section_path: Analytic results/Equations/Polar coordinates
section_name: Polar coordinates
paragraph_text: Polar coordinates
  
  In polar coordinates, the equation of a circle is r 2 − 2 r r 0 cos (θ − ϕ) + r 0 2 = a 2, {\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi)+r_{0}^{2}=a^{2},}
  
  where a is the radius of the circle, (r, θ) {\displaystyle (r,\theta)} are the polar coordinates of a generic point on the circle, and (r 0, ϕ) {\displaystyle (r_{0},\phi)} are the polar coordinates of the centre of the circle (i.e., r is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of the circle). For a circle centred on the origin, i.e. r = 0, this reduces to r = a. When r = a, or when the origin lies on the circle, the equation becomes r = 2 a cos (θ − ϕ). {\displaystyle r=2a\cos(\theta -\phi).}
  
  In the general case, the equation can be solved for r, giving r = r 0 cos (θ − ϕ) ± a 2 − r 0 2 sin 2 (θ − ϕ). {\displaystyle r=r_{0}\cos(\theta -\phi)\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi)}}.} Without the ± sign, the equation would in some cases describe only half a circle.