td> </tr></table>
+
$$
 +
+
 +
G x + H y + M z + N  = 0 .
 +
$$
  
 
The principal method of study and classification of such surfaces is to select a Cartesian coordinate system in which the equation of the surface assumes its simplest form, the latter being then studied. Cf. [[Surface of the second order|Surface of the second order]].
 
The principal method of study and classification of such surfaces is to select a Cartesian coordinate system in which the equation of the surface assumes its simplest form, the latter being then studied. Cf. [[Surface of the second order|Surface of the second order]].
Line 21: Line 100:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Descartes,  "La géometrie" , Leiden  (1637)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Wieleitner,  "Die Geschichte der Mathematik von Descartes bis zum Hälfte des 19. Jahrhunderts" , de Gruyter  (1923)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.V. Efimov,  "A short course of analytical geometry" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Analytical geometry" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.S. Aleksandrov,  "Lectures on analytical geometry" , Moscow  (1968)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.M. Postnikov,  "Analytic geometry" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.V. Bakhvalov,  P.S. Modenov,  A.S. Parkhomenko,  "Collection of problems on analytic geometry" , Moscow  (1964)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Descartes,  "La géometrie" , Leiden  (1637)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Wieleitner,  "Die Geschichte der Mathematik von Descartes bis zum Hälfte des 19. Jahrhunderts" , de Gruyter  (1923)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.V. Efimov,  "A short course of analytical geometry" , Moscow  (1967)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Il'in,  E.G. Poznyak,  "Analytical geometry" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.S. Aleksandrov,  "Lectures on analytical geometry" , Moscow  (1968)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.M. Postnikov,  "Analytic geometry" , Moscow  (1973)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.V. Bakhvalov,  P.S. Modenov,  A.S. Parkhomenko,  "Collection of problems on analytic geometry" , Moscow  (1964)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Bol,  "Elemente der analytischen Geometrie" , Vandenhoeck &amp; Ruprecht  (1948)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,  "Analytic geometry" , PWN  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.J. Struik,  "Lectures on analytic and projective geometry" , Addison-Wesley  (1953)  pp. 157–160</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.A. Todd,  "Projective and analytical geometry" , Pitman  (1947)  pp. Chapt. VI</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Bol,  "Elemente der analytischen Geometrie" , Vandenhoeck &amp; Ruprecht  (1948)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Borsuk,  "Analytic geometry" , PWN  (1969)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.J. Struik,  "Lectures on analytic and projective geometry" , Addison-Wesley  (1953)  pp. 157–160</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.A. Todd,  "Projective and analytical geometry" , Pitman  (1947)  pp. Chapt. VI</TD></TR></table>

Revision as of 18:47, 5 April 2020


A branch of geometry. The fundamental concepts of analytic geometry are the simplest geometric elements (points, straight lines, planes, second-order curves and surfaces). The principal means of study in analytic geometry are the method of coordinates and the methods of elementary algebra. The genesis of the method of coordinates is closely linked with the intense development of astronomy, mechanics and technology in the 17th century. In his Géometrie, R. Descartes (1637) gave a clear and exhaustive account of this method and of the foundations of analytic geometry. P. Fermat, a contemporary of Descartes, was also familiar with the principles of this method. Subsequent development of analytic geometry is due to the studies of G. Leibniz, I. Newton and, particularly, L. Euler. The tools of analytic geometry were utilized by J.L. Lagrange in his construction of analytic mechanics and by G. Monge in differential geometry. In our own days, analytic geometry has no significance as an independent branch of science, but its methods are extensively employed in various fields of mathematics, mechanics, physics and other sciences.

The principle of the method of coordinates is as follows. Consider, for example, two mutually-perpendicular straight lines $ Ox $ and $ Oy $ in a plane $ \pi $. These lines, including their direction, the coordinate origin $ O $ and the selected scale unit $ e $, form a so-called Cartesian orthogonal system $ Oxy $ of coordinates on the plane. The straight lines $ Ox $ and $ Oy $ are called the abscissa (axis) and the ordinate (axis) respectively. The location of any point $ M $ in the plane relative to this system $ Oxy $ can be determined as follows. Let $ M _ {x} $ and $ M _ {y} $ be the projections of $ M $ onto $ Ox $ and $ Oy $, and let the numbers $ x $ and $ y $ be the magnitudes of the segments $ O M _ {x} $ and $ O M _ {y} $( the magnitude $ x $ of the segment $ O M _ {x} $, for example, is equal to the length of this segment, taken with the plus sign if the direction from $ O $ to $ M _ {x} $ is the same as the direction of the straight line $ Ox $, and taken with the minus sign in the opposite case). The numbers $ x $( the abscissa) and $ y $( the ordinate) are said to be the Cartesian orthogonal coordinates of the point $ M $ in the system $ Oxy $. A point $ M $ with abscissa $ x $ and ordinate $ y $ is denoted by the symbol $ M (x, y) $. The coordinates of a point $ M $ clearly determine its location with respect to the system $ Oxy $.

