Investigating Polar Graphs

Madiha Baloch
mbaloch100@qc.cuny.edu

Objective : 
This Demonstration plots the polar graphs of r (θ) = a + fcn[b*θ] for various trig functions. where fcn represents the list of the functions such as Sin, Cos and a represents the lenth of the function and b is the constant variable.The main objective of doing this demonstration  first is to get more familiar with the command Manipulate and second is to explore or investigate the Polar cordinate system.
The graph of a polar equation r=f(θ) consists of all points P that have atleast one polar representation (r,θ) whose cordinate satisfy the equation.

Manipulate[Show[PolarPlot[a+fcn[b*θ],{b,-2,5},(*A polar equation where 'a' represents the length and theta is an angle from the fixed direction*) AxesLabel->{Style["θ",15,Italic],Style["r",15,Italic]},PlotRange->zoom,PlotStyle->{Thick,RGBColor[.25,.44,.92]},ImagePadding->20(*ImagePadding specifies what extra padding should be left for extended objects such as axis labels*),ImageSize->300,PlotLabel->Style["Polar Graph",14,Black,Bold,Italic],Ticks->{{-Pi,-Pi/2,-Pi/4,0,Pi/4,Pi/2,Pi,3Pi/2,2Pi},{-2,-1,0,1,2}}],TicksStyle->Directive[Purple,10]](*Ticks is an option for graphics functions that specifies the ticks marks for axes, whereas, ticksStyle specifies the ticks qualities*) ,{θ,0,3Pi/2},{a,-1,3},{{zoom,3},2,4,.01,Appearance->"Labeled",ImageSize->Tiny},Button["reset",θ=0,ImageSize->50,Appearance->"DialogBox"],{fcn,{Sin,Cos,Tan,Cot,Sec,Csc},ControlType->PopupMenu}]

For better understanding of the Polar graphs, i manipulated the cartesian coordinate system.In Cartesian system every point has only one representation, but in the polar coordinate system each point has many representation.like (1, 3 Pi/4), (1, 2 Pi).

Manipulate[(* Sin Function in cartesian coordinates*) Column[{Row[{"x = ",x}],Row[{"1+Sin[4*x] = ",1+Sin[4*x]}],Show[Plot[1+Sin[4x],{x,-Pi/4,2Pi},PlotStyle->{Blue,ImageSize->300},Ticks->{{-Pi,-Pi/2,-Pi/4,0,Pi/4,Pi/2,Pi,3Pi/2,2Pi},{-2,-1,0,1,2}}],Graphics[{Purple,PointSize[0.05],Point[{x,1+Sin[4*x]}]}]]
}],{{x,0},-Pi/4,2Pi},TrackedSymbols->{x}]

from the cartesian plot, we can easily plot the polar function in terms of r (θ) and can check that how both are corelated.

Show[PolarPlot[1+Sin[4x],{x,-Pi/4,2Pi},PlotStyle->{Blue,ImageSize->300},Ticks->{{-Pi,-Pi/2,-Pi/4,0,Pi/4,Pi/2,Pi,3Pi/2,2Pi},{-2,-1,0,1,2}}]]

In conclusion, as the Polar coordinate system was inveted by Newton, and is designed for various purposes such as to find the area with in the spiral.In this demonstration, I learned to manipulate the polar coordinate, which is some times really hard to figure out for some functions like csc and sec and i also learned that we can easily do the polar plot  of the cartesian equation with in some seconds.