page_title: Circle
section_title: In other p-norms
hierarchical_section_title: Circle / In other p-norms
context_page_description: A circle is a shape consisting of all points in a plane that are a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius.
  Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.
  A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.
context_section_description: 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
  In Euclidean geometry, p = 2, giving the familiar
  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  using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to  is 4 in this geometry. The formula for the unit circle in taxicab geometry is  in Cartesian coordinates and
  in polar coordinates.
  A circle of radius 1 (using this distance) is the von Neumann neighborhood of its center.
  A circle of radius r for the Chebyshev distance (L∞ metric) on a plane is also a square with side length 2r 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 L₁ and L∞ metrics does not generalize to higher dimensions.
caption: Illustrations of unit circles (see also superellipse) in different p-norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding p).