SECTION THREE:
Limaçons and Cardioids

Introduction

The graph of r = 2 - Cosθ  is one type of limaçon.

Clear[r,t,x,y,p,g]
r[t_] = 2 - Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

p=ParametricPlot[{x[t],y[t]},{t,0,2Pi},
     				AxesLabel->{"x","y"},
					PlotRange->All,
					PlotStyle->{RGBColor[1,0,0]},
     				AspectRatio->Automatic,
					DisplayFunction->Identity];

g=Graphics[Text["r = 2 -  Cos[t]",{1,2.1}]];

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

Could you have used your knowledge of the cosine function to predict the 
shape of r = 2 - Cosθ?

Now let's try a similar function;  the graph of r = 1 - Cosθ  is a cardioid.

Clear[r,t,x,y,p,g]
r[t_] = 1 - Cos[t];
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->{{-2.1,1.5},{-1.5,1.5}},
     				AspectRatio->Automatic,
					DisplayFunction->Identity];

g=Graphics[Text["r = 1 - Cos[t]",{.77,1.1}]];

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

Could you have used your knowledge of the cosine function to predict the 
shape of r = 1 - Cosθ?

Now another variation; the graph of r = 1 -2Cosθ  is another type of limaçon.

Clear[r,t,x,y,p,g]
r[t_] = 1 - 2 Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

p=ParametricPlot[{x[t],y[t]},{t,0,2Pi},
     				AxesLabel->{"x","y"},
					PlotRange->All,
					PlotStyle->{RGBColor[1,0,0]},
     				AspectRatio->Automatic,
					DisplayFunction->Identity];

g=Graphics[Text["r = 1 - 2Cos[t]",{.9,1.3}]];

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

Could you have used your knowledge of the cosine function to predict the 
shape of r = 1 - 2Cosθ?

The graphs of  r = 2 - Cosθ, r = 1 - Cosθ, and r = 1 - 2Cosθ, are examples of the three basic types of limaçons and cardioids. Let's look at each type in more detail.

Limaçons of the Type r = a - bCosθ, for a > b.

The graph of  r = a - bCosθ, for a > b is one type of limaçon.  
First let's look at how the graph of r = 2 - Cosθ is traced out as θ varies 
from 0 to 2`.   Select and animate the following graphics.

Clear[r,t,x,y]
r[t_] = 2 - Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

Do[ParametricPlot[{x[t],y[t]},{t,0,k Pi/6},           
     PlotRange->{{-3.5,1.5},{-3,3}},                  
	 PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},          
     AspectRatio->Automatic],                         
     {k,1,12}];

Select the above cells and  press command-y to animate

Now show how varying a affects the graph of r = a - bCosθ, for 
			b = 1, a > b.  	
	Plot		r = a - Cosθ for a = 1.25, 1.5, 1.75,and 2.

Clear[r,t,x,y]
r[t_] = (.25 k) - Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

Do[ParametricPlot[{x[t],y[t]},{t,0,2Pi},
     PlotRange->{{-3,1},{-2.5,2.5}},
	 PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     AspectRatio->Automatic],
     {k,5,8}];

To get a better idea of what happens to  r = a - Cosθ as a varies from 1.25 to 2, select and animate the above graphics.

Next show how varying b affects the graph of r = a - bCosθ, for a = 2, a > b.  	
Plot		r = 2 - bCosθ for b = 1, 1.25, 1.5, and 1.75.

Clear[r,t,x,y]
r[t_] = 2 - (.25 k) Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

Do[ParametricPlot[{x[t],y[t]},{t,0,2Pi},
     PlotRange->{{-4.5,1.5},{-3,3}},
   	 PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     AspectRatio->Automatic],
     {k,4,7}];

To get a better idea of what happens to  r = 2 - bCosθ as b varies from 
1 to 1.75, select and animate the above graphics.

By now you have a pretty good idea of how varying b affects the graph 
		of r = a - bCosθ, for a > b.  Now let's look at what happens if a = b.

Cardiods:  Functions of the Type r = a - bCosθ, 
 for a = b.

Let's look at r = 1 - Cosθ, r = 2 - 2Cosθ, and r = 3 - 3Cosθ.

Clear[r,t,x,y,p1l]
r1c[t_] = 1 - Cos[t];
r2c[t_] = 2 - 2Cos[t];
r3c[t_] = 3 - 3Cos[t];
x1[t_] = r1c[t] Cos[t];
y1[t_] = r1c[t] Sin[t];
x2[t_] = r2c[t] Cos[t];
y2[t_] = r2c[t] Sin[t];
x3[t_] = r3c[t] Cos[t];
y3[t_] = r3c[t] Sin[t];

p1l=ParametricPlot[{x1[t],y1[t]},{t,0,2Pi},
     PlotRange->{{-6.5,3.5},{-5,5}},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotLabel->"r = 1 - Cos[t]",
     AspectRatio->Automatic,
     DisplayFunction->Identity];

p2l=ParametricPlot[{x2[t],y2[t]},{t,0,2Pi},
     PlotRange->{{-6.5,3.5},{-5,5}},
	 PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     PlotLabel->"r = 2 - 2Cos[t]",
     AspectRatio->Automatic,
     DisplayFunction->Identity];

p3l=ParametricPlot[{x3[t],y3[t]},{t,0,2Pi},
     PlotRange->{{-6.5,3.5},{-5,5}},
	 PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},
     PlotLabel->"r = 3 - 3Cos[t]",
     AspectRatio->Automatic,
     DisplayFunction->Identity];

Show[GraphicsArray[{p1l,p2l,p3l},
					DisplayFunction->$DisplayFunction]];

The graph of r = a - bCosθ, for a = b, is a cardioid.  As you can see, other than size, cardioids don't exhibit much variety.

