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From Rectangle Graphs to Polar Graphs 
With Highlights of Polar Functions
By Kathy Hammad

Introducing Rectangle
and Polar Graphs

Rectangle Graphs do a fine job of investigating many different kind of curves.

Some equations that offer a variety of curves that we commonly examine are found in  parabolic, exponential,  rational, and periodic equations.

Lets take a look at them.

Parabolic

The good old quadratic equation will produce a fine parabolic curve.

Clear[x]
parabolic[x_] = (x^2) - 25;
par = 
        Plot[parabolic[x], {x, -10, 10}, 
          PlotStyle -> {Red,  Thickness[ .02]}];

Exponential

This doubling function is a good example of an exponential curve.

Clear[x]
exponential[x_] = 3 (((2))^x);
exp = 
        Plot[exponential[x], {x, -10, 10}, 
          PlotStyle -> {Orange,  Thickness[ .02]}];

Rational

This rational equations clearly defines where the asymptotes are.

Clear[x]
rational[x_]  =  (x/((x^2) - 2));
rat = 
        Plot[rational[x], {x, -10, 10}, 
          PlotStyle -> {Hue[ .18],  Thickness[ .02]}];

Periodic

This periodic function demonstrates the cyclic nature of the Sine wave.

Clear[x]
periodic[x_] =    Sin [2 x];
per = 
        Plot[periodic[x], {x, 0, 2 π}, 
          PlotStyle -> {Green,  Thickness[ .02]}];

Another way to investigate curves is with Polar Graphs, as they offer another perspective.

These same equations, parabolic, exponential,  rational, and periodic can also be graphed around a pole.

Lets take a look at the same curves in polar form.

Parabolic

Clear[x]
parabolic1[x_] = (x^2) - 25;
parpol = 
        PolarPlot[parabolic[x], {x, 0, 2  π}, 
          PlotStyle -> {Red,  Thickness[ .02]}];

Exponential

Clear[x]
exponential1[x_] = 3 (((2))^x);
exppol = 
        PolarPlot[exponential1[x], {x, 0, 2  π}, 
          PlotStyle -> {Orange,  Thickness[ .02]}];

Rational

Clear[x]
rational1[x_]  =  (x/((x^2) - 2));
ratpol = 
        PolarPlot[rational1[x], {x, 0, 2  π}, 
          PlotStyle -> {Hue[ .18],  Thickness[ .03]}];

Periodic

Clear[x]
periodic1[x_] =    Sin [2 x];
perpol = 
        PolarPlot[periodic1[x], {x, 0, 2  π}, 
          PlotStyle -> {Green,  Thickness[ .02]}];

The Rectangle form of these functions follow a predictable pattern, but the Polar form of parabolic, exponential,  rational functions appears to have no predictable pattern.

Lets take a look at both forms side by side.

Show[GraphicsArray[{par, parpol}]];

Show[GraphicsArray[{exp, exppol}]];

Show[GraphicsArray[{rat, ratpol}]];

Show[GraphicsArray[{per, perpol}]];

There is one function which seems to follow a pattern from the rectangle form to the polar form.  It is the periodic function shown above in green,  y = Sin (2 x).   It is the periodic function in polar coordinates which we will investigate next as it offers a variety of curves to study.

Polar Coordinates  consist of (r, θ), r being the radial coordinate defined by θ (theta), and θ being the variable.

In other words, r is like a rotated number line that can rotate at any angle, and a point on that number line can be positive traveling outward on the rotated number line, or the point can be graphed negative, going backwards of the preferred angle.  

To see how a Sin function is graphed around the pole of a polar coordinate graph, we will need to set up a list of numbers for  θ and r (θ).

table1 = 
      Table[ {θ, Sin[θ]}, {θ, 0, 
          2  π, π/6}]

For r = Sin θ, when θ = 0, r = 0.  When θ = π / 6, r = 1/2. When θ = π / 3,
 r = (Sqrt[3])/ 2.   This means that when the angle is π / 6 radians, the point will be at a radius of 1/2 units out from the origin.   When the angle is π / 3 radians, the point will be at a radius of (Sqrt[3])/2 units out from the origin.  As we keep plotting these points around the origin, the functions takes shape.
 
