SECTION FOUR:
Other Polar Graphs

Introduction

Roses, limaçons, and cardioids are examples of graphs whose equations look particularly nice in polar coordinates.  We will now look at some other graphs whose equations also have this property.  We will specifically not consider lines, parabolas, ellipses, and hyperbolas at this point since they tend to be handled more easily in rectangular coordinates.

Circles

Certain types of circles have nice-looking equations in polar coordinates.  The most obvious type is a circle with center at the pole (or "Origin" in rectangular coordinates).  Consider the graph of r = 3.

Clear[r,t,x,y]
r[t_] = 3;
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p = ParametricPlot[{x[t],y[t]},{t,0,2Pi},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
t = Graphics[Text["r = 3",{3,3}]];

Show[p,t, DisplayFunction->$DisplayFunction];

You should also be able to write the equation of this circle in rectangular coordinates.

Other types of circles with simple polar representation are those of form 
r = aSinθ and r = aCosθ.  Consider two examples.

Clear[r1,r2,t,x1,y1,x2,y2,a,b]
r1[t_] = Sin[t];
x1[t_] = r1[t] Cos[t];
y1[t_] = r1[t] Sin[t];
r2[t_] = Cos[t];
x2[t_] = r2[t] Cos[t];
y2[t_] = r2[t] Sin[t];

p1 = ParametricPlot[{x1[t],y1[t]},{t,0,2Pi},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
t1 = Graphics[Text["r = Sin[t]",{.8,.9}]];
a=Show[p1,t1];
p2 = ParametricPlot[{x2[t],y2[t]},{t,0,2Pi},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
t2 = Graphics[Text["r = Cos[t]",{1.05,.5}]];
b=Show[p2,t2];
Show[GraphicsArray[{a,b},
	DisplayFunction->$DisplayFunction]];

Given what you know about the sine and cosine functions, these results are probably not surprising.  Try a few more examples if you want; there is not really too much more to the story on r = aSinθ and r = aCosθ.  Circles other than the types already mentioned have much more complicated equations in polar coordinates.  For most purposes they are more easily handled in rectangular coordinates.

Lemniscates

The graphs of the polar equations r¤ = a¤Sin2θ and r¤ = a¤Cos2θ
are lemniscates. Consider two examples.

First, r¤ = 9Sin2θ.

Clear[t,r1,r2,x1,y1,x2,y2,t1]
r1[t_] = 3Sqrt[Sin[2t]];
x1[t_] = r1[t] Cos[t];
y1[t_] = r1[t] Sin[t];
r2[t_] = -3Sqrt[Sin[2t]];
x2[t_] = r2[t] Cos[t];
y2[t_] = r2[t] Sin[t];
r3[t_] = 3Sqrt[Cos[2t]];
x3[t_] = r3[t] Cos[t];
y3[t_] = r3[t] Sin[t];
r4[t_] = -3Sqrt[Cos[2t]];
x4[t_] = r4[t] Cos[t];
y4[t_] = r4[t] Sin[t];
p1 = ParametricPlot[{x1[t],y1[t]},{t,0.0001,Pi/2},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
p2 = ParametricPlot[{x2[t],y2[t]},{t,0.0001,Pi/2},
     AxesLabel->{"x","y"},
     PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
t1 = Graphics[Text["r^2 = 9Sin[2t]",{3,2.5}]];
Show[p1,p2,t1,DisplayFunction->$DisplayFunction];

Now, r¤ = 9Cos2θ.

Clear[t,r3,r4,r5,r6,x3,y3,x4,y4,x5,y5,x6,y6,t3]
r3[t_] = 3Sqrt[Cos[2t]];
x3[t_] = r3[t] Cos[t];
y3[t_] = r3[t] Sin[t];
r4[t_] = -3Sqrt[Cos[2t]];
x4[t_] = r4[t] Cos[t];
y4[t_] = r4[t] Sin[t];
r5[t_] = 3Sqrt[Cos[2t]];
x5[t_] = r5[t] Cos[t];
y5[t_] = r5[t] Sin[t];
r6[t_] = -3Sqrt[Cos[2t]];
x6[t_] = r6[t] Cos[t];
y6[t_] = r6[t] Sin[t];
p3 = ParametricPlot[{x3[t],y3[t]},{t,0,.999 Pi/4},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
p4 = ParametricPlot[{x4[t],y4[t]},{t,0,.999 Pi/4},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
p5 = ParametricPlot[{x5[t],y5[t]},
			{t,1.0001 (3 Pi/4),Pi},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
p6 = ParametricPlot[{x6[t],y6[t]},
			{t,1.0001 (3 Pi/4),Pi},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotRange->All,
     AspectRatio->Automatic,
     DisplayFunction->Identity];
t3 = Graphics[Text["r^2 = 9Cos[2t]",{1.5,1.8}]];
Show[p3,p4,p5,p6,t3,DisplayFunction->$DisplayFunction];

You should consider the following questions about these two graphs:
								i.	Why do the graphs need to be generated in seperate
								 	sections?
								ii.	Why is it necessary to be very careful in choosing
								 		ranges for θ in the various sections?

