Circle in the plane

A generalization to higher dimensions is sphere in Euclidean space

Definition

General definition

Consider $R^{2}$ , the Euclidean plane. Let $p \in R^{2}$ be a point and $r > 0$ be a positive real number. The circle with center $p$ and radius $r$ is the set of all points in $R^{2}$ that have distance exactly $r$ from $p$ .

Some easy facts:

Two circles centered at the same point are termed concentric circles. Given two concentric circles, there is a dilation, or scaling, about the common center that takes one circle to the other
Given two circles of the same radius but with different centers, there is a translation of $R^{2}$ that sends one circle to the other. Namely, choose the translation that sends the center of the first circle, to the center of the second circle.
The group of all orthogonal motions fixing the origin, sends each circle centered at the origin, to itself

Variant definitions

In complex analysis, it is sometimes convenient to view a line as a circle. We think of the center of the line as being a point at infinity, and the radius as infinity. This makes the theory of inversion, the geometric intuition behind complex analysis, as well as coaxial systems of circles, easier to comprehend.

Equational descriptions

Parametric description

The circle can be viewed as a parametrized curve. In Cartesian coordinates, if the center of the circle is given by $(x_{0}, y_{0})$ , then the parametrization is:

$t \mapsto (x_{0} + r \cos t, y_{0} + r \sin t)$

Here $t \in R$ , though we only need $t$ to vary over an interval of length $2 π$ or more.

Equational description

In Cartesian coordinates, the equational description of a circle with center $(x_{0}, y_{0})$ and radius $r$ is given by:

$(x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}$

Implicit description

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