 After evaluating the following cell, grab the cells and animate to see how the function grows.  Slow down the animation to see how each π / 6 interval is formed.  To further investigate these functions, look at each successive graph.  The change between each graph represents adding an angle of π / 6 radians to the functions.

Exploring Functions:
Polar Form and Rectangle Form

This section is more about viewing the differences between polar and rectangle graphs.  Many of the functions will be viewed again in more detail later on.  In this section we will be concerned with learning a few basic facts.

Polar Graphing

Let's take a look at a simple polar function, r = sin θ.  Notice how each graph adds another section of (π/6).  Intervals can make a difference on how a function will look.

Clear[θ, u, r]
r[θ_] :=  
      Sin[ θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], DodgerBlue},AxesLabel->{"","r = Sin θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.1,1.1  π,π/6}];

Compare this to the functions rectangle counterpart below as it is being graphed.  Grab and animate the cells to see if being formed.

Rectangle Graphing

Notice that when the cell is run slowly, each new section that is added is θ = (π/6).    Also notice how the function crossed through the x-axis and comes back to the starting point of the pattern.  The polar graph completes a picture over an interval of 0 to π, where the rectangle graph completes a picture over an interval of 0 to 2π.

Clear[x, y, u]
y[x_] := Sin[x]
Table[Plot[y[x],{x, 0, u},PlotStyle -> {Thickness[0.02], DarkSlateBlue},AxesLabel->{"","y = Sin x"},PlotRange -> {{-2 π, 2  π}, {-2, 
                    2}}],{u,0.1,2.1  π,π/6}];

If you noticed, the title of a polar graph used the variables  r and θ  while the title of a rectangle graph used the variables y and x .  For the remainder of this program, you will know whether a function is being graphed a polar axis or a rectangle axis by the use of variables.

What about Constants?

One should not forget about constants as they tell the story of the beginning.

Polar Constant

Clear[θ, u, 
      r]
r[θ_] :=   1
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Blue},AxesLabel->{"","r = 1"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,  .1, 2.1  π, π/6}];

Notice how r = 1 is a circle about the origin with radius one, but r = Sinθ is a circle sitting on top of the origin with center at (0, 0.5) and radius 0.5.  When the function is a constant, such as r = 1, it is saying that the radius should always be length of unit one away from the origin.

Rectangle Constant

Clear[x, y, u]
y[x_] := 1
Table[Plot[y[x],{x, 0, u},PlotStyle -> {Thickness[0.02], Navy},AxesLabel->{"","y = 1"},PlotRange -> {{-2 π, 2  π}, {-2, 
                    2}}],{u,  .1, 2.2  π, π/6}];

Notice how the rectangle constant is a line parallel to the x-axis.  When the function is a constant, such as y = 1, it is saying that no matter how far one moves on the x-axis, to always be one unit above it.

There are many functions to graph that make interesting polar functions.  Lets compare their polar form to their rectangle form

Polar Form

Notice that this function has eight loops.  Also, notice that the coefficient in front of Sin is 2, and the radius of each loop is also 2.

Clear[θ, r, u]
r[θ_] := 
      2 Sin[ 4  θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], DeepSkyBlue},AxesLabel->{"","r = 2 Sin 4θ"},PlotRange -> {{-2, 2}, {-2, 2}}],{u,  .1, 2.1  π, π/6}];

Rectangle Form

How many bends does the rectangle form have?  Also, the coefficient in front of Sin is 2 and the height of the waves are 2.

Clear[x, y, u]
y[x_] := 2  Sin[4  x]
Table[Plot[y[x],{x, 0, u},PlotStyle -> {Thickness[0.02], Cobalt},AxesLabel->{"","y = 2 Sin 4x"},PlotRange -> {{-2 π, 2  π}, {-2, 
                    2}}],{u,  .1, 2.5  π, π/6}];

Polar Form

Notice how the function has change slightly.

