Essay about Conic Sections in Taxicab Geometry

Essay about Conic Sections in Taxicab Geometry

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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...


... middle of paper ...


... (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

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