# Circle in the plane

A generalization to higher dimensions is sphere in Euclidean space

## Contents

## Definition

### General definition

Consider , the Euclidean plane. Let be a point and be a positive real number. The circle with center and radius is the set of all points in that have distance exactly from .

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 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 , then the parametrization is:

Here , though we only need to vary over an interval of length or more.

### Equational description

In Cartesian coordinates, the equational description of a circle with center and radius is given by:

### Implicit description

*Fill this in later*

## The plane with respect to the circle

### Global study

The circle divides its complement in the Euclidean plane into two connected components. One component, called the *interior* of the circle, and also called an *open disc*, is the set of points that are at a distance from the center *less* than the radius. The other component, called the *exterior* of the circle, is the set of points whose distance from the center is greater than the radius.

The interior of a circle is contractible (in fact, we can contract it to the center by radial shrinking) whereas the exterior is *not*: it is homotopy-equivalent to the circle.

### Pointwise study

Some notions that come up when studying single points with respect to a circle include:

- Power of a point with respect to the circle
- Distance from the point to the center of the circle

Notions that relate multiple points:

- Inversion
- Conjugate points: Pairs of points satisfying a pole-polar relationship

### Points and lines

- Reciprocation is a map that sends points to lines and lines to points, sending a point to its
*polar*and a line to its*pole*

## Families of circles

Important families of circles that come up:

- Family of concentric circles
- Family of coaxial circles with intersection points
- Family of coaxial circles with limit points