Analytic geometry

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 Géométrie, 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 géométrie" , 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)
[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