Conic Sections

SetOptions[PolarPlot,BaseStyle->{FontFamily->"Arial",FontSize->16},PlotStyle->{{Black,AbsoluteThickness[1.5]}},AxesLabel->{"x","y"},AxesStyle->Directive[FontFamily->"Arial",FontSize->16,Black,AbsoluteThickness[0.5],Arrowheads[0.04]],TicksStyle->{{Black,AbsoluteThickness[0.5]},{Black,AbsoluteThickness[0.5]}},ImageSize->500];
SetOptions[Manipulate,BaseStyle->{FontSize->20}];
SetOptions[Manipulator,Appearance->{"Labeled"}];

In this worksheet you will look at the graphs of a family of curves written in polar coordinates r, ϕ.

1) How are r and ϕ represented on a polar plot?

Below, we plot the function, 

		r(ϕ) = (α/(1+ϵ cos(ϕ+δ))),

where ϵ, α, and δ are adjustable parameters.

Manipulate[PolarPlot[α/(1+ϵ*Cos[ϕ+δ]), {ϕ, 0,2* π},PlotRange->{{-5,5},{-5,5}}],{{ϵ,.5},-2,2,0.01},{{α,1},0.5,2,0.01},{δ,0,2*Pi}]

2) How do each of the parameters (ϵ, α, and δ) effect the plot?

3) Over what ranges of ϵ do you get different shapes?

4) What is one thing that you would like to remember from this activity?

By Corinne Manogue, Mike Vignal, and Ian Founds