Spirals

Spirals tend to have nice equations in polar coordinates.  One famous spiral is the Spiral of Archimedes,
 r = 2θ.

Clear[x,y,r,t,p,tex]
r[t_] = 2 t;
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p = ParametricPlot[{x[t],y[t]},{t,-10 Pi, 10Pi},
		PlotStyle->{RGBColor[1,0,0]},
		AspectRatio->Automatic,
		DisplayFunction->Identity];
tex = Graphics[Text["r = 2t",Scaled[{.9,.9}]]];
Show[p,tex,DisplayFunction->$DisplayFunction];

This is the correct graph of the Spiral of Archimedes, but it is visually somewhat overwhelming.  Splitting the θ range into two parts may help to clarify things.

First let's look at positive values of θ:

Clear[x,y,r,t,p1,tex1]
r[t_] = 2 t;
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p1 = ParametricPlot[{x[t],y[t]},{t,0, 10Pi},
		AspectRatio->Automatic,
		PlotStyle->{RGBColor[1,0,0]},
		DisplayFunction->Identity];
tex1 = Graphics[Text["r = 2t, t > 0",Scaled[{.85,.9}]]];
Show[p1,tex1,DisplayFunction->$DisplayFunction];

Now consider negative values of θ:

Clear[x,y,r,t,p2,tex2]
r[t_] = 2 t;
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p2 = ParametricPlot[{x[t],y[t]},{t,-10Pi,0},
		PlotStyle->{RGBColor[1,0,0]},
		AspectRatio->Automatic,
		DisplayFunction->Identity];
tex2 = Graphics[Text["r = 2t, t < 0",Scaled[{.15,.9}]]];
Show[p2,tex2,DisplayFunction->$DisplayFunction];

Another interesting type of spiral is the Logarithmic Spiral, r = e˘‚∕ˇ

Clear[x,y,r,t,p,tex]
r[t_] = E^((.01) t);
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p = ParametricPlot[{x[t],y[t]},{t,0, 10Pi},
		AspectRatio->Automatic,
		PlotStyle->{RGBColor[1,0,0]},
		PlotRange->All,
		DisplayFunction->Identity];
tex = Graphics[Text["r = E^((.01) t)",{1.4,1.3}]];
Show[p,tex,DisplayFunction->$DisplayFunction];

The interesting things to experiment with on this one include:
				i.	Varying the value of the parameter "a".
				ii.	Varying the range for θ.

A third type of spiral is the Hyperbolic Spiral, r = a/θ, a > 0.

For example, consider  r = 2/θ.

Clear[x,y,r,t,p,tex]
r[t_] = 2/t;
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p = ParametricPlot[{x[t],y[t]},{t,Pi/6, 5Pi},
		PlotStyle->{RGBColor[1,0,0]},
		PlotRange->{{-3.6,3.6},{-2,2}},
		AspectRatio->Automatic,
		DisplayFunction->Identity];
tex = Graphics[Text["r = 2/t",Scaled[{.9,.9}]]];
Show[p,tex,DisplayFunction->$DisplayFunction];

To get a different view, we can try changing the range for θ.

First try 6`< θ < 12`

Clear[x,y,r,t,p,tex,tex2]
r[t_] = 2/t;
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p = ParametricPlot[{x[t],y[t]},{t,6Pi, 12Pi},
		PlotStyle->{RGBColor[1,0,0]},
		AspectRatio->Automatic,
		DisplayFunction->Identity];
tex = Graphics[Text["r = 2/t",
				Scaled[{.9,.95}]]];
tex2 = Graphics[Text["6Pi<t<12Pi",
				Scaled[{.9,.92}]]];
Show[p,tex,tex2,DisplayFunction->$DisplayFunction];

This shows a part of the graph nearer the pole.

Now try `/12 < θ < 4`

Clear[x,y,r,t,p,tex,tex2]
r[t_] = 2/t;
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];
p = ParametricPlot[{x[t],y[t]},{t,Pi/12, 4Pi},
		AspectRatio->Automatic,
		PlotStyle->{RGBColor[1,0,0]},
		PlotRange->{{-2,6},{-2,2}},
		DisplayFunction->Identity];
tex = Graphics[Text["r = 2/t",Scaled[{.9,.9}]]];
tex2 = Graphics[Text["Pi/12 <t< 4Pi",
				Scaled[{.9,.8}]]];

Show[p,tex,tex2,DisplayFunction->$DisplayFunction];

This seems to indicate that y = 2 might be a horizontal asymptote.  If you think about the behaviour of (cosθ)/θ and (sinθ)/θ near θ = 0, you might be able to figure out what is going on.  What would be the effect of changing the value of "a"
in r = a/θ?