section_path: Generalizations/In other p -norms
section_name: In other p -norms
paragraph_text: In other p -norms
  
  Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In p -norm, distance is determined by ‖ x ‖ p = (| x 1 | p + | x 2 | p + ⋯ + | x n | p) 1 / p. {\displaystyle \left\|x\right\|_{p}=\left(\left|x_{1}\right|^{p}+\left|x_{2}\right|^{p}+\dotsb +\left|x_{n}\right|^{p}\right)^{1/p}.} In Euclidean geometry, p = 2, giving the familiar ‖ x ‖ 2 = | x 1 | 2 + | x 2 | 2 + ⋯ + | x n | 2. {\displaystyle \left\|x\right\|_{2}={\sqrt {\left|x_{1}\right|^{2}+\left|x_{2}\right|^{2}+\dotsb +\left|x_{n}\right|^{2}}}.}
  
  In taxicab geometry, p = 1. Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes. While each side would have length 2 r {\displaystyle {\sqrt {2}}r} using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2 r. Thus, a circle's circumference is 8 r. Thus, the value of a geometric analog to π {\displaystyle \pi } is 4 in this geometry. The formula for the unit circle in taxicab geometry is | x | + | y | = 1 {\displaystyle |x|+|y|=1} in Cartesian coordinates and r = 1 | sin θ | + | cos θ | {\displaystyle r={\frac {1}{\left|\sin \theta \right|+\left|\cos \theta \right|}}} in polar coordinates.
  
  A circle of radius 1 (using this distance) is the von Neumann neighborhood of its centre.
  
  A circle of radius r for the Chebyshev distance (L metric) on a plane is also a square with side length 2 r parallel to the coordinate axes, so planar Chebyshev distance can be viewed as equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L and L metrics does not generalize to higher dimensions.