Clear[θ, r, u]
r[θ_] := 
      1 - Cos[θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], CornflowerBlue},AxesLabel->{"","r = 1 - Cos θ"},PlotRange -> {{-2, 2}, {-2, 2}}],{u,  .1, 2.1  π, π/6}];

Rectangle Form

Clear[x, y, u]
y[x_] := 1 - Cos[x]
Table[Plot[y[x],{x, 0, u},PlotStyle -> {Thickness[0.02], SteelBlue},AxesLabel->{"","y = 1 - Cos x"},PlotRange -> {{-2 π, 2  π}, {-2, 
                    2}}],{u,  .1, 2.1  π, π/6}];

Polar Form

Notice how this function is similar the previous one in polar form.

Clear[θ, r, u]
r[θ_] := 
      1  + 2  Sin[θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], LightSkyBlue},AxesLabel->{"","r = 1 + 2 Sin θ"},PlotRange -> {{-2, 2}, {-2, 3}}],{u,  .1, 2.1  π, π/6}];

Rectangle Form

Clear[x, y, u]
y[x_] := 1  + 2  Sin[x]
Table[Plot[y[x],{x, 0, u},PlotStyle -> {Thickness[0.02], MediumBlue},AxesLabel->{"","y = 1 + 2 Sin x"},PlotRange -> {{-2 π, 2  π}, {-2, 
                    3}}],{u,  .1, 2.1  π, π/6}];

Notice that all of the functions used contained either Sine or Cosine. These two trigonometric functions make great polar graphs as they naturally restrict growth.  The polar graphing wrapped around the origin, while the rectangle form wrapped around the x-axis.

Now it is time to have some fun.  Functions can be combined to produce even more intricate graphs.

The Basics of a Polar Function

The following will show what defines the basic polar function

Loops

In order to graph a polar function, we need three things: a function, an axis and an interval for θ.  If you had noticed above, some functions finish a cycle (back where they started from in  π units, while others finish in  2π..  Some patterns have an odd number of leaves, while others have an even number.

 Why is this?  For Polar Roses, when r = a sin n θ,  or r = a cos n θ,  the value of n determines how many loops there will be.  If n is odd, then there will be n loops, where the petals will be traced out once as θ varies over zero to π.   If n is even,  then there will be 2n loops, where the petals will be traced out once as θ varies over zero to 2π.
 
 For example,  in the following rose, the number next to θ is assumed one.  So I can predict, since one is odd, that there will be n loops or one.

Clear[θ, u, r]
r[θ_]:= Sin[ θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], ChromeOxideGreen},AxesLabel->{"n=1","r = Sin θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,1.01π,π/6}];

In the following rose,the number next to θ is two. So I can predict,since two is even,that there will be 2n loops, or 4.

Clear[θ, u, r]
r[θ_]:= Sin[2 θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], DarkGreen},AxesLabel->{"2n=4","r = Sin 2 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,2.01π,π/6}];

Yes, just like I predicted.

In the following rose, the number next to θ is three, odd, so I can predict that there will be exactly n petals, or three.

Clear[θ, u, r]
r[θ_]:= Sin[3 θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], CinnabarGreen},AxesLabel->{"n=3","r = Sin 3 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,1.01π,π/6}];

In the rose below, the number next to θ is four, even, so there will be 2n or eight petals.

Clear[θ, u, r]
r[θ_]:= Sin[4 θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], EmeraldGreen},AxesLabel->{"2n=8","r = Sin 4 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,2.01π,π/6}];

In the following rose, the number next to θ is five, odd, so there will be exactly n petals, or five.

Clear[θ, u, r]
r[θ_]:=Sin[5 θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], SeaGreen},AxesLabel->{"n=5","r = Sin 5 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,1.01π,π/6}];

In the rose below, the number next to θ is six, even, so there will be 2n or 12 petals.

Clear[θ, u, r]
r[θ_]:= Sin[6 θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], DarkSeaGreen},AxesLabel->{"2n=12","r = Sin 6 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,2.01π,π/6}];

Clear[θ, u, r]
r[θ_]:= Sin[7 θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], DarkTurquoise},AxesLabel->{"n=7","r = Sin 7 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,1.01π,π/6}];

Look how beautifully the polar function divides 360 degrees into 7 neat and tidy sections.  Here is a table which summarizes the above roses.

