Equation of a Circle

Cartesian coordinates

Circle of radius r = 1, centre (a, b) = (1.2, −0.5)

In an x–y Cartesian coordinate system, the circle with center coordinates (a, b) and radius r is the set of all points (x, y) such that

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

This equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simplifies to

$x^{2}+y^{2}=r^{2}.$

Parametric form

The equation can be written in parametric form using the trigonometric functions sine and cosine as

$x=a+r\,\cos t,$

$y=b+r\,\sin t,$

where t is a parametric variable in the range 0 to $2\pi$, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x axis.

An alternative parametrisation of the circle is

$x=a+r{\frac {1-t^{2}}{1+t^{2}}},$

$y=b+r{\frac {2t}{1+t^{2}}}.$

In this parameterisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x axis (see Tangent half-angle substitution). However, this parameterisation works only if t is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.

Resourcs

• Circle Equation Review, Khan Academy
• Standard Form of Circle Equation, Khan Academy
• Circle, Wikipedia