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
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
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?
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.
It is very likely that this rule is the same for any shape but I must
As stated by John Fuchs, “The triangle is the only two dimensional polygon that if constructed of rigid members with hinged corners is absolutely
This design brief is intended to take into account the results of Heath’s VAST heuristic (relating to activities) (1989, in Elliott-Burns, 2003) and a selection of Lackney’s Design Principles (n.d.). Hennah’s (2007) concepts of traffic flow and layout will be incorporated. The design will also consider the requirements of integrating a guided inquiry pedagogy. The users of the space will also be given the opportunity to take part in the process.
...the other end there are only two. Such a figure is an impossible figure. It is possible, because, the depth cues and shape cues of one part of the figure do not have to be consistent with the depth and shape cues of another part of the same figure.
Whebell, C.F. “Corridors: a theory of urban systems.” Annals of the Association of American Geographers. Vol.59; No.1. (March) (1969): 1-26.
The spatial structure of the city has a huge impact on how transport networks function. In the case of South Africa with the effects of apartheid planning still looming in the background the ordinary citizen is the one that is affected by this in the end. Public transportation in South Africa has faced a lot problems in the past, some areas in the country are better off than others. The current government is still perpetuating elements of apartheid style planning in the new South Africa. To this day people are still being pushed to the fringes of the city, increasing the pressure on the state to provide better access to public transportation for these newly developed communities (Lucas, 2011:2). The government faces the challenge of providing safe, efficient, accessible and affordable means of public transportation. With high levels of urbanisation and inward migration from other African countries taking place, it is becoming a major problem for the state to provide adequate services for its people. The physical structure of South African cities could be argued to be the main source of all the issue facing the country’s public transportation system. The physical structure of the country and the way in which transportation planning was approached in the past could be seen as the stumbling block that prevents meaningful change in improving public transportation in the future. In attempting to answer the question this paper will be evaluating which mode of public transportation is more superior, either buses or mini buses (taxi). The paper will start off with a short history of public transportation in South Africa, followed by the social, economic and environmental factors that would be used to assess which mode of ...
We all use vehicles for transportation. People usually go to their desired destination either by driving their own cars or traveling in public transportation. Actually, it might be tough to choose that which one is the best selection for people to travel. Many people choose one of them according to their comfort while traveling and both of them have advantages and disadvantages in different conditions. Public transportation and driving own car both shares differences and similarities in many aspects, such as facility, cost, and comfort as well as traffic jam and accident occurrence.
The road can take many shapes and many forms, but for today we will talk of a more simple kind of road. A road that guides pedestrians through a park. The city park is the perfect intercourse between nature and the urban. It is the unspoken hero of the urbanite. As the seasons wax and wane as do the nature of the surrounding nature. Even the surrounding wildlife will come and go as all that is green turns gray. Although the road is an ageless guardian that will lead urbanites through the
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.
A. Transportation elements, which represents the impedance or difficulty of travel, usually measured by travel distance, time, or costs