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Geometric Figures > Circles

Having changed the Euclidean metric, even very simple geometric figures take on new forms. The circle is the poster child for taxicab geometric figures.

One description of a circle is the set of all points equidistant from a fixed point. To begin investigating the shape of a circle in taxicab geometry, we could begin with two points and try to understand how to move off of one of the points to another point while keeping the distance to the second point constant.

Therefore, begin with two points (Figure 1). If we begin moving from one of the points in a horizontal direction towards the other point, the taxicab distance will get smaller. To keep the distance constant, we must move in the vertical direction an equal amount to increase the distance and compensate for the horizontal movement. So, distance is kept constant by moving along lines of slope 1 or -1.

FIGURE 1: Finding a path in taxicab geometry to keep
the distance to a fixed point constant.

Extending this observation, we see that the set of all points equidistant from a central point is now a square with its edges oriented 45 degrees to the horizontal (Figure 2).

FIGURE 2: A taxicab circle of radius 1.

The Equation of a Circle

For a circle of radius r centered at a point (h,k), any point (x,y) on the circle must be a distance r from the center. This is simply the definition of a circle. So, using the taxicab distance formula, the distance from (h,k) to a point (x,y) on the circle is

This is the general equation for a taxicab circle of radius r centered at (h,k).

The Value of π

If we maintain the Euclidean definition of the constant π being the ratio of the circumference of a circle to its diameter, we see from the unit circle in Figure 2 that the value of π in taxicab geometry is πt = 8/2 = 4.

Circumference of Circle

The circumference of the unit circle in Figure 2 is C = 2 + 2 + 2 + 2 = 8. If the radius of the circle is r, we have the general formula for the circumference of a taxicab circle which strikingly resembles the Euclidean formula for a Euclidean circle:

References
[1] Euler, Russell and Jawad Sadek. The πs Go Full Circle, Mathematics Magazine, Vol. 72, No. 1 (Feb 1999), pp. 59-63.
[2] Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry, 1975.

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