section_path: Analytic results/Equations/Cartesian coordinates/Parametric form
section_name: Parametric form
paragraph_text: 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, {\displaystyle {\begin{aligned}x&=a+r\,\cos t,\\y&=b+r\,\sin t,\end{aligned}}} where t is a parametric variable in the range 0 to 2 π, 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 1 − t 2 1 + t 2, y = b + r 2 t 1 + t 2. {\displaystyle {\begin{aligned}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{2}}}.\end{aligned}}}
  
  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.