Let a line $ L $ be given on a plane $ \pi $ with a given Cartesian orthogonal coordinate system $ Oxy $. Using the concept of coordinates of a point it is possible to introduce the concept of the equation of the line $ L $ with respect to system $ Oxy $ as an equation of the type $ F (x, y) = 0 $, which will be satisfied by the coordinates $ x $ and $ y $ of any point $ M $ on $ L $, but not by those of any point which does not lie on $ L $.

The basic idea of the method of coordinates on a plane is that the geometric properties of the line $ L $ can be clarified by studying its equation $ F(x, y) = 0 $ by analytic and algebraic tools. Thus, for instance, the problem of the number of intersection points between a straight line and a circle is reduced to the analytic problem of the number of solutions of the set of equations of the line and of the circle.

This was in fact the method of study of the properties of an ellipse, a hyperbola and a parabola, which are the lines of intersection between a circular cone and planes which do not pass through its vertex (cf. Conic sections).

The so-called algebraic curves of the first and second orders are a subject of systematic study in plane analytic geometry; in Cartesian orthogonal coordinates these are described by algebraic equations of the first and second degree, respectively. The curves of the first order are rectilinear and, conversely, any straight line is described by an algebraic equation $ Ax + By + C = 0 $ of the first degree. The curves of the second order are described by equations of the form $ A x ^ {2} + Bxy + C y ^ {2} + Dx + Ey + F = 0 $. The principal technique in the study and classification of such curves is to select a Cartesian orthogonal coordinate system in which the equation assumes its simplest form, after which the simple equation is studied. Cf. Second-order curve.

In the analytic geometry of space, the Cartesian orthogonal coordinates are $ x, y $ and $ z $( the abscissa, the ordinate and the applicate), and points $ M $ are described exactly as in plane analytic geometry. Any given surface $ S $ in space has its own equation $ F(x, y, z) = 0 $ in the coordinate system $ Oxyz $, and its geometric properties are studied by studying its equation by analytic and algebraic means. A curve $ L $ in space is given as the intersection of two surfaces $ S _ {1} $ and $ S _ {2} $. If the respective equations of $ S _ {1} $ and $ S _ {2} $ are $ F _ {1} (x, y, z) = 0 $ and $ F _ {2} (x, y, z) = 0 $, these equations, considered together, are the equation of the curve $ L $. Thus, a straight line $ L $ in space may be considered as the intersection of two planes. In analytic geometry of space a systematic study is made of the so-called algebraic surfaces of the first and second orders. It was found that only planes are algebraic surfaces of the first order. Surfaces of the second order have equations of the type

$$ A x ^ {2} + B y ^ {2} + C z ^ {2} + D x y + E y z + F x z + $$

$$ + G x + H y + M z + N = 0 . $$

The principal method of study and classification of such surfaces is to select a Cartesian coordinate system in which the equation of the surface assumes its simplest form, the latter being then studied. Cf. Surface of the second order.

References

[1] R. Descartes, "La géometrie" , Leiden (1637)
[2] H. Wieleitner, "Die Geschichte der Mathematik von Descartes bis zum Hälfte des 19. Jahrhunderts" , de Gruyter (1923)
[3] N.V. Efimov, "A short course of analytical geometry" , Moscow (1967) (In Russian)
[4] V.A. Il'in, E.G. Poznyak, "Analytical geometry" , MIR (1984) (Translated from Russian)
[5] P.S. Aleksandrov, "Lectures on analytical geometry" , Moscow (1968) (In Russian)
[6] M.M. Postnikov, "Analytic geometry" , Moscow (1973) (In Russian)
[7] S.V. Bakhvalov, P.S. Modenov, A.S. Parkhomenko, "Collection of problems on analytic geometry" , Moscow (1964) (In Russian)

Comments

References

[a1] G. Bol, "Elemente der analytischen Geometrie" , Vandenhoeck & Ruprecht (1948)
[a2] K. Borsuk, "Analytic geometry" , PWN (1969)
[a3] D.J. Struik, "Lectures on analytic and projective geometry" , Addison-Wesley (1953) pp. 157–160
[a4] J.A. Todd, "Projective and analytical geometry" , Pitman (1947) pp. Chapt. VI
How to Cite This Entry:
Analytic geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_geometry&oldid=15081
This article was adapted from an original article by E.G. Poznyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article