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 theCantor–Dedekind axiom.
TheGreek mathematicianMenaechmus 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, inOn 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 theConics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work ofDescartes 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 (expressed in words) 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 curvea posteriori instead ofa 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 11th-century Persian mathematicianOmar Khayyam saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical andgeometric algebra[4] with his geometric solution of the generalcubic equations,[5] but the decisive step came later with Descartes.[4] Omar Khayyam is credited with identifying the foundations ofalgebraic geometry, and his bookTreatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe.[6] Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.[7]: 248
Analytic geometry was independently invented byRené Descartes andPierre de Fermat,[8][9] although Descartes is sometimes given sole credit.[10][11]Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes.
Descartes made significant progress with the methods in an essay titledLa Géométrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with hisDiscourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to asDiscourse on Method.La Geometrie, written in his nativeFrench tongue, and its philosophical principles, provided a foundation forcalculus 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 intoLatin and the addition of commentary byvan Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.[12]
Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form ofAd 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.[13][14][15] Clearly written and well received, theIntroduction 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 that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves.[12] 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. It wasLeonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.
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.
In analytic geometry, theplane is given a coordinate system, by which everypoint has a pair ofreal number coordinates. Similarly,Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a variety of coordinate systems used, but the most common are the following:[16]
The most common coordinate system to use is theCartesian coordinate system, where each point has anx-coordinate representing its horizontal position, and ay-coordinate representing its vertical position. These are typically written as anordered pair (x, y). This system can also be used for three-dimensional geometry, where every point inEuclidean space is represented by anordered triple of coordinates (x, y, z).
Inpolar coordinates, every point of the plane is represented by its distancer from the origin and itsangleθ, withθ normally measured counterclockwise from the positivex-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: This system may be generalized to three-dimensional space through the use ofcylindrical orspherical coordinates.
Incylindrical coordinates, every point of space is represented by its heightz, itsradiusr from thez-axis and theangleθ its projection on thexy-plane makes with respect to the horizontal axis.
In spherical coordinates, every point in space is represented by its distanceρ from the origin, theangleθ its projection on thexy-plane makes with respect to the horizontal axis, and the angleφ that it makes with respect to thez-axis. The names of the angles are often reversed in physics.[16]
In analytic geometry, anyequation involving the coordinates specifies asubset of the plane, namely thesolution set for the equation, orlocus. For example, the equationy = x corresponds to the set of all the points on the plane whosex-coordinate andy-coordinate are equal. These points form aline, andy = x is said to be the equation for this line. In general, linear equations involvingx andy specify lines,quadratic equations specifyconic sections, and more complicated equations describe more complicated figures.[17]
Usually, a single equation corresponds to acurve on the plane. This is not always the case: the trivial equationx = x specifies the entire plane, and the equationx2 + y2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives asurface, and a curve must be specified as theintersection of two surfaces (see below), or as a system ofparametric equations.[18] The equationx2 + y2 = r2 is the equation for any circle centered at the origin (0, 0) with a radius of r.
Lines in aCartesian plane, or more generally, inaffine coordinates, can be described algebraically bylinear equations. In two dimensions, the equation for non-vertical lines is often given in theslope-intercept form:where:
In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (thenormal vector) to indicate its "inclination".
Specifically, let be the position vector of some point, and let be a nonzero vector. The plane determined by this point and vector consists of those points, with position vector, such that the vector drawn from to is perpendicular to. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points such that(The dot here means adot product, not scalar multiplication.)Expanded this becomeswhich is thepoint-normal form of the equation of a plane.[citation needed] This is just alinear equation:Conversely, it is easily shown that ifa,b,c andd are constants anda,b, andc are not all zero, then the graph of the equationis a plane having the vector as a normal.[citation needed] This familiar equation for a plane is called thegeneral form of the equation of the plane.[19]
In three dimensions, lines cannot be described by a single linear equation, so they are frequently described byparametric equations:where:
x,y, andz are all functions of the independent variablet which ranges over the real numbers.
(x0,y0,z0) is any point on the line.
a,b, andc are related to the slope of the line, such that thevector (a,b,c) is parallel to the line.
In theCartesian coordinate system, thegraph of aquadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the formAs scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensionalprojective space
The conic sections described by this equation can be classified using thediscriminant[20]
The distance formula on the plane follows from the Pythagorean theorem.
In analytic geometry, geometric notions such asdistance andangle measure are defined usingformulas. These definitions are designed to be consistent with the underlyingEuclidean geometry. For example, usingCartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the formulawhich can be viewed as a version of thePythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formulawherem is theslope of the line.
