section_path: Other representations/Polar coordinates
section_name: Polar coordinates
paragraph_text: Polar coordinates
  
  In a Cartesian plane, polar coordinates (r, θ) are related to Cartesian coordinates by the parametric equations: x = r cos θ, y = r sin θ. {\displaystyle x=r\cos \theta,\quad y=r\sin \theta.}
  
  In polar coordinates, the equation of a line not passing through the origin —the point with coordinates (0, 0) —can be written r = p cos (θ − φ), {\displaystyle r={\frac {p}{\cos(\theta -\varphi)}},} with r > 0 and φ − π / 2 < θ < φ + π / 2. {\displaystyle \varphi -\pi /2<\theta <\varphi +\pi /2.} Here, p is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and φ {\displaystyle \varphi } is the (oriented) angle from the x -axis to this segment.
  
  It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between the x -axis and the line. In this case, the equation becomes r = p sin (θ − α), {\displaystyle r={\frac {p}{\sin(\theta -\alpha)}},} with r > 0 and 0 < θ < α + π. {\displaystyle 0<\theta <\alpha +\pi.}
  
  These equations can be derived from the normal form of the line equation by setting x = r cos θ, {\displaystyle x=r\cos \theta,} and y = r sin θ, {\displaystyle y=r\sin \theta,} and then applying the angle difference identity for sine or cosine.
  
  These equations can also be proven geometrically by applying right triangle definitions of sine and cosine to the right triangle that has a point of the line and the origin as vertices, and the line and its perpendicular through the origin as sides.
  
  The previous forms do not apply for a line passing through the origin, but a simpler formula can be written: the polar coordinates (r, θ) {\displaystyle (r,\theta)} of the points of a line passing through the origin and making an angle of α {\displaystyle \alpha } with the x -axis, are the pairs (r, θ) {\displaystyle (r,\theta)} such that r ≥ 0, and θ = α or θ = α + π. {\displaystyle r\geq 0,\qquad {\text{and}}\quad \theta =\alpha \quad {\text{or}}\quad \theta =\alpha +\pi.}