Exploring the Limaçon
by Ginnie Teuscher

Introduction

The Limaçon (pronounced lee muh SOHN) was first studied and named by Etienne Pascal(father of Blaise Pascal).  The word limaçon comes from theLatin word meaning "a snail."  

The general forms of the equations that graph as a limaçon are:
    r= a + b cosθ     r=a - b cosθ     r=a + b sinθ     r=a - b sinθ 
    
We will explore the graphs for various values of a and b.  Can we draw some generalizations to connect the value of the ratio a : b to different shapes of the limaçon?

The Inner Loop

You will easily see how this curve got its name.  Let's graph  
1 + 2 cosθ and see what it looks like.

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ Cos[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[0, 1, 0]}, 
        PlotRange -> {{\(-4\), 4}, {\(-4\), 4}}]\)\(\(Do[p[a, b], {b, 2, 2, 1}, {a, 1, 1, 1}]; \)
\)

After we look at several equations and their graphs, you will record the observations on the chart.  Pay attention to the orientation of the graph, its shape and size, and the ratio of  a:b  in our pattern equations 
 a + b cosθ,  a + b sinθ,  a - b cosθ,  and  a - b sinθ .

Let's look at the graph of 3 + 5 cosθ

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ \ Cos[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 0]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 5, 5, 1}, {a, 3, 3, 1}]; \)
\)

Now, let's look at the graph of 4 + 7 sinθ.  How do you think the graph might change?

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ \ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[0, 0, 1]}, 
        PlotRange -> {{\(-8\), 8}, {\(-2\), 12}}]\)\(\(Do[p[a, b], {b, 7, 7, 1}, {a, 4, 4, 1}]; \)
\)

Are you game for one more?   2 - 3 sinθ  Let's notice what effect the negative has on the graph.  Can you make a prediction?

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a - b\ \ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 1]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 3, 3, 1}, {a, 2, 2, 1}]; \)
\)

Stop now and record the information on your chart from the first four graphs.  When you have recorded your observations, proceed with the cardioid.

The Cardioid

Another type of limaçon is the cardioid.  You can send this one to someone special on Valentine's Day.  If he/she recognizes the equation, you know you have a keeper!     Here's one to try:    4 + 4 cosθ

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ Cos[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 0]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 4, 4, 1}, {a, 4, 4, 1}]; \)
\)

It is a little on the plump side perhaps but a heart-shape nevertheless!

Let's try 4 + 4 sinθ and see what it looks like.

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ \ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 0]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 4, 4, 1}, {a, 4, 4, 1}]; \)
\)

Oops!  How could we alter the equation so that the heart shape looks right?

Let's try 4 - 4 sinθ

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a - b\ \ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 0]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 4, 4, 1}, {a, 4, 4, 1}]; \)
\)

Still a little plump, but at least it is oriented in the traditional way!

Stop and record your observations about these three cardioid curves and their equations.

The Dimpled

This limaçon is called "dimpled" because its indentation is not as pronounced as that of the cardioid.  It is still a rather unique curve.  Let's try a graph of   4 + 3 cosθ

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ Cos[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 0]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 3, 3, 1}, {a, 4, 4, 1}]; \)
\)

You can see that it is not as heart-shaped as the cardioid.  Nevertheless, it is a nice curve!

Another one might have this equation  8 + 5sinθ   Graph it.

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[0, 1, 0]}, 
        PlotRange -> {{\(-10\), 10}, {\(-4\), 14}}]\)\(\(Do[p[a, b], {b, 5, 5, 1}, {a, 8, 8, 1}]; \)
\)

How could we alter the equation to reflect the graph around the x-axis so it would look rather like an apple?

If you guessed  8 - 5 sinθ   you were correct!

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a - b\ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[0, 1, 0]}, 
        PlotRange -> {{\(-10\), 10}, {\(-14\), 4}}]\)\(\(Do[p[a, b], {b, 5, 5, 1}, {a, 8, 8, 1}]; \)
\)

Stop and record your observations about these three dimpled limaçons and their equations.

The Convex

The last type of limaçon we will look at is the convex.  If you have noticed, the curves have gotten progressively more "circular".  The convex  has an equation such as 
 5 + 2 cosθ.  Let's try graphing it.

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ Cos[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 1]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 2, 2, 1}, {a, 5, 5, 1}]; \)
\)

Do you notice the slightly flattened side?

Let's try   5 - 3 sinθ

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 0]}, 
        PlotRange -> {{\(-8\), 8}, {\(-3\), 8}}]\)\(\(Do[p[a, b], {b, 3, 3, 1}, {a, 5, 5, 1}]; \)
\)

By now you have discovered how to reflect the curve across the x-axis or y-axis.

Stop and record your observations about the convex limaçons on your table.

Conclusion

We have looked at the four major types of limaçons.  The type of graph depended on the ratio of a:b.  Let's watch what happens as the ratio changes from  a:b < 1, a:b = 1, 1 < a:b < 2, 
and a:b ≥ 2 for the  general form a + b cos θ

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ \ Cos[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 0]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 2, 2, 1}, {a, 1, 5, 1}]; \)
\)

What fun!  Let's watch again as we do the same with 
 a + b sinθ

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ \ Sin[t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[0, 0, 1]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 2, 2, 1}, {a, 1, 5, 1}]; \)
\)

Be sure your observations are recorded for the four types of limaçons on the chart below.  Highlight the cells and use the "Print Selection" option to print the completed chart.  You can sketch in the graphs after you print the chart.   After the chart is complete and printed,  proceed to the section "More interesting polar graphs" and create some graphs of your own.  Print them and submit them with your completed chart.  Enjoy!

equation        a:b       type of curve            sketch of curve

More interesting polar graphs

Just for fun, we could combine what we learned about the Rose graphs [when we put a coefficient  n in front of the θ] with what we now know about the limaçon graphs to create some unique and rather pretty graphs!   Graph
a + b sin 3θ  as a ranges in value from 1 to 5.

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ \ Sin[3\ \ t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[1, 0, 1]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 2, 2, 1}, {a, 1, 5, 1}]; \)
\)

Let's do the same thing except use the coefficient of 4 rather than 3.  How would the graphs change?

Clear[x, r, t, a, b, n, p]\(r[t_, a_, b_] = a + b\ \ Sin[4\ t]; 

    p[a_, b_] := 
      ParametricPlot[{r[t, a, b] Cos[t], r[t, a, b] Sin[t]}, {t, 0, 
          2  π}, AspectRatio -> Automatic, 
        PlotStyle -> {RGBColor[0, 1, 1]}, 
        PlotRange -> {{\(-8\), 8}, {\(-8\), 8}}]\)\(\(Do[p[a, b], {b, 2, 2, 1}, {a, 1, 5, 1}]; \)
\)

Use the copy and paste option to print some graphs of your own.  Be creative and have fun!  Include your creations when you turn in your chart of recorded observations.