Analytic geometry

Cartesian coordinates

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.

In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on axioms and theorems to derive truth. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry.

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.

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 introduced analytic geometry.[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 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.[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.[4]

Analytic geometry has traditionally been attributed to René Descartes.[4][6][7] 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.[8]

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.[9] 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.[8] 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.

Basic principles

Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.

Coordinates

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 (xy). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (xyz).

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.

Equations of curves

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.

The distance formula on the plane follows from the Pythagorean theorem.

Distance and angle

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 (x1y1) and (x2y2) is defined by the formula

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2},\!$

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

$\theta = \arctan(m)\!$

where m is the slope of the line.

Transformations

Transformations are applied to parent functions to turn it into a new function with similar characteristics. For example, the parent function y=1/x 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 y = f(x), then it can be transformed into y = af(b(x − k)) + h. In the new transformed function, a 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 a values, the function is reflected in the x-axis. The b value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like a, reflects the function in the y-axis when it is negative. The k and h values introduce translations, h, vertical, and k horizontal. Positive h and k 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 R(x,y) is a relation in the xy plane. For example

x2 + y2 -1= 0

is the relation that describes the unit circle. The graph of R(x,y) is changed by standard transformations as follows:

Changing x to x-h moves the graph to the right h units.

Changing y to y-k moves the graph up k units.

Changing x to x/b stretches the graph horizontally by a factor of b. (think of the x as being dilated)

Changing y to y/a stretches the graph vertically.

Changing x to xcosA+ ysinA and changing y to -xsinA + ycosA rotates the graph by an angle A.

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.

Intersections

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 P(x,y) and Q(x,y) the intersection is the collection of all points (x,y) which are in both relations. For example, P might be the circle with radius 1 and center (0,0): P = {(x,y) | x2+y2=1} and Q might be the circle with radius 1 and center (1,0): Q = {(x,y) | (x-1)2+y2=1}. The intersection of these two circles is the collection of points which make both equations true. Does the point (0,0) make both equations true? Using (0,0) for (x,y), the equation for Q becomes (0-1)2+02=1 or (-1)2=1 which is true, so (0,0) is in the relation Q. On the other hand, still using (0,0) for (x,y) the equation for P becomes (0)2+02=1 or 0=1 which is false. (0,0) is not in P so it is not in the intersection.

The intersection of P and Q can be found by solving the simultaneous equations:

x2+y2 = 1

(x-1)2+y2 = 1

Traditional methods include substitution and elimination.

Substitution: Solve the first equation for y in terms of x and then substitute the expression for y into the second equation.

x2+y2 = 1

y2=1-x2 We then substitute this value for y2 into the other equation:

(x-1)2+(1-x2)=1 and proceed to solve for x:

x2 -2x +1 +1 -x2 =1

-2x = -1

x=½

We next place this value of x in either of the original equations and solve for y:

½2+y2 = 1

y2 = ¾

$y = \frac{\pm \sqrt{3}}{2}$

So that our intersection has two points:

$\left(1/2,\frac{+ \sqrt{3}}{2}\right) \;\; \mathrm{and} \;\; \left(1/2,\frac{-\sqrt{3}}{2}\right)$

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: (x-1)2-x2=0 The y2 in the first equation is subtracted from the y2 in the second equation leaving no y term. y has been eliminated. We then solve the remaining equation for x, in the same way as in the substitution method. x2 -2x +1 +1 -x2 =1 -2x = -1 x=½ We next place this value of x in either of the original equations and solve for y: ½2+y2 = 1

y2 = ¾

$y = \frac{\pm \sqrt{3}}{2}$

So that our intersection has two points:

$\left(1/2,\frac{+ \sqrt{3}}{2}\right) \;\; \mathrm{and} \;\; \left(1/2,\frac{-\sqrt{3}}{2}\right)$

For conic sections, as many as 4 points might be in the intersection.

Intercepts

One type of intersection which is widely studied is the intersection of a geometric object with the x and y coordinate axes.

The intersection of a geometric object and the y-axis is called the y-intercept of the object. The intersection of a geometric object and the x-axis is called the x-intercept of the object.

For the line y=mx+b, the parameter b specifies the point where the line crosses the y axis. Depending on the context, either b or the point (0,b) is called the y-intercept.

Themes

Important themes of analytical geometry are

Ax2 + Bxy + Cy2 +Dx + Ey + F = 0. If the Bxy term is considered, rotations are generally used.

Many of these problems involve linear algebra.

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 $ABCDE$, the sides have lengths $1$, $2$, $3$, $4$, and $5$, though not necessarily in that order. Let $F$, $G$, $H$, and $I$ be the midpoints of the sides $AB$, $BC$, $CD$, and $DE$, respectively. Let $X$ be the midpoint of segment $FH$, and $Y$ be the midpoint of segment $GI$. The length of segment $XY$ is an integer. Find all possible values for the length of side $AE$.

Solution: Without loss of generality, 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 $XY$ has to be an integer,

$AE\equiv 0\pmod{4}$

(see modular arithmetic) so $AE=4$.

Modern analytic geometry

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.

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.

Notes

1. ^ Boyer, Carl B. (1991). "The Age of Plato and Aristotle". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. pp. 94–95. ISBN 0-471-54397-7. "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."
2. ^ Boyer, Carl B. (1991). "Apollonius of Perga". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 142. ISBN 0-471-54397-7. "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."
3. ^ Boyer, Carl B. (1991). "Apollonius of Perga". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 156. ISBN 0-471-54397-7. "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 of 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."
4. ^ a b c Boyer (1991). "The Arabic Hegemony". pp. 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.""
5. ^ Glen M. Cooper (2003). "Omar Khayyam, the Mathmetician", The Journal of the American Oriental Society 123.
6. ^ Stillwell, John (2004). "Analytic Geometry". Mathematics and its History (Second Edition ed.). Springer Science + Business Media Inc. p. 105. ISBN 0-387-95336-1. "the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments."
7. ^ Cooke, Roger (1997). "The Calculus". The History of Mathematics: A Brief Course. Wiley-Interscience. p. 326. ISBN 0-471-18082-3. "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."
8. ^ a b Katz 1998, pg. 442
9. ^ Katz 1998, pg. 436

References

• Katz, Victor J. (1998), A History of Mathematics: An Introduction (2nd Ed.), Reading: Addison Wesley Longman, ISBN 0-321-01618-1