                                     ({{function, number of

               loops}, {r = sin θ, 1}, {r = sin 2  θ, 4}, {r = sin 3  θ, 3}, {r = sin 4  θ, 8}, {r = sin 5  θ, 5}, {r = sin 6  θ, 12}, {r = sin 7  θ, 7}})

Notice that we cannot make a 2 looped polar function, or a 6 looped polar function, etc.  There are ways to do it.  In the next section we shall find out, and see some of the classical polar functions with fancy names.

Translation

Just like a function on a rectangle graph can be translated, so can one on a polar graph.  Notice how the function shifts clockwise when numbers are added to θ.  Looks like translation and rotation are the same polar-wise.

Clear[θ, u, r]
r[θ_] :=  
      Sin[ θ]
T1=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Coral},AxesLabel->{"","r = Sin θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.1,1.1π,π/6}];

Clear[θ, u, r]
r[θ_] :=  
      Sin[ θ + 1]
T2=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], GoldOchre},AxesLabel->{"","r = Sin [θ+1]"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.1,1.1π,π/6}];

Clear[θ, u, r]
r[θ_] :=  
      Sin[ θ + 2]
T3=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Apricot},AxesLabel->{"","r = Sin [θ+2]"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.1,1.1π,π/6}];

Clear[θ, u, r]
r[θ_] :=  
      Sin[ θ + 3]
T4=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], PeachPuff},AxesLabel->{"","r = Sin [θ+3]"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.1,1.1π,π/6}];

By translating, I can turn a 3 loop function into a six loop function by graphing two functions at the same time:  one with out translating, and one with.

Show[T1, T2, T3, T4]

Dilation

Clear[θ, u, r]
r[θ_] :=  
      Sin[ θ]
TT=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Gold},AxesLabel->{"","r = Sin θ"},PlotRange -> {{-4, 4}, {-4, 4}}],{u,0.1,1.1π,π/6}];

Clear[θ, u, r]
r[θ_] := 
      2 Sin[ θ]
TT1=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Wheat},AxesLabel->{"","r = 2 Sin θ"},PlotRange -> {{-4, 4}, {-4, 4}}],{u,0.1,1.1π,π/6}];

Clear[θ, u, r]
r[θ_] := 
      3 Sin[ θ]
TT2=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], PaleGoldenrod},AxesLabel->{"","r = 3 Sin θ"},PlotRange -> {{-4, 4}, {-4, 4}}],{u,0.1,1.1π,π/6}];

Clear[θ, u, r]
r[θ_] := 
      4 Sin[ θ]
TT3=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], YellowGreen},AxesLabel->{"","r = 4 Sin θ"},PlotRange -> {{-4, 4}, {-4, 4}}],{u,0.1,1.1π,π/6}];

In order to make the best looking graph, the axis are fitted to the function. Putting these all together we get a nested looking function.

Show[TT, TT1, TT2, TT3]

To summarize this section, a function can contain  a differing number of loops, which can be lengthened and rotated counterclockwise.  Functions can also be put together.  Later on function will be combined to produce some pretty fancy graphs.

Classical Polar Functions

The Greeks came to favor some simple, but enlightening shapes as we shall see below.  When a shape is simple, we can come up with basic rules that govern that shape.

The following two function are of the same form called limacons (from the Greek word "slug"):  r = a + b sin θ. The first one is a limacon with an inner loop, and the second one is a cardoid (from the Greek word "heart").

Limacon with an Inner Loop

Clear[θ, r, u]
r[θ_] := 
      1  + 2  Sin[θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Plum},AxesLabel->{"","r = 1 + 2 Sin θ"},PlotRange -> {{-2, 2}, {-2, 3}}],{u,  .1, 2.1  π, π/6}];

Cardoid

Clear[θ, r, u]
r[θ_] := 
      1 + Sin[θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Purple},AxesLabel->{"","r = 1 + Sin θ"},PlotRange -> {{-2, 2}, {-2, 2}}],{u,  .1, 2.1  π, π/6}];

The next function is of the form (r^2)=(a^2)Sin 2θ.  Here is the two loops we have been looking for.

Lemniscate

It would be good to turn off the ParametricPlot message in this part, to avoid having to answer questions about it.

