User:GaiaSilvestri
Hi everybody!
My name is Gaia, and I came to canada from Milan, Italy. This is my first year at UBC and my intended major is Cognitive Systems.
ELLIPSES IN THE PHYSICAL WORLD
In mathematics conic sections are curves obtained by the intersection of a plane with one or two pieces of a double cone (two cones placed apex to apex). A circle is generated if the plane intersects a single nappe of the double cone being perpendicular to its axis. When the plane intersects one nappe of the double cone without being perpendicular to its axis, and without crossing its apex, a parabola or an ellipse is generated. The conic section that is produced when the plane intersects both nappes is the hyperbola. The ellipse is a central conic and a closed curve, unlike the parabola. An ellipse is produced by planes that intercept one nappe, and that form angles greater of θ and less or equal to π/2 with the axis of the double cone.
The conic sections have been accurately studied during the Hellenistic age, especially by the mathematicians and astronomists Menecmus and Apollonius of Perga in 200 a.C, before their application to the planetary orbits was known. Apollonius of Perga was the inventor of the specific terminology that is still used in this field nowadays: παραβολή “comparable” (parabola), ἔλλειψις “deficient” (ellipse), ὑπερβολή “excessive” (hyperbola). Apollonius’s terminology might refer to the eccentricity of the curves: greater than one (hyperbola), one (parabola), and less than one (ellipse). The two foci of an ellipse (F1 and F2) are located on its major axis (whose midpoint is at the center of the ellipse) and are equidistant from the center. The sum of the distances from any point P on the closed curve to the foci is equal to 2a (where a is the semimajor axis).
Conic sections are fundamental in astronomy: the orbits of two bodies (with significant masses) that interact according to the universal law of gravitation are conic sections . If the attraction between the two bodies is sufficient, they will both display elliptic orbits with the common barycenter being one of the foci of each ellipse. Depending on the energy of the orbiting body, the orbits can be any of the four types of conic sections.Elliptic orbits were discovered by the astronomist J. Kepler in the 17th century. For the very first time in history the belief that planets were revolving around the sun by describing circular orbits was abandonedAccording to his first law of planetary motion the orbits of the planets moving around the Sun are ellipses with the Sun being one of the two foci.
Where: a= major semiaxis b= minor semiaxis c= focal semi-distance e= eccentricity (~0.5)
BIBLIOGRAPHY: Weisstein, Eric W. "Ellipse." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Ellipse.html Weisstein, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConicSection.html Math DL The MAA Mathematica Sciences Digital Library, The Mathematical Association of America www.wikipedia.com
MATHEMATICS AND COMPUTER SCIENCE
Mathematics and computer science have an extremely close relationship. The discipline of computer science has its roots in the work of some of the most successful mathematicians of all times (Alan Turing to cite its founder) and there is a continuous interchange of ideas between these two fields.
What do computer science and mathematics share? Let’s try to use a metaphor here. When you were a child you could do some simple operations like addition and subtraction. As time went by, with the help of your teachers you learnt more mathematical operations. But you also learnt how to describe the rules of computation: you were able to manipulate the data to work on through a set of instructions.
Your math teacher trained you and transformed you into a “computer”, someone capable of executing a program to do computations. Yes, computers kind of work the same way as a child who develops these skills does. Amazing right? But there is more to it. Computers are better than people at this job. They are faster and they can perform millions of operations while we might still be struggling with the first one. They manipulate mathematical data in a very efficient way and beat humans in this task. However, what is really mind blowing about computers is that they can do more than just processing numbers. They can beat you at chess. They can draw pictures. They can drive a car. They can look up things for you in the WEB. In a few words, they can manipulate information of many different forms. Sadly though computers are extremely good only at working with numbers. They can barely understand human language (it is a very confusing mean of communication!) but they are indeed awesome at mathematics (pure and logical).
Every task carried out by a computer, whether we consider the hardware or the software, involves a great deal of mathematical ideas: the people who program computers need to know the foundations of logic and calculus very well if they want their final product to be working. Logical reasoning, counting, geometry, conversions, functions, lambda calculus, boolean algebra, proofs, algorithms and much more are central to the basics of computer science and many research areas. For example, proofs are really important when it comes to computer programming. We need them when we want to verify that computer programs produce the correct output for all possible input values, when we want to show that algorithms always produce the correct result and when we create artificial intelligence. Most computer programs can be divided in functional and logic programming. Functional programming treats computation as the evaluation of mathematical functions. A program uses a data definition, parameters and a main function (with secondary functions) to carry out its task. Every problem is broken up in smaller units, and processed by numerous functions that take in different inputs and produce outputs that are used in higher-order functions of the same program. Functional programming has its roots in lambda calculus (introduced by the mathematician Alonzo Church in the 1930s). Discrete mathematics and logic are used to translate English sentences into a language that can be understood by computers (sequences of bits, truth values). Binary format is also used to produce digital images in sequences of 1s and 0s in the field of computer graphics, where geometry is obviously the main focus of study. This list is just to name a few applications of mathematics in computer science.