Deforming a Circle

Geometry Level 2

In taxicab geometry, Euclidean distance for a point ( x 1 , y 1 ) (x_1,y_1) and a point ( x 2 , y 2 ) (x_2,y_2) d = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 d=\sqrt {(x_1-x_2)^2+(y_1-y_2)^2} is replaced by Manhattan distance d = ∣ x 1 − x 2 ∣ + ∣ y 1 − y 2 ∣ d=|x_1-x_2|+|y_1-y_2| and the shape of circles may change as well.

What is the shape of taxicab circles?

Square Circle Hexagon Equilateral triangle

Brian Lie by Brian Lie , 26, China

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1 solution

Brian Lie
May 26, 2018

A circle is a set of points with a fixed distance, so the formula for the circle of radius r r centered at the orgin in taxicab geometry is ∣ x ∣ + ∣ y ∣ = r |x|+|y|=r in Cartesian coordianates whose graph is a square \boxed{\text{square}} , as shown in the picture.

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