Analytic 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.
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 of measurement. 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 geometrical shapes in a numerical way and extracting numerical information from that representation. 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 Term Paper on Analytic Geomety
The History of Geometry1 may be roughly divided into the four periods: (1) The synthetic geometry of the Greeks, practically closing with Archimedes; (2) The birth of analytic geometry, in which the synthetic geometry of Guldin, Desargues, Kepler, and Roberval merged into the coordinate geometry of Descartes and Fermat; (3) 1650 to 1800, characterized by the application of the calculus to ...
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 than 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 Geometry, one of the three accompanying essays published in 1637 together with his Discourse on the Method of Rightly Conducting the Reason in the Search 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 the foundation for Infinitesimal calculus in Europe.
The Term Paper on Relationship and Differences Between the Central and Peripheral Nervous System
“From the brain, and from the brain alone, arise our pleasure, joys, laughter, and jokes, as well as our sorrows, pains, griefs, and tears. Through it, in particular, we see, hear, and distinguish the ugly from the beautiful, the bad from the good, the pleasant from the unpleasant. ”(Attributed to Hippocrates, 5th century BCE, as quoted by Kandel et al. , 2000, as cited in Baars and Gage, 2012). ...
Abraham de Moivre also pioneered the development of analytic geometry.
Basic principles
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 (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 rfrom 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 Essay on Analytic Geometry Descartes Rene Formulas
Analytic geometry was brought fourth by the famous French mathematician Rene' Descartes in 1637. Descartes did not start his studying and working with geometry until after he had retired out of the army and settled down. If not for Descartes great discovery then Sir Isaac Newton might not have ever invented the concept of calculus. Descartes concept let to calculus and Newton and G. W. Leibniz ...
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 (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
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.
Themes
Important themes of analytical geometry are
vector space
definition of the plane
distance problems
the dot product, to get the angle of two vectors
the cross product, to get a perpendicular vector of two known vectors (and also their spatial volume)
intersection problems
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:
The Essay on Analytic Geometry and Unit
his unit goes from natural numbers to solving everyday problems and the need to gradually increase the numerical sets shown. Operations with real numbers are formalized and the existence of imaginary and complex numbers is mentioned. Operating with absolute value, scientific notation and logarithms begins. At the end of this unit we will be moving from numerical to symbolic representation to ...
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
and
Using the distance formula,
and
Since XY has to be an integer,
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. Anycomplex 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 it has been shown by Jean-Pierre Serre in his paper GAGA, whose name is, in French, Algebraic geometry and analytic geometric. Nevertheless, the two fields remain distinct, as the methods of proof are quite different and algebraic geometry includes also geometry in finite characteristic.
