Circles Taxicab geometry

circles in discrete , continuous taxicab geometry

a circle set of points fixed distance, called radius, point called center. in taxicab geometry, distance determined different metric in euclidean geometry, , shape of circles changes well. taxicab circles squares sides oriented @ 45° angle coordinate axes. image right shows why true, showing in red set of points fixed distance center, shown in blue. size of city blocks diminishes, points become more numerous , become rotated square in continuous taxicab geometry. while each side have length





2


r


{\displaystyle {\sqrt {2}}r}

using euclidean metric, r circle s radius, length in taxicab geometry 2r. thus, circle s circumference 8r. thus, value of geometric analog



π


{\displaystyle \pi }

4 in geometry. formula unit circle in taxicab geometry




|

x

|

+

|

y

|

=
1


{\displaystyle |x|+|y|=1}

in cartesian coordinates and

r
=


1


|

sin
⁡
θ

|

+

|

cos
⁡
θ

|






{\displaystyle r={\frac {1}{|\sin \theta |+|\cos \theta |}}}

in polar coordinates.

a circle of radius 1 (using distance) von neumann neighborhood of center.

a circle of radius r chebyshev distance (l∞ metric) on plane square side length 2r parallel coordinate axes, planar chebyshev distance can viewed equivalent rotation , scaling planar taxicab distance. however, equivalence between l1 , l∞ metrics not generalize higher dimensions.

whenever each pair in collection of these circles has nonempty intersection, there exists intersection point whole collection; therefore, manhattan distance forms injective metric space.