Name:   Marcella Cremer

Title:  Rose Curves

Description: The following notebook allows students to explore 
Rose Curves by changing constants in the equations:          
r = a Sin(nt), r = a Cos(nt).  After running the animations and
 looking at the graphs, the students should be able to generalize 
 how the constants change the graph of the curve.

Explanation

A Rose Curve in polar graphing is a type of curve that when 
     graphed from 0 to 2` , forms "petals" - thus the name Rose 
     Curve.  
     The two main equations for Rose Curves are:
                       r = a Sin(nt)
                       r = a Cos(nt)  .
The following notebook will help you explore the relationships 
between the constants in the two equations, and the shape of the 
"Rose".

r = a Sin(nt)

Changing "a"

The graph of a Rose Curve varies when "a" changes.  How do you think the 
     graph will change as "a" is changed?  The following animation will graph 
     the curve r = a Sin(2t), and "a" will vary from 1,3,5, and 7.  Watch what 
     happens to the Rose.  [Remember:  To run the animation, click on the 
     bracket that contains all of the graphics.  Then press Apple-y.  Press "q" 
     to stop.]

Clear[x,r,t,a,n,p]
r[t_,a_]=a Sin[2 t];
p[a_]:=ParametricPlot[{r[t,a]Cos[t],r[t,a]Sin[t]},
      {t,0,2 Pi},AspectRatio->Automatic,
    		PlotStyle->{RGBColor[1,0,0]},
       PlotRange->{{-8,8},{-8,8}}]
Do[p[a],{a,1,7,2}]

Making "a" negative

The graph of the Rose Curve changes when "a" is a negative value.  The 
     following will graph r = a Sin(2t), and   r = a Sin(3t) while "a" varies from 
     positive and negative,3 and 5.  Watch what happens to the Rose Curve, 
     first with Sin2t.

Clear[x,r1,r2,t,a,n,p]
r1[t_,a_]=a Sin[2 t];
r2[t_,a_]=-a Sin[2 t]; 
p[a_]:=ParametricPlot[{r1[t,a] {Cos[t],Sin[t]},
                       r2[t,a] {Cos[t],Sin[t]}},
      {t,0,2 Pi},AspectRatio->Automatic,
    		PlotStyle->{RGBColor[1,0,0]},
       PlotRange->{{-8,8},{-8,8}}];
Do[p[a],{a,1,3,2}]

Here it is with Sin3t:

Clear[x,r1,r2,t,a,n,p]
r1[t_,a_]=a Sin[3 t];
r2[t_,a_]=-a Sin[3 t]; 
p[a_]:=ParametricPlot[{r1[t,a] {Cos[t],Sin[t]},
                       r2[t,a] {Cos[t],Sin[t]}},
                {t,0,2 Pi},AspectRatio->Automatic,
   PlotRange->{{-8,8},{-8,8}},
   PlotStyle->{{Thickness[.008],RGBColor[1,0,0]},
   {Thickness[.002]}}];
Do[p[a],{a,3,5,2}]

Changing "n"

The graph of the Rose Curve varies when "n" is changed.  How do you think the 
graph will change as "n" is changed?  The following animation will graph the 
curve r = 6 Sin(nt), and "n" will vary from 2,3,4,and 5.  Watch what happens 
to the number of Rose petals.

Clear[x,r,t,a,n,p]
r[t_,n_]=6 Sin[n t];
p[n_]:=ParametricPlot[{r[t,n]Cos[t],r[t,n]Sin[t]},
      {t,0,2 Pi},AspectRatio->Automatic,
    		PlotStyle->{RGBColor[1,0,0]},
       PlotRange->{{-8,8},{-8,8}}]
Do[p[n],{n,2,5,1}]

r = a Cos(nt)

Changing "a"

The graph of a Rose Curve varies when "a" changes.  How do you think the 
graph will change as "a" is changed?  The following animation will graph the 
curve r = a Cos(2t), and "a" will vary from 1,3,5, and 7.  Watch what happens 
to the Rose.  Note the differences between the graph of  r = a Sin(2t), 
shown earlier, and the graph of r = a Cos(2t).

Clear[x,r,t,a,n,p]
r[t_,a_]=a Cos[2 t];
p[a_]:=ParametricPlot[{r[t,a]Cos[t],r[t,a]Sin[t]},
      {t,0,2 Pi},AspectRatio->Automatic,
    		PlotStyle->{RGBColor[1,0,0]},
       PlotRange->{{-8,8},{-8,8}}];
Do[p[a],{a,1,7,2}];

Changing "n"

The graph of the Rose Curve varies when "n" is changed.  How do you think the
 graph will change as "n" is changed?  The following will graph the curve 
 r = 6 Cos(nt), and "n" will vary from 2,3,4,and 5.  Watch what happens to the 
 number of Rose petals?  Note the differences between the graph of                   r = 6 Sin(nt), shown earlier, and the graph of r = 6 Cos(nt).

Clear[x,r,t,a,n,p]
r[t_,n_]=6 Cos[n t];
p[n_]:=ParametricPlot[{r[t,n]Cos[t],r[t,n]Sin[t]},
      {t,0,2 Pi},AspectRatio->Automatic,
    		PlotStyle->{RGBColor[1,0,0]},
       PlotRange->{{-8,8},{-8,8}}];
Do[p[n],{n,2,5,1}];

Adding constants

In this section you will be looking at the polar graph of 
r = 6 Sin(3t).  In the two subsections that follow, constants will be added to 
change the graph of the Rose Curve.  Observe how the addition of the constants 
changes the graph.

Adding to the argument

In this section, a constant "b" will be added to the argument of the equation: 
r = 6 Sin(3t + b).  Notice how the graph of the Rose changes when "b" varies 
from `/4, `/2, 3`/4, `.

Clear[x,r,t,a,n,p,b]
r[t_,b_]=6 Sin[3 t + b];
p[b_]:=ParametricPlot[{r[t,b]Cos[t],r[t,b]Sin[t]},
      {t,0,2 Pi},AspectRatio->Automatic,
    		PlotStyle->{{RGBColor[1,0,0]}},
       PlotRange->{{-8,8},{-8,8}}];
Do[p[b],{b,Pi/4,Pi,Pi/4}];

Translating the Rose

In this section, a constant "c" {1,2,3,4} will be added to translate the Rose. 
     Notice where the "c" is added in the animation instructions, and notice 
     what direction the Rose translates.

Clear[x,r,t,a,n,c,p]
r[t_]=6 Sin[5 t];
p[c_]:=ParametricPlot[{r[t]Cos[t]+c,r[t]Sin[t]},
      {t,0,2 Pi},AspectRatio->Automatic,
    		PlotStyle->{{RGBColor[1,0,0]}},
       PlotRange->{{-8,8},{-8,8}}];
Do[p[c],{c,1,4,1}];

Summary

A rose curve is any curve in the form:
r = a Sin(nt), or r = a Cos(nt).  The Rose has "n" equally spaced petals if "n" is 
odd, and "2n" equally spaced petals if "n" is even.  The orientation of the Rose is
 effected by the sign of the constant, and whether  the curve includes Sine or
  Cosine.