Clear[θ, r, 
      u]
r[θ_] := (Sqrt[Sin])[2  θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Violet},AxesLabel->{"","r = \!\(\*
StyleBox[
SqrtBox[
StyleBox[\(Sin\\ 2  θ\),
FontSize->18,
FontColor->RGBColor[1, 0, 0]]],
FontColor->RGBColor[1, 0, 0]]\)"},PlotRange -> {{-1, 1}, {-1, 1}}],{u, 0.1, 1.8  π, π/6}];

This next function does not use a Sine of Cosine function. As you can see, since θ has no bounds it just keeps going.

Spiral of Archimedes

Clear[θ, r, 
      u]
r[θ_] := θ
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], Raspberry},AxesLabel->{"","r = θ"},PlotRange -> {{-20, 20}, {-20, 20}}],{u,  .1, 6.5  π, π/3}];

In order to get the full view of the behavior of this function, I increased the interval to zero to 6π.

The Final Polar Product

Now we will combine all that we know about graphing polar function: how many petals we want, how large to make the radius of the petals, and rotating the petals.    But first, we need to learn how to affect the picture by changing width and color of line.

Width and Color of Line

Clear[θ, u, r]
r[θ_]:= Sin[2 θ]
TS1=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle->{Thickness[0.1], IndianRed},AxesLabel->{"","r = Sin 2 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u, 0.01, 2.01  π, π/6}];

Clear[θ, u, r]
r[θ_]:= Sin[2 θ]
TS2=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle->{Thickness[0.05], EnglishRed},AxesLabel->{"","r = Sin 2 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,2.01π,π/6}];

Clear[θ, u, r]
r[θ_]:= Sin[2 θ]
TS3=Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle->{Thickness[0.015], Red},AxesLabel->{"","r = Sin 2 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u, 0.01, 2.01  π, π/6}];

Show[TS1, TS2, TS3]

Same function. The only changes were the width of the line and the color of the line.  The edges flake a bit as the line gets rather large.

Restricting Parameters

If I graph a function that completes its picture in  the interval over zero to 2π, but restrict the interval to zero to π, a different picture is graphed.

Clear[θ, u, r]
r[θ_]:= Sin[2 θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], LightSalmon},AxesLabel->{"2n=4","r = Sin 2 θ"},PlotRange -> {{-1, 1}, {-1, 1}}],{u,0.01,1.01  π,π/6}];

Combining many different changes in a function as well as graphing parameters produce a variety of effects.

Clear[θ, r, u]
r[θ_] := 
      2 Sin[ 4  θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], LimeGreen},AxesLabel->{"","r = 2 Sin 4θ"},PlotRange -> {{-3, 3}, {-3, 3}}],{u,  .1, 2.1  π, π/6}];

Clear[θ, r, u]
r[θ_]:=2 Sin[ 4  θ]  -  Sin [θ]
Table[PolarPlot[r[θ],{θ, 0, u},PlotStyle -> {Thickness[0.02], LightSeaGreen},AxesLabel->{"","r = 2 Sin 4θ"},PlotRange -> {{-3, 3}, {-3, 3}}],{u,  .1, 2.1  π, π/6}];

The changes that were made between the two functions is an alteration of the function and color.

We will also use other changes:altering basic functions,how many times to rap the function around the pole,and putting different functions together on the same axis. (not pictures)

Highlights of Creativity

Chrysanthemum

Translate the same function three times and vary the color of the line.

Clear[θ, u,  r]
r3[θ_] := Sin[2  θ + 1]
r4[θ_] :=  Sin[2  θ + 2]
r5[θ_] :=  Sin[2  θ + 3]
Table[
        PolarPlot[{r3[θ], r4[θ],  r5[θ]}, {θ, 0, 
            u}, PlotStyle -> {{Thickness[0.03], 
                Orange}, {Thickness[0.03], CadmiumLemon}, {Thickness[0.03], 
                CadmiumOrange}}, 
          AxesLabel -> {"", "r = Sin [2θ+1]"}, 
          PlotRange -> {{-1, 1}, {-1, 1}}], {u,  .1, 
          2.2  π, π/6}];

Petunia

Vary the rotation, length, width and color.

