Math 2.0

Geometry Definition

Circle

Builds on Euclidean planeEuclidean distance

Definition

Let dd be the Euclidean distance on the Euclidean plane ℝ2\mathbb{R}^2. Let c∈ℝ2c \in \mathbb{R}^2 be a point and r>0r > 0 a real number. The circle with the centre cc and the radius rr is the set { x∈ℝ2  :  d(x,c)=r }.\{\, x \in \mathbb{R}^2 \;:\; d(x, c) = r \,\}. With c=(a,b)c = (a, b), it is the set of the points (x,y)(x, y) with (x−a)2+(y−b)2=r2(x - a)^2 + (y - b)^2 = r^2.

Remarks

A chord is a segment between two points of the circle. A diameter is a chord through the centre, and its length is 2r2r. The circle has the length 2πr2 \pi r and bounds a disc of area πr2\pi r^2. A circle in ℝn\mathbb{R}^n is a circle in some plane of ℝn\mathbb{R}^n. Mathlib's EuclideanGeometry.Sphere is the sphere of a centre and a radius in a metric space; in the plane it is a circle.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Euclidean distance, concept, no evidence levelEuclidean distanceconceptEuclidean plane, object, no evidence levelEuclidean planeobjectCircle, definition, no evidence levelCircledefinition
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  1. Euclidean distance concept
  2. Euclidean plane object
  3. Circle definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Circle, lead section
  2. Web https://en.wikipedia.org/wiki/Circle, Terminology
  3. Web https://en.wikipedia.org/wiki/Circle, Equation of a circle
About this page
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Kinddefinition
WritingW1 stated
MSC51M04
DefinesCircle
UsesEuclidean plane, Euclidean distance

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