Lab 05: Polar Curves

Introduction

In this lab we will use Mathematica to explore polar equations.  The primary way to  graph polar equations in Mathematica is through the use of the PolarPlot command.  First let us look at an example:

Manipulate[
p1=PolarPlot[Sin[3*t],{t,0,tf},PlotRange->{{-1,1},{-1,0.6}}];
p2=Graphics[{PointSize[Large],Red,Point[{Sin[3*tf]*Cos[tf],Sin[3*tf]*Sin[tf]}]}];
Show[p1,p2],
{tf,Pi/48,2*Pi,Pi/48}
]

Notice in this example, we are graphing the polar function r=Sin[3θ] , on an interval [0, 2π] using increments of (π/48).  The red point indicated the value of θ.  In this example, the graph is drawn twice when we use an interval of [0, 2π].
The Manipulate command is used to view how the graph progresses and θ increases.

Tasks:

In the following tasks, use the PolarPlot and/or Show commands with or without a Manipulate command to explore the graphs of polar equations.

1.  How are the graphs of r=1+2Sin[θ-(π/6)] and r=1+2Sin[θ-(π/3)] related to the graph of r=1+2Sin[θ] ?  In general, how is the graph of r=f(θ-ϕ) related to the graph of r=f(θ)?

2.  A family of polar curves has equations r=(((Sin[n θ]Cos[n θ]))^2)(n θ) for θ in the interval [-2π, 2π].  Plot the graph of this equation for several different positive interger values of n.

(a)  Clearly explain how the graph changes as n changes and is positive.

(b)  Clearly explain how the graph chages for negative values of n.

(c)  Plot the graph for n = (1/2), (1/4), and (1/8).  Do these graphs fit the above pattern?  Explain.

3. Graph the following functions and explain your choice of domain for each graph.

r=(1/(Sqrt[θ])), r=θ Sin[θ] , and r=Sin[(θ/2)]