5.3 The circle and the ellipse

The equation of a circle can be formed easily if we recall the Pythagoras' theorem. again. A circle with centre at point $C=(c_1,c_2)$ and radius $r>0$ is the set of all points $(x,y)$ whose distance from $C$ is equal to $r$. Hence

\begin{equation*} (x-c_1)^2 + (y-c_2)^2 = r^2 \end{equation*}

This situation is shown in Figure 5.3.

Figure 5.3: Circle with centre at point $(c_1,c_2) \in \mathbb{R}^2$ and radius $r>0$.

The equation of an ellipse is given by

\begin{equation*} \frac{(x-c_1)^2}{a^2} + \frac{(y-c_2)^2}{b^2} = 1, \end{equation*}

where $a$ and $b$ are positive parameters and $A = (c_1,c_2)$ is the centre of the ellipse. The parameters $a$ and $b$ define the length of the semi-major axis and the semi-minor axis. If $a=b$ then we get a circle. A typical ellipse is depicted in Figure 5.4.

Figure 5.4: Ellipse with centre at $(c_1,c_2) \in \mathbb{R}^2$, semi-major axis $b$ and semi-minor axis $a$, $0 < a < b$.