Clear[θ, u,  r]
r6[θ_] := 5  Sin[2  θ + 1]
r7[θ_] := 5 Sin[2  θ + 2]
r8[θ_] :=  4  Sin[2  θ + 1]
r9[θ_] := 4  Sin[2  θ + 2]
r10[θ_] := 2 Sin[2  θ + 2]
r11[θ_] :=  2  Sin[2  θ + 1]
Table[
        PolarPlot[{r6[θ], r7[θ],  r8[θ], r9[θ], 
            r10[θ],  r11[θ]}, {θ, 0, u}, 
          PlotStyle -> {{Thickness[0.1], Lavender}, {Thickness[0.1], 
                Lavender}, {Thickness[0.04], LightBlue}, {Thickness[0.04], 
                LightBlue}, {Thickness[0.03], 
                MediumPurple}, {Thickness[0.03], MediumPurple}}, 
          AxesLabel -> {"", "r = 5 Sin [2θ +1]"}, 
          PlotRange -> {{-6, 6}, {-6, 6}}], {u,  .1, 
          2.2  π, π/6}];

Chrysanthemum

Vary only translation and color, but add a lot more functions.

Clear[θ, u,  r]
r30[θ_] := Sin [2  θ + 1]
r31[θ_] := Sin [2  θ + 2]
r32[θ_] := Sin [2  θ + 3]
r33[θ_] := Sin [2  θ + 1.5]
r34[θ_] := Sin [2  θ + 2.5]
r35[θ_] := Sin [2  θ + 3.5]
Table[
        PolarPlot[{r30[θ], r31[θ], r32[θ], 
            r33[θ], r34[θ], r35[θ]}, {θ, 0, u}, 
          PlotStyle -> {{Thickness[0.02],  
                RoseMadder}, {Thickness[0.02], Tomato}, {Thickness[0.02], 
                VenetianRed}, {Thickness[0.02],  
                RoseMadder}, {Thickness[0.02], Tomato}, {Thickness[0.02], 
                VenetianRed}}, 
          AxesLabel -> {"", "r = Sin [2θ+1]"}, 
          PlotRange -> {{-1.2, 1.2}, {-1.2, 1.2}}], {u,  .1, 
          2.1  π, π/6}];

Dahlia

Vary length, translation and color in a specific pattern.

Clear[θ, u,  r]
r36[θ_] := 3 Sin [2  θ + 3]
r37[θ_] := 3 Sin [2  θ + 2]
r38[θ_] := 3 Sin [2  θ + 1]
r39[θ_] := 
      2.5 Sin [2  θ + 1]
r40[θ_] := 2.5 Sin [2  θ + 2]
r41[θ_] := 
      2.5 Sin [2  θ + 3]
Table[
        PolarPlot[{r36[θ], r37[θ], r38[θ], 
            r39[θ], 
            r40[θ], 
r41[θ]}, {θ, 0, 
            u}, PlotStyle -> {{Thickness[0.03],  
                LavenderBlush}, {Thickness[0.03], Pink}, {Thickness[0.03], 
                MistyRose}, {Thickness[0.03], Pink}, {Thickness[0.03], 
                LightPink}, {Thickness[0.03], LemonChiffon}}, 
          AxesLabel -> {"", "r = 3 Sin [2θ+1]"}, 
          PlotRange -> {{-3.5, 3.5}, {-3.5, 3.5}}], {u,  .1, 
          2.1  π, π/6}];

Butterfly

Reflect, change length and color, and restrict interval.

Clear[θ, u,  r]
r12[θ_] := 5 Sin[2  θ] + 2
r13[θ_] := 5 Sin[2  θ] + 1
r14[θ_] := -5 Sin[2  θ] + 
        2
r15[θ_] := -5 Sin[2  θ] + 
        1
r16[θ_] := 5 Sin[2  θ]
r17[θ_] := -5 Sin[2  θ]
Table[
        PolarPlot[{r12[θ], r13[θ],  r14[θ],  
            r15[θ],  r16[θ], r17[θ]}, {θ, 0, u}, 
          PlotStyle -> {{Thickness[0.02], 
                ManganeseBlue}, {Thickness[0.05], 
                Aquamarine}, {Thickness[0.02], 
                ManganeseBlue}, {Thickness[0.05], 
                Aquamarine}, {Thickness[0.03], LightPink}, {Thickness[0.03], 
                LightPink}}, 
          AxesLabel -> {"", "r = 5 Sin [2θ]+2"}, 
          PlotRange -> {{-7, 7}, {-7, 7}}], {u,  .1, 
          1.05  π, π/6}];

Spiky Star

Increase amount of petals to 2n=60, throw in a vivid color and combine it with a similar function.

