In this essay the conic sections in taxicab geometry will be researched. The area of mathematics used is geometry. I have chosen this topic because it seemed interesting to me. I have never heard for this topic before, but then our math teacher presented us mathematic web page and taxicab geometry was one of the topics discussed there. I looked at the topic before and it encounter problems, which seemed interesting to explore. I started with a basic example, just to compare Euclidean and taxicab distance and after that I went further into the world of taxicab geometry. I explored the conic sections (circle, ellipse, parabola and hyperbola) of taxicab geometry. All pictures, except figure 12, were drawn by me in the program called Geogebra.
DEFINING THE PROBLEM
Problem given by teacher was:
A probe on the surface of planet Mars has a limited amount of fuel left. Because of broken rudder it can only move north- south and east- west. Which points can it reach if it must return back to its base?
I solved the problem hence finding a shape of a circle in so-called taxicab geometry.
My research question was:
How would other conic sections look like in taxicab geometry?
INTRODUCING TAXICAB AND EUCLIDEAN DISTANCE
I think it is my duty to define the two distances that the essay is going to talk about: Euclidian and taxicab distance, as many people as well as I, probably have never heard about the taxicab distance.
EUCLIDEAN DISTANCE
Euclidean distance was proposed by Greek mathematician Euclid of Alexandria. In mathematics, the Euclidean distance or Euclidean metric is the distance between two points, which is shown as a length of a line segment and is given by the Pythagorean theorem. The formula of Euclidean distance is a squ...
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... (2014) Feature Column from the AMS. [online] Available from: http://www.ams.org/samplings/feature-column/fcarc-taxi [Accessed: 02/03/14].
P. Thompson, K. (2010) Taxicab Geometry - General Info. [online] Available from: http://www.taxicabgeometry.net/general/index.html [Accessed 02/ 03/14].
RAKANOVIČ, A. and ŠURC, D. and MITROVIČ, B. (2007) Stožnice v taksi razdalji.
WIKIPEDIA. (2014) Conic section. [online] Available from: http://en.wikipedia.org/wiki/Conic_section. [Accessed 02/03/14].
COMPONENT, C. (2007) Taxicab Geometry: Not the Shortest Ride Across Town. Iowa: Iowa State University.
BUCHANAN, L. Et al. (2012) Mathematics Standard Level. Oxford: Oxford University
Image:
PBROKS13 (2009) The three conic sections that are created when a double cone is intersected with a plane. [image] Available from: http://en.wikipedia.org/wiki/File:Conic_sections_with_plane.svg
While his works of symmetry are ingenious, this paper investigates mostly those that depict the impossible. M. C. Escher created two types of impossible artwork— impossible geometries and impossible perspectives. Impossible geometries are all possible at any given point, and also have only one meaning at any given point, but are impossible on a higher level. Roger Penrose (the British mathematician) described the second type—impossible perspectives—as being “rather than locally unambiguous, but globally impossible, they are everywhere locally ambiguous, yet globally impossible” (Quoted from Coxeter, 154).
Cities are built in different ways and one Question I have always wanted to know is what is the design of roman cities. This paper will tell you the answer to this question in form of three other questions. How did they build their cities? Why did they build them this way? and Do we still build cities this way?
It is very likely that this rule is the same for any shape but I must
Step 4: Stand with the first mark (west) to your left and the second mark to your right--you are now facing north. This fact is true everywhere on earth.
The journey matters more than the destination when you gain as much knowledge as possible, but keep the destination in mind to keep you moving forward. The definition of a journey is, “The act of traveling from one place to another while acquiring as much knowledge as possible.” This
we are designing for. It has 20 parking slots in which all are available for customers.
Always ,roads have had a dual function : as traffic routes and as means of access to dwellings and other buildings ; it is only since the vast growth of transport which has arisen from the invention of the internal combustion that these have been seriously in conflict with each other . Today this conflict is very grave , and leads to terrible injury and loss of life and to grossly wasteful delay in the transport of worker and goods.
As stated by John Fuchs, “The triangle is the only two dimensional polygon that if constructed of rigid members with hinged corners is absolutely
The layout of modern day airport terminals are more than simple buildings offering protection from the weather as it was in the early days of air travel. In the early 1920’s the terminal was a small building mostly used for cargo and airmail, with very little amenities for passengers. As the demand for air travel increased the necessity for larger air facilities grew, soon airport style copied the main transportations of the time, the railroads. The smaller buildings became bigger art deco style of building, holding ticket offices, baggage handling areas and waiting areas for travelers. These early terminals most often were single level facilities that opened to the tarmac. As the number of passengers increased, these airports became inefficient and it became unsafe to have passengers walk across the apron. It was apparent that the ...
A. Transportation elements, which represents the impedance or difficulty of travel, usually measured by travel distance, time, or costs
Whebell, C.F. “Corridors: a theory of urban systems.” Annals of the Association of American Geographers. Vol.59; No.1. (March) (1969): 1-26.
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