# circle

Another way of defining the circle is thus: Given $A$ and $P$ as two points and $O$ as another point, the circle with center $O$ is the locus of points $X$ with $OX$ congruent   to $AP$. (Hilbert, 1927)

Notice there is no definition of distance needed to make that definition and so it works in many geometries  , even ones with no distance function. Hilbert uses it in his Foundations of Geometry book Also used by Forder in his Foundations of Euclidean geometry book, c 1927

In this entry, we only work with the standard Euclidean norm in the plane. A circle has one center only.

A circle determines a closed curve in the plane, and this curve is called the perimeter or circumference of the circle. If the radius of a circle is $r$, then the length of the perimeter is $2\pi r$. Also, the area of the circle is $\pi r^{2}$. More precisely, the interior of the perimeter has area $\pi r^{2}$. The diameter   of a circle is defined as $d=2r$.

## 1

Let us next derive an analytic equation for a circle in Cartesian coordinates  $(x,y)$. If the circle has center $(a,b)$ and radius $r>0$, we obtain the following condition for the points of the sphere,

 $(x-a)^{2}+(y-b)^{2}=r^{2}.$ (1)

In other words, the circle is the set of all points $(x,y)$ that satisfy the above equation. In the special case that $a=b=0$, the equation is simply $x^{2}+y^{2}=r^{2}$. The unit circle is the circle $x^{2}+y^{2}=1$.

It is clear that equation 1 can always be reduced to the form

 $x^{2}+y^{2}+Dx+Ey+F=0,$ (2)

where $D,E,F$ are real numbers. Conversely, suppose that we are given an equation of the above form where $D,E,F$ are arbitrary real numbers. Next we derive conditions for these constants, so that equation (2) determines a circle . Completing the squares yields

 $x^{2}+Dx+\frac{D^{2}}{4}+y^{2}+Ey+\frac{E^{2}}{4}=-F+\frac{D^{2}}{4}+\frac{E^{% 2}}{4},$

whence

 $\left(x+\frac{D}{2}\right)^{2}+\left(y+\frac{E}{2}\right)^{2}=\frac{D^{2}-4F+E% ^{2}}{4}.$

There are three cases:

1. 1.

If $D^{2}-4F+E^{2}>0$, then equation (2) determines a circle with center $(-\frac{D}{2},-\frac{E}{2})$ and radius $\frac{1}{2}\sqrt{D^{2}-4F+E^{2}}$.

2. 2.

If $D^{2}-4F+E^{2}=0$, then equation (2) determines the point $(-\frac{D}{2},-\frac{E}{2})$.

3. 3.

If $D^{2}-4F+E^{2}<0$, then equation (2) has no (real) solution in the $(x,y)$ - plane.

## 2 The circle in polar coordinates

Using polar coordinates for the plane, we can parameterize the circle. Consider the circle with center $(a,b)$ and radius $r>0$ in the plane $\mathbb{R}^{2}$. It is then natural to introduce polar coordinates $(\rho,\phi)$ for $\mathbb{R}^{2}\setminus\{(a,b)\}$ by

 $\displaystyle x(\rho,\phi)$ $\displaystyle=a+\rho\cos\phi,$ $\displaystyle y(\rho,\phi)$ $\displaystyle=b+\rho\sin\phi,$

with $\rho>0$ and $\phi\in[0,2\pi)$. Since we wish to parameterize the circle, the point $(a,b)$ does not pose a problem; it is not part of the circle. Plugging these expressions for $x,y$ into equation (1) yields the condition $\rho=r$. The given circle is thus parameterization by $\phi\mapsto(a+\rho\cos\phi,b+\rho\sin\phi)$, $\phi\in[0,2\pi)$. It follows that a circle is a closed curve in the plane.

## 3 Three point formula for the circle

Suppose we are given three points on a circle, say $(x_{1},y_{1})$, $(x_{2},y_{2})$, $(x_{3},y_{3})$. We next derive expressions for the parameters $D,E,F$ in terms of these points. We also derive equation (3), which gives an equation for a circle in terms of a determinant   .