In three dimensions, distance is given by the generalization of the Pythagorean theorem:while the angle between two vectors is given by thedot product. The dot product of two Euclidean vectorsA andB is defined by[22]whereθ is theangle betweenA andB.
a) y = f(x) = |x| b) y = f(x+3) c) y = f(x)-3 d) y = 1/2 f(x)
Transformations are applied to a parent function to turn it into a new function with similar characteristics.
The graph of is changed by standard transformations as follows:
Changing to moves the graph to the right units.
Changing to moves the graph up units.
Changing to stretches the graph horizontally by a factor of. (think of the as being dilated)
Changing to stretches the graph vertically.
Changing to and changing to rotates the graph by an angle.
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 onaffine transformations.
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.
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.[23]
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 for finding intersections 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:
Next, we place this value of in either of the original equations and solve for:
So 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 then place this value of in either of the original equations and solve for:
So 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.
Axis in geometry is the perpendicular line to any line, object or a surface.
Also for this may be used the common language use as a: normal (perpendicular) line, otherwise in engineering asaxial line.
Ingeometry, anormal is an object such as a line or vector that isperpendicular to a given object. For example, in the two-dimensional case, thenormal line to a curve at a given point is the line perpendicular to thetangent line to the curve at the point.
In the three-dimensional case asurface normal, or simplynormal, to asurface at a pointP is avector that isperpendicular to thetangent plane to that surface atP. The word "normal" is also used as an adjective: aline normal to aplane, the normal component of aforce, thenormal vector, etc. The concept ofnormality generalizes toorthogonality.
Ingeometry, thetangent line (or simplytangent) to a planecurve at a givenpoint is thestraight line that "just touches" the curve at that point. Informally, it is a line through a pair ofinfinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curvey =f(x) at a pointx =c on the curve if the line passes through the point(c,f(c)) on the curve and has slopef'(c) wheref' is thederivative off. A similar definition applies tospace curves and curves inn-dimensionalEuclidean space.
As it passes through the point where the tangent line and the curve meet, called thepoint of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
Similarly, thetangent plane to asurface at a given point is theplane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions indifferential geometry and has been extensively generalized; seeTangent space.
^Boyer, Carl B. (1991)."The Age of Plato and Aristotle".A History of Mathematics (Second ed.). John Wiley & Sons, Inc. pp. 94–95.ISBN0-471-54397-7.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.
^Boyer, Carl B. (1991)."Apollonius of Perga".A History of Mathematics (Second ed.). John Wiley & Sons, Inc. pp. 142.ISBN0-471-54397-7.The Apollonian treatiseOn 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.
^Boyer, Carl B. (1991)."Apollonius of Perga".A History of Mathematics (Second ed.). John Wiley & Sons, Inc. pp. 156.ISBN0-471-54397-7.The method of Apollonius in theConics 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 superimposeda 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 downa 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.
^abBoyer (1991)."The Arabic Hegemony".A History of Mathematics. pp. 241–242.ISBN9780471543978.Omar Khayyam (ca. 1050–1123), the "tent-maker," wrote anAlgebra 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."
^Cooper, G. (2003). Journal of the American Oriental Society,123(1), 248-249.
^Stillwell, John (2004). "Analytic Geometry".Mathematics and its History (Second ed.). Springer Science + Business Media Inc. p. 105.ISBN0-387-95336-1.the two founders of analytic geometry, Fermat and Descartes, were both strongly influenced by these developments.
^Cooke, Roger (1997)."The Calculus".The History of Mathematics: A Brief Course. Wiley-Interscience. pp. 326.ISBN0-471-18082-3.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.
^Pierre de Fermat,Varia Opera Mathematica d. Petri de Fermat, Senatoris Tolosani (Toulouse, France: Jean Pech, 1679), "Ad locos planos et solidos isagoge,"pp. 91–103.Archived 2015-08-04 at theWayback Machine
^"Eloge de Monsieur de Fermat"Archived 2015-08-04 at theWayback Machine (Eulogy of Mr. de Fermat),Le Journal des Scavans, 9 February 1665, pp. 69–72. From p. 70:"Une introduction aux lieux, plans & solides; qui est un traité analytique concernant la solution des problemes plans & solides, qui avoit esté veu devant que M. des Cartes eut rien publié sur ce sujet." (An introduction to loci, plane and solid; which is an analytical treatise concerning the solution of plane and solid problems, which was seen before Mr. des Cartes had published anything on this subject.)
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