Analytical Geometry Essay

1544 words - 7 pages

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, ...view middle of the document...

[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.
Abraham de Moivre also pioneered the development of analytic geometry.
Basic principles
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...

Other Papers Like Analytical Geometry

Indian Streams Research Journals Essay

3602 words - 15 pages in the analysis were investigated. Results from FE analysis were verified by strain gages attached to several locations on the crankshaft. Results achieved from aforementioned analysis can be used in fatigue life calculation and optimization of this component. INTRODUCTION Crankshaft is a large component with a complex geometry in the engine, which converts the reciprocating displacement of the piston to a rotary motion with a four link mechanism

Probability And Genetics Essay

1984 words - 8 pages (3). Pascal then began to turn his attention toward analytical geometry and physics. He repeated many of the experiments of Toricelli, the inventor of the barometer, and confirmed the theory of barometrical variations. Then in the middle of this research, in 1650, Pascal abandoned his mathematical pursuits to study religion. After his father's death in 1653, Pascal had to administer his father's estate, and this reignited his interest in

Dynamic Mechanical Analysis

5007 words - 21 pages to the development of new materials. Dynamic Mechanical Analysis (DMA) measures the mechanical properties of materials as function of temperature, frequency and time and also it is a thermal analytical method by which the mechanical response of a sample subjected to a specific temperature program is investigated under periodic stress. Dynamic mechanical analyzer is a thermal analytical instrument used to test the mechanical properties of many

Content and Description of Courses in Economics Department at the University of Jos

1194 words - 5 pages central tendency and dispersion; the unit normal distribution. Simple tests of hypotheses; Simple regression analysis; Index Numbers ECO 113 & 123 – Introductory Math for Economists Mathematical concepts in the social sciences; set theory; factors and exponents, logarithms and trigonometry, equations, functions and progressions. Coordinates geometry, trigonometric functions and their inverse; inequalities; matrix algebra; differentiation

Buet Wre Syllabus

5652 words - 23 pages on the design of roadway 60 geometry and pavements, application of analytical, simulation and statistical packages. CE 481 : Foundation Engineering 3.00 Credit, 3 hrs/week Soil investigation techniques; types of foundations; bearing capacity of shallow and deep foundations; settlement and distortion of foundations; design and construction of footings, rafts and piles; slope stability analyses. CE 424 : Concrete Structures Design

Guiding and Confining Light in Nanostructures

2315 words - 10 pages June 1, 2004 / Vol. 29, No. 11 / OPTICS LETTERS 1209 Guiding and confining light in void nanostructure Vilson R. Almeida, Qianfan Xu, Carlos A. Barrios, and Michal Lipson School of Electrical and Computer Engineering, Cornell University, 411 Phillips Hall, Ithaca, New York 14853 Received January 20, 2004 We present a novel waveguide geometry for enhancing and confining light in a nanometer-wide low-index material. Light enhancement

Isaac Newton

2472 words - 10 pages must first understand the period of time in which he lived. Before Newton's birth, many scientists had began to make discoveries that would affect his thought. Galileo wrote on mechanics, kinematics and astronomy. Descartes studied analytical geometry and optics. Hooke examined elasticity (Anthony, 89). At the same time, many others scientists were advancing other fields. The seventeenth century was truly a highly creative period. However, this


2400 words - 10 pages effects. However, when we look at atomic particles (i.e., protons, electrons, neutrons, etc.), we can no longer describe motion and energy using Newtonian physics. The characteristics of atomic particles are described using the theorems of quantum mechanics. Quantum mechanics--or quantum chemistry--describes the geometry of atoms and molecules in terms of complex mathematical expressions. It also describes the relative states of atomic matter. The

The Unconditional Bond of Family

2414 words - 10 pages . It isn’t a circle -- it is simply a long line – as in geometry, you know, one that reaches into infinity. And because we cannot see the end -- we also cannot see how it changes. And it is very odd but those who see the changes – who dream, who will not give up – are called idealists… and those who see only the circle -- we call them the “realists”! (674). The Younger’s, the African-American community, as well as the white community all

Mr and Mrs

2249 words - 9 pages determined by solving a distance geometry problem. Dual polarisation interferometry is a quantitative analytical method for measuring the overall protein conformation and conformational changes due to interactions or other stimulus. Circular dichroism is another laboratory technique for determining internal beta sheet/ helical composition of proteins. Cryoelectron microscopy is used to produce lower-resolution structural i

Postcolonial Literature

4905 words - 20 pages –terature. In what follows, I will explore a less obvious but equally significant aspect оf this project: thе tendency, in efforts tо expand our understanding оf lÑ–terature, tо make thаt expansion contingent on a corresponding expansion оf analytical focus. "Globalizing lÑ–terature" frequently goes hand in hand wÑ–th globalizing commentary about lÑ–terature; indeed, much оf thе project оf postcolonial lÑ–terary histоry depends on a set Ð

Related Essays

Foundations Of Psychology Essay

602 words - 3 pages etc ever since. He also developed the theory of consciousness, defining psychology as the science that studies the conscious. His many accomplishments were in the fields of Algebra, Geometry, Calculus, and Analysis; because of this he was one of the figures in the scientific revolution. ("Rene Decartes", 2012) According to “Rene Descartes” (2012) “He is credited as the father of analytical geometry, the bridge between algebra and geometry, crucial

Development And Validation Of Instructional Material Using Mathematics Software

3324 words - 14 pages tools to help students with math and science studies. The researcher chose this study to develop and validate an instructional material utilizing Microsoft Mathematics Software in teaching selected topics on geometry. The researcher get into this subject because she believes that this endeavour will provide a set of mathematical tools that will help students get school work done quickly and easily. With this instructional material, pupils can

Humanities Essay

972 words - 4 pages 1. Define humanities: The humanities are academic disciplines that study human culture, using methods that are primarily analytical,[citation needed] critical, or speculative, and having a significant historical element,[1] as distinguished from the mainly empirical approaches of the natural sciences.[1] The humanities include ancient and modern languages, literature, philosophy, religion, and visual and performing arts such as music and

Computer Since Essay

1049 words - 5 pages computations have existed since antiquity, even before the development of sophisticated computing equipment. The ancient Sanskrit treatise Shulba Sutras, or "Rules of the Chord", is a book of algorithms written in 800 BC for constructing geometric objects like altars using a peg and chord, an early precursor of the modern field of computational geometry. Blaise Pascal designed and constructed the first working mechanical calculator, Pascal's