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Circle

Circle

Object type: Plane curve

Definition

In $\mathbb{R}^2$, the circle $C$ of radius $r > 0$ centred at the origin is the set of all points $(x, y)$ satisfying the equation $$x^2 + y^2 = r^2.$$ In particular, if $r = 1$, we have the unit circle, shown below.

The length of the curve (the circumference of the circle) is $2 \pi r$ and the enclosed area is $A = \pi r^2$. The diameter $D = 2 r$ may be defined as the greatest (Euclidean) distance between two points on the circle.

Parameterisations

From the definition of the trigonometric functions, it is immediate that $C = \mathbf{r} \left(\left[0, 2\pi\right[\right)$ where $$\mathbf{r}\left(t\right) = \underline{\mathbf{e}} \begin{pmatrix}r \cos{t} \\ r \sin{t}\end{pmatrix},\quad\quad \forall t \in \left[0, 2\pi \right[.$$ It is also evident that every $\mathbf{r}(t) \in C$ because of the trigonometric identity $\sin^2 {t} + \cos^2 {t} = 1$.

The defining equation of the circle is equivalent to $y = \pm \sqrt{r^2 - x^2}$. Hence, the upper part ($y \ge 0$) of the circle can also be parameterised using $x \in \left[ -r, r\right]$ as parameter by putting $y = \sqrt{r^2 - x^2}$, or, more explicitly, $$\mathbf{r}\left(t\right) = \underline{\mathbf{e}} \begin{pmatrix}t \\ \sqrt{r^2 - t^2} \end{pmatrix}, \quad\quad \forall t \in \left[-r, r\right].$$ In other words, the upper part is the graph of the function $x \mapsto \sqrt{r^2 - x^2}$. The lower part of the circle can be parameterised by putting $y = -\sqrt{r^2 - x^2}$.

A General Circle

More generally, the circle of radius $r > 0$ centred at $(x_0, y_0) \in \mathbb{R}^2$ is the set of points $(x, y)$ satisfying $$(x-x_0)^2 + (y-y_0)^2 = r^2$$ which is also the image of the parameterisation map $$t \mapsto \underline{\mathbf{e}}\begin{pmatrix}x_0 + r \cos{t}\\ y_0 + r \sin{t}\end{pmatrix}, \quad\quad \forall t \in \left[0, 2\pi\right[.$$ It can also be written $$C = \left\{ \mathbf{x} \in \mathbb{R}^2: \| \mathbf{x} − \mathbf{x}_0 \| = r\right\}$$ where $\|\cdot\|$ is the Euclidean norm and $\mathbf{x}_0 = \left(x_0, y_0\right)$.

Curvature

The curvature function of a circle of radius $r > 0$ using the standard trigonometric parameterisation (with any translation) is $$\kappa = \frac{1}{r}.$$ Hence, a circle is a curve of constant positive curvature.

Generalisations

If one applies a linear transformation $\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with matrix $$\begin{pmatrix}a & 0 \\ 0 & b\end{pmatrix}$$ ($a, b > 0$) to a circle centred at the origin, one obtains an ellipse that is not a circle if $a \ne b$, namely, the ellipse $$\left(\frac{x}{a}\right)^2 + \left(\frac{y} {b}\right)^2 = 1.$$