Polar Curves and Cartesian Graphs
Author: Jamie Mulholland
Date: November 28, 2008

To sketch the graph of a polar equation a good first step is to sketch the graph in the cartesian coordinate system.  This will give a graphical depiction of how r changes with theta.  
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The information about how r changes with theta can then be used to sketch the graph of the equation in the polar coordinate system.

You can change the polar equation here.  (Use x instead of theta in the input.  Technically we should be using theta so this is a bug that needs to be fixed.)

radius & angle

Cartesian 
Coordinate
Graph

Polar
Coordinate
Graph

θ = \frac{\pi}{6}

θ = \frac{\pi}{3}

θ = \frac{\pi}{2}

θ = \frac{2\pi}{3}

θ = \frac{5\pi}{6}

θ = \pi

θ = \frac{7\pi}{6}

θ = \frac{3\pi}{2}

θ = \frac{5\pi}{3}

θ = \frac{4\pi}{3}

θ = \frac{11\pi}{6}

θ = 0, 2\pi

Move Me!

\frac{\pi}{2}

\pi

\frac{3\pi}{2}

2\pi

Polar Equation: 

polarlength = 3.5
pole = (0, 0)
xCartLength = 12.566370614359172
xCartLeftEnd = -3
yCartLength = 4
yCartHeight = 8
f(x) = 1 - 2 cos(x)
xCartRightEnd: xCartLeftEnd + xCartLength
fCartFunction = Function(0.5 f(x - xCartLeftEnd) + yCartHeight, xCartLeftEnd, xCartRightEnd)
r(x) = f(x)
T = 1.4000000000000001 (slider from 0 to 12.6, step 0.1)
P': (abs(r(T)) cos(T), abs(r(T)) sin(T))
C = CurveCartesian(r(theta) cos(theta), r(theta) sin(theta), theta, 0, T)
P: (r(T) cos(T), r(T) sin(T))
ThetaRayPositive = Ray((0, 0), P')
T1: " θ = " + T
T2: "r = " + (r(T))
θ = Angle(xAxis, ThetaRayPositive)
T3: "θ = " + T
T4: " r = " + r
ThetaRayNegative = Ray((0, 0), -P')
R = Segment((0, 0), P)
a = true
T5: "r = " + (r(T))
fCart = CurveCartesian(t, fCartFunction(t), t, xCartLeftEnd, xCartRightEnd)
PCart: (T + xCartLeftEnd, fCartFunction(T + xCartLeftEnd))
xAxisCart = CurveCartesian(t - 1, yCartHeight, t, xCartLeftEnd, xCartRightEnd)
yAxisCart = CurveCartesian(xCartLeftEnd, t, t, yCartHeight - 0.5 yCartLength, yCartHeight + 0.5 yCartLength)
polaraxis = Segment(pole, polarlength (1, 0))
ThetaRay3 = Segment(pole, polarlength (cos(3.141592653589793 / 3), sin(3.141592653589793 / 3)))
ThetaRay1 = Segment(pole, polarlength (cos(3.14 / 6), sin(3.14 / 6)))
ThetaRay4 = Segment(pole, polarlength (cos(3.141592653589793 / 2), sin(3.141592653589793 / 2)))
ThetaRay5 = Segment(pole, polarlength (cos(2 * 3.141592653589793 / 3), sin(2 * 3.141592653589793 / 3)))
ThetaRay7 = Segment(pole, polarlength (cos(5 * 3.14 / 6), sin(5 * 3.14 / 6)))
ThetaRay8 = Segment(pole, polarlength (cos(3.14), sin(3.14)))
ThetaRay9 = Segment(pole, polarlength (cos(7 * 3.14 / 6), sin(7 * 3.14 / 6)))
ThetaRay10 = Segment(pole, polarlength (cos(8 * 3.14 / 6), sin(8 * 3.14 / 6)))
ThetaRay12 = Segment(pole, polarlength (cos(10 * 3.14 / 6), sin(10 * 3.14 / 6)))
ThetaRay11 = Segment(pole, polarlength (cos(9 * 3.14 / 6), sin(9 * 3.14 / 6)))
ThetaRay13 = Segment(pole, polarlength (cos(11 * 3.14 / 6), sin(11 * 3.14 / 6)))
rCartSegment = Segment(PCart, (x(PCart), yCartHeight))
T6: "θ = " + T
T7: "r = " + (f(T))