Clear[θ, u,  r]
r1[θ_] := 5  Sin[30  θ]
r2[θ_] := 2 Sin[21  θ]
Table[
        PolarPlot[{r1[θ], r2[θ]}, {θ, 0, u}, 
          PlotStyle -> {{Thickness[0.02], Blue}, {Thickness[0.02], 
                Yellow}}, 
          AxesLabel -> {"", "r = 5 Sin 30 θ"}, 
          PlotRange -> {{-6, 6}, {-6, 6}}], {u,  .1, 
          2.2  π, π/6}];

Oyster Shell

Change length, color and restrict interval.

Clear[θ, u,  r]
r18[θ_] := 5 Sin[2  θ]
r19[θ_] := 4 Sin[2  θ]
r20[θ_] := 3 Sin[2  θ]
r21[θ_] := 2 Sin[2  θ]
r22[θ_] :=  Sin[2  θ]
r23[θ_] :=  .5 Sin[2  θ]
Table[
        PolarPlot[{r18[θ], r19[θ],  r20[θ],  
            r21[θ],  r22[θ], r23[θ]}, {θ, 0, u}, 
          PlotStyle -> {{Thickness[0.02], LightGray}, {Thickness[0.03], 
                LightGray}, {Thickness[0.04], Gray}, {Thickness[0.02], 
                Burlywood}, {Thickness[0.01], Peru}, {Thickness[0.01], 
                LightGray}}, 
          AxesLabel -> {"", "r =5 Sin [2θ]"}, 
          PlotRange -> {{-4.5, 4.5}, {-4.5, 4.5}}], {u,  .1, 
          1.05  π, π/6}];

Clam Shell

Alter function, length and color.

Clear[θ, u,  r]
r24[θ_] :=  .3 θ  Sin[θ]
r25[θ_] :=  .4  θ  Sin[θ]
r26[θ_] :=  .25  θ  Sin[θ]
Table[
        PolarPlot[{r24[θ], r25[θ], r26[θ]}, {θ, 
            0, u}, PlotStyle -> {{Thickness[0.05],  
                Melon}, {Thickness[0.04], YellowOchre}, {Thickness[0.03], 
                SandyBrown}}, 
          AxesLabel -> {"", "r = .3 θ Sin θ"}, 
          PlotRange -> {{-2, 2}, {-2, 2}}], {u,  .1, 
          2.1  π, π/6}];

Frog

Alter function, loops, length and color.

Clear[θ, u,  r]
r27[θ_] := 
      5  Sin[θ]  -  Sin[9  θ]
r28[θ_] :=  3  Sin[θ]  -  Sin [5  θ]
r29[θ_] :=  
      Sin[θ]  -  Sin[5  θ]
Table[
        PolarPlot[{r27[θ], r28[θ], r29[θ]}, {θ, 
            0, u}, PlotStyle -> {{Thickness[0.04],  
                SapGreen}, {Thickness[0.05], Mint}, {Thickness[0.03], 
                MediumSeaGreen}}, 
          AxesLabel -> {"", "r = 5 Sin θ - Sin 9 \
θ"}, PlotRange -> {{-6, 6}, {0, 6}}], {u,  .1, 
          1.1  π, π/6}];

No Name

Here is where you can invent a graph using all of the suggestions throughout this program.  I am sure you can be much more creative than I.

Clear[θ, u,  r]
r30[θ_] :=  9 Sin[3  θ + 3]
r31[θ_] := 6 Sin[101  θ]
r32[θ_] :=  1.3
Table[
        PolarPlot[{r30[θ], r31[θ], r32[θ]}, {θ, 
            0, u}, PlotStyle -> {{Thickness[0.1],  
                Pink}, {Thickness[0.001], Blue}, {Thickness[0.02], 
                LightBlue}}, 
          AxesLabel -> {"", "r = .3 θ Sin θ"}, 
          PlotRange -> {{-10, 10}, {-10, 10}}], {u,  .1, 
          2.1  π, π/6}];