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.…show more content… All of them are distant from the origin or 5 units (kilometers) and that is where the Euclidian and taxi distance match each other. To find other points, we should move among x-axis and then up and down as far as possible. We have enough fuel for five kilometers in one-way, which means that:
|x|+|y|=5
When solving the equation we get: y=5-x for x,y>0 y=x-5 for x>0; y≤0 y=-x-5 for x,y≤0 y=x+5 for x≤0;y>0
So when all the functions are drawn we get the final picture of taxi circle.

Figure 4: Picture showing a taxicab circle of radius 5.

CIRCLE

A circle is a set of all points that are given distance, called radius, usually denoted by r, away from the center.

An equation of Euclidean circle:

〖(x-h)〗^2+〖(y-k)〗^2=r^2

Figure 5: An Euclidean circle

Figure 6: A taxicab circle

CONIC SECTIONS OF TAXICAB GEOMETRY

In the following part of an essay, the conic sections in taxicab distance are researched, as they must vary from the Euclidean distance as well as circle of taxicab distance from circle of Euclidean distance.

WHAT ARE CONIC SECTIONS the conic section is the locus of a point, such that its distance from focus is in the constant ratio to its distance from directrix ratio, called e; if e<1, the conic section is an ellipse, if e =1 a conic section is a parabola and if e >1 the conic section is a