Editing Circle (section)

===Equations===
==== Cartesian coordinates ====
[[Image:Circle center a b radius r.svg|thumb|right|Circle of radius ''r''&nbsp;=&nbsp;1, centre (''a'',&nbsp;''b'') =&nbsp;(1.2,&nbsp;−0.5)]]
===== Equation of a circle =====
In an ''x''–''y'' [[Cartesian coordinate system]], the circle with centre [[Coordinate system|coordinates]] (''a'', ''b'') and radius ''r'' is the set of all points (''x'', ''y'') such that
<math display="block">(x - a)^2 + (y - b)^2 = r^2.</math>

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,&nbsp;0), then the equation simplifies to
<math display="block">x^2 + y^2 = r^2.</math>

===== One coordinate as a function of the other =====
[[File:Circle derivative.png|thumb| Upper semicircle with radius {{math|1}} and center {{math|(0, 0)}} and its derivative.]]

The circle of radius {{tmath|r}} with center at {{tmath|(x_0, y_0)}} in the {{tmath|x}}–{{tmath|y}} plane can be broken into two semicircles each of which is the [[graph of a function]], {{tmath|y_+(x)}} and {{tmath|y_-(x)}}, respectively:
<math display=block>\begin{align}
y_+(x) = y_0 + \sqrt{ r^2 - (x - x_0)^2}, \\[5mu]
y_-(x) = y_0 - \sqrt{ r^2 - (x - x_0)^2},
\end{align}</math>
for values of {{tmath|x}} ranging from {{tmath|x_0 - r}} to {{tmath|x_0 + r}}.

===== Parametric form =====
The equation can be written in [[parametric equation|parametric form]] using the [[trigonometric function]]s sine and cosine as
<math display="block">\begin{align}
x &= a + r\,\cos t, \\
y &= b + r\,\sin t,
\end{align}</math>
where ''t'' is a [[parametric variable]] in the range 0 to 2{{pi}}, interpreted geometrically as the [[angle]] that the ray from (''a'',&nbsp;''b'') to (''x'',&nbsp;''y'') makes with the positive ''x''&nbsp;axis.

An alternative parametrisation of the circle is
<math display="block">\begin{align}
x &= a + r \frac{1 - t^2}{1 + t^2}, \\
y &= b + r \frac{2t}{1 + t^2}.
\end{align}</math>

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''&nbsp;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.

===== 3-point form =====
The equation of the circle determined by three points <math>(x_1, y_1), (x_2, y_2), (x_3, y_3)</math> not on a line is obtained by a conversion of the [[Ellipse#Circles|''3-point form of a circle equation'']]:
<math display="block">
\frac{({\color{green}x} - x_1)({\color{green}x} - x_2) + ({\color{red}y} - y_1)({\color{red}y} - y_2)}
     {({\color{red}y} - y_1)({\color{green}x} - x_2) - ({\color{red}y} - y_2)({\color{green}x} - x_1)} =
\frac{(x_3 - x_1)(x_3 - x_2) + (y_3 - y_1)(y_3 - y_2)}
     {(y_3 - y_1)(x_3 - x_2) - (y_3 - y_2)(x_3 - x_1)}.</math>

===== Homogeneous form =====
In [[homogeneous coordinates]], each [[conic section]] with the equation of a circle has the form
<math display="block">x^2 + y^2 - 2axz - 2byz + cz^2 = 0.</math>

It can be proven that a conic section is a circle exactly when it contains (when extended to the [[complex projective plane]]) the points ''I''(1: ''i'': 0) and ''J''(1:&nbsp;−''i'':&nbsp;0). These points are called the [[circular points at infinity]].

====Polar coordinates====
In [[polar coordinates]], the equation of a circle is
<math display="block">r^2 - 2 r r_0 \cos(\theta - \phi) + r_0^2 = a^2,</math>

where ''a'' is the radius of the circle, <math>(r, \theta)</math> are the polar coordinates of a generic point on the circle, and <math>(r_0, \phi)</math> are the polar coordinates of the centre of the circle (i.e., ''r''<sub>0</sub> is the distance from the origin to the centre of the circle, and ''φ'' is the anticlockwise angle from the positive ''x''&nbsp;axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. {{nowrap|''r''<sub>0</sub> {{=}} 0}}, this reduces to {{nowrap|''r'' {{=}} ''a''}}. When {{nowrap|''r''<sub>0</sub> {{=}} ''a''}}, or when the origin lies on the circle, the equation becomes
<math display="block">r = 2 a\cos(\theta - \phi).</math>

In the general case, the equation can be solved for ''r'', giving
<math display="block">r = r_0 \cos(\theta - \phi) \pm \sqrt{a^2 - r_0^2 \sin^2(\theta - \phi)}.</math>
Without the ± sign, the equation would in some cases describe only half a circle.

====Complex plane====
In the [[complex plane]], a circle with a centre at ''c'' and radius ''r'' has the equation
<math display="block">|z - c| = r.</math>

In parametric form, this can be written as
<math display="block">z = re^{it} + c.</math>

The slightly generalised equation
<math display="block">pz\overline{z} + gz + \overline{gz} = q</math>

for real ''p'', ''q'' and complex ''g'' is sometimes called a [[generalised circle]]. This becomes the above equation for a circle with <math>p = 1,\ g = -\overline{c},\ q = r^2 - |c|^2</math>, since <math>|z - c|^2 = z\overline{z} - \overline{c}z - c\overline{z} + c\overline{c}</math>. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a [[line (geometry)|line]].