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|History of geometry|
Analytic geometry, or analytical geometry, has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties. This article focuses on the classical and elementary meaning.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. The numerical output, however, might also be a vector or a shape. 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.
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 introduced analytic geometry. 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. 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 from 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.
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 with his geometric solution of the general cubic equations, but the decisive step came later with Descartes.
Analytic geometry has traditionally been attributed to René Descartes. Descartes made significant progress with the methods in an essay entitled La Geometrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching 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 a foundation for Infinitesimal calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van Schooten in 1649 (and further work thereafter) did Descarte's masterpiece receive due recognition.
Pierre Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse. Clearly written and well received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint. Fermat always started with an algebraic equation and then described the geometric curve which satisfied it, while Descartes starts with geometric curves and produces their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree.
In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z).
Other coordinate systems are possible. On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ. In three dimensions, common alternative coordinate systems include cylindrical coordinates and spherical coordinates.
In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.
Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x2 + y2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. The equation x2 + y2 = r2 is the equation for any circle with a radius of r.
In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the formula
which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula
where m is the slope of the line.
Transformations are applied to parent functions to turn it into a new function with similar characteristics. For example, the parent function has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote,and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if , then it can be transformed into . In the new transformed function, is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative values, the function is reflected in the -axis. The value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like , reflects the function in the -axis when it is negative. The and values introduce translations, , vertical, and horizontal. Positive and values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.
Transformations can be applied to any geometric equation whether or not the equation represents a function. Transformations can be considered as individual transactions or in combinations.
Suppose that is a relation in the plane. For example
is the relation that describes the unit circle. The graph of is changed by standard transformations as follows:
There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered. For more information, consult the Wikipedia article on affine transformations.
While this discussion is limited to the xy-plane, it can easily be extended to higher dimensions. For two geometric objects P and Q represented by the relations and the intersection is the collection of all points which are in both relations. For example, might be the circle with radius 1 and center : and might be the circle with radius 1 and center . The intersection of these two circles is the collection of points which make both equations true. Does the point make both equations true? Using for , the equation for becomes or which is true, so is in the relation . On the other hand, still using for the equation for becomes or which is false. is not in so it is not in the intersection.
The intersection of and can be found by solving the simultaneous equations:
Traditional methods include substitution and elimination.
Substitution: Solve the first equation for in terms of and then substitute the expression for into the second equation.
We then substitute this value for into the other equation:
and proceed to solve for :
We next place this value of in either of the original equations and solve for :
So that our intersection has two points:
Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, If we subtract the first equation from the second we get: The in the first equation is subtracted from the in the second equation leaving no term. The variable has been eliminated. We then solve the remaining equation for , in the same way as in the substitution method. We next place this value of in either of the original equations and solve for :
So that our intersection has two points:
For conic sections, as many as 4 points might be in the intersection.
One type of intersection which is widely studied is the intersection of a geometric object with the and coordinate axes.
The intersection of a geometric object and the -axis is called the -intercept of the object. The intersection of a geometric object and the -axis is called the -intercept of the object.
For the line , the parameter specifies the point where the line crosses the axis. Depending on the context, either or the point is called the -intercept.
Important themes of analytical geometry are
. If the term is considered, rotations are generally used.
Many of these problems involve linear algebra.
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 , the sides have lengths , , , , and , though not necessarily in that order. Let , , , and be the midpoints of the sides , , , and , respectively. Let be the midpoint of segment , and be the midpoint of segment . The length of segment is an integer. Find all possible values for the length of side .
Solution: Without loss of generality, let , , , , and be located at , , , , and .
Using the midpoint formula, the points , , , , , and are located at
Using the distance formula,
Since has to be an integer,
(see modular arithmetic) so .
An analytic variety is defined locally as the set of common solutions of several equations involving analytic functions. It is analogous to the included concept of real or complex algebraic variety. Any complex manifold is an analytic variety. Since analytic varieties may have singular points, not all analytic varieties are manifolds.
Modern analytic geometry is essentially equivalent to real and complex algebraic geometry, as has been shown by Jean-Pierre Serre in his paper GAGA, the name of which is French for Algebraic geometry and analytic geometry. Nevertheless, the two fields remain distinct, as the methods of proof are quite different and algebraic geometry includes also geometry in finite characteristic.