Limaçons of the Type r = a - bCosθ, for a < b.

The graph of r = a - bCosθ, for a < b, is a different type of limaçon.  
Recall our earlier example, r = 1 - 2Cosθ.

Clear[r,t,x,y]
r[t_] = 1 - 2 Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

ParametricPlot[{x[t],y[t]},{t,0,2Pi},
     AxesLabel->{"x","y"},
	 PlotStyle->{RGBColor[1,0,0]},
     PlotLabel->"r = 1 - 2Cos[t]",
     AspectRatio->Automatic];

Now look at how this graph is traced out as θ varies from 0 to 2`.

Clear[r,t,x,y]
r[t_] = 1 - 2Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

Do[ParametricPlot[{x[t],y[t]},{t,0,k Pi/6},        
     PlotRange->{{-3.5,1},{-3,3}},                 
	 PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},       
     PlotLabel->"r = 1 - 2Cos[t]",                 
     AspectRatio->Automatic],                      
     {k,1,12}];

Let's look at how varying a affects the graph of r = a - bCosθ, for b = 2,
 a < b.  	
Plot		r = a - 2Cosθ for a = 1, 1.25, 1.5, and 1.75.

Clear[r,t,x,y]
r[t_] = (.25 k) - 2 Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

Do[ParametricPlot[{x[t],y[t]},{t,0,2Pi},												
     PlotRange->{{-4,1.5},{-3.5,3.5}},              
	 PlotStyle->{RGBColor[1,0,0]},
     AxesOrigin->{0,0},AxesLabel->{"x","y"},        
     AspectRatio->Automatic],                       
     {k,4,7}];

For more help in understanding what is happening, animate the above graphics.

Next show how varying b affects the graph of r = a - bCosθ, fora = 1, a < b.  	
Plot		r =1 -bCosθ for b =  1.25, 1.5, 1.75, and 2.

Clear[r,t,x,y]
r[t_] = 1 - .25k Cos[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

Do[ParametricPlot[{x[t],y[t]},{t,0,2Pi},												
     	PlotRange->All,              
     	AxesOrigin->{0,0},
		PlotStyle->{RGBColor[1,0,0]},
		AxesLabel->{"x","y"},        
     	AspectRatio->Automatic],                       
     	{k,5,8}];

Again,animation may help you to see what is going on.

Limaçons and Cardioids

What you have seen so far probably leads you to believe that there is a strong relatioship between the two types of limacons and cardioids.  The following may help to reinforce the relationship.  Select and animate the graphics below.  They show what happens to the the graph of r = a - bCosθ, first for a varying from 2 to 1 with b = 1, then for b varying from 1 to 2 with a = 1.

Explain why the two types of limacons and the cardioids are all special cases of the same concept.

Changing Cosine to Sine

Only one major issue remains in our investigation of limacons and the cardioids:  suppose we change the cosine function to the sine function in r = a - bCosθ.  How will this effect the graphs?
		Let's make that change in our original three graphs.

i.	Graph of r = 2 - Sinθ .

Clear[r,t,x,y,p,g]
r[t_] = 2 - Sin[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

p=ParametricPlot[{x[t],y[t]},{t,0,2Pi},
				PlotRange->All,
     			AxesLabel->{"x","y"},
				DisplayFunction->Identity,
     			AspectRatio->Automatic];

g=Graphics[Text["r = 2 - Sin[t]",{1.3,1.2}]];

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

ii.	Graph r = 1 - Sinθ.

Clear[r,t,x,y,p,g]
r[t_] = 1 - Sin[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

p=ParametricPlot[{x[t],y[t]},{t,0,2Pi},
				PlotRange->All,
     			AxesLabel->{"x","y"},
				DisplayFunction->Identity,
     			AspectRatio->Automatic];

g=Graphics[Text["r = 1 - Sin[t]",{.8,.4}]];

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

iii.	Graph r = 1 -2Sinθ.

Clear[r,t,x,y,p,g]
r[t_] = 1 - 2Sin[t];
x[t_] = r[t] Cos[t];
y[t_] = r[t] Sin[t];

p=ParametricPlot[{x[t],y[t]},{t,0,2Pi},
				PlotRange->All,
     			AxesLabel->{"x","y"},
				DisplayFunction->Identity,
     			AspectRatio->Automatic];

g=Graphics[Text["r = 1 - 2Sin[t]",{1.2,.35}]];

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

Explain why you could have used your knowledge of the cosine function and the sine function to predict the above changes.