First, from equation (2), we have

 $\displaystyle x_{1}^{2}+y_{1}^{2}+Dx_{1}+Ey_{1}+F$ $\displaystyle=0,$ $\displaystyle x_{2}^{2}+y_{2}^{2}+Dx_{2}+Ey_{2}+F$ $\displaystyle=0,$ $\displaystyle x_{3}^{2}+y_{3}^{2}+Dx_{3}+Ey_{3}+F$ $\displaystyle=0.$

These equations form a linear set of equations for $D,E,F$, i.e.,

 $\begin{pmatrix}x_{1}&y_{1}&1\\ x_{2}&y_{2}&1\\ x_{3}&y_{3}&1\end{pmatrix}\cdot\begin{pmatrix}D\\ E\\ F\end{pmatrix}=-\begin{pmatrix}x_{1}^{2}+y_{1}^{2}\\ x_{2}^{2}+y_{2}^{2}\\ x_{3}^{2}+y_{3}^{2}\\ \end{pmatrix}.$

Let us denote the matrix on the left hand side by $\Lambda$. Also, let us assume that $\det\Lambda\neq 0$. Then, using Cramer’s rule, we obtain

 $\displaystyle D$ $\displaystyle=-\frac{1}{\det\Lambda}\det\begin{pmatrix}x_{1}^{2}+y_{1}^{2}&y_{% 1}&1\\ x_{2}^{2}+y_{2}^{2}&y_{2}&1\\ x_{3}^{2}+y_{3}^{2}&y_{3}&1\\ \end{pmatrix},$ $\displaystyle E$ $\displaystyle=-\frac{1}{\det\Lambda}\det\begin{pmatrix}x_{1}&x_{1}^{2}+y_{1}^{% 2}&1\\ x_{2}&x_{2}^{2}+y_{2}^{2}&1\\ x_{3}&x_{3}^{2}+y_{3}^{2}&1\\ \end{pmatrix},$ $\displaystyle F$ $\displaystyle=-\frac{1}{\det\Lambda}\det\begin{pmatrix}x_{1}&y_{1}&x_{1}^{2}+y% _{1}^{2}\\ x_{2}&y_{2}&x_{2}^{2}+y_{2}^{2}\\ x_{3}&y_{3}&x_{3}^{2}+y_{3}^{2}\\ \end{pmatrix}.$

These equations give the parameters $D,E,F$ as functions  of the three given points. Substituting these equations into equation (2) yields

 $\displaystyle(x^{2}+y^{2})\det\begin{pmatrix}x_{1}&y_{1}&1\\ x_{2}&y_{2}&1\\ x_{3}&y_{3}&1\\ \end{pmatrix}$ $\displaystyle\mathbin{-}x\det\begin{pmatrix}x_{1}^{2}+y_{1}^{2}&y_{1}&1\\ x_{2}^{2}+y_{2}^{2}&y_{2}&1\\ x_{3}^{2}+y_{3}^{2}&y_{3}&1\\ \end{pmatrix}$ $\displaystyle\mathbin{-}y\det\begin{pmatrix}x_{1}&x_{1}^{2}+y_{1}^{2}&1\\ x_{2}&x_{2}^{2}+y_{2}^{2}&1\\ x_{3}&x_{3}^{2}+y_{3}^{2}&1\\ \end{pmatrix}$ $\displaystyle\mathbin{-}\det\begin{pmatrix}x_{1}&y_{1}&x_{1}^{2}+y_{1}^{2}\\ x_{2}&y_{2}&x_{2}^{2}+y_{2}^{2}\\ x_{3}&y_{3}&x_{3}^{2}+y_{3}^{2}\\ \end{pmatrix}=0.$

Using the cofactor expansion, we can now write the equation for the circle passing through $(x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})$ as [3, 4]

 $\det\begin{pmatrix}x^{2}+y^{2}&x&y&1\\ x_{1}^{2}+y_{1}^{2}&x_{1}&y_{1}&1\\ x_{2}^{2}+y_{2}^{2}&x_{2}&y_{2}&1\\ x_{3}^{2}+y_{3}^{2}&x_{3}&y_{3}&1\\ \end{pmatrix}=0.$ (3)

## References

 Title circle Canonical name Circle Date of creation 2013-03-22 13:36:23 Last modified on 2013-03-22 13:36:23 Owner PrimeFan (13766) Last modified by PrimeFan (13766) Numerical id 14 Author PrimeFan (13766) Entry type Definition Classification msc 51-00 Synonym circular Related topic SqueezingMathbbRn Related topic CurvatureOfACircle Defines unit circle Defines radius Defines radii Defines perimeter Defines circumference Defines three point formula for the circle Defines center