Definition of a circle

In the Cartesian plane the circle is defined, as in Euclidean geometry, through a condition on distances: all and only the points that lie at a fixed distance from a given point.

Definition — Circle

The circle with centre $C(x_0;y_0)$ and radius $R>0$ is the locus of the points of the plane that lie at exactly distance $R$ from $C$: $P \in \text{circonferenza} \iff d(P, C) = R.$

The centre $C$ and the radius $R$ completely determine the circle: knowing these two data is enough to draw it and, as we shall see, to write its equation.

Explore the definition with the simulation: drag the centre $C$ and adjust the radius $r$ to see how the equation of the circle changes.

Drag the centre $C$ and move the slider $r$: the equation updates in real time.

Links

Topics: Analytical circle
Concepts: Centre · Circle · Distance between points · Locus · Cartesian plane · Radius
Skills: Analytical geometry