Off[General::"spell1"];

beginAssignment;

Make a cog

In working through this notebook you will make a cog with sinusoidal teeth and a daisy.

The curves will be plotted parametrically and the main Mathematica function used will be ParametricPlot.

?ParametricPlot

Open each closed group of cells below
by double-clicking on the outer bracket.

Sinusoids

Curves of the same general shape as the graph of the function Sin are called sinusoids.

Plot[Sin[t],{t,-2 π,2 π},PlotRange->{{-2 π,2 π},{-1,1}}]

Sin has period 2Pi and amplitude 1.

You can get a sinusoid with different period or amplitude by modifing Sin:

Plot[Sin[8 t],{t,0,2 π},PlotRange->{{0,2 π},{-1,1}}]

Plot[0.1` Sin[8 t],{t,0,2 π},PlotRange->{{0,2 π},{-1,1}}]

The period and amplitude of the next curve are unchanged, but it has a different location.

Plot[1+0.1` Sin[8 t],{t,0,2 π},PlotRange->{{0,2 π},{-0.2`,1.2`}},AspectRatio->1]

Circles

The length of the vector from the origin to the point {x,y} may be defined as follows:

length[{x_,y_}]:=(Sqrt[(x^2)+(y^2)])

length[{(3/5),(4/5)}]

length[{(1/(Sqrt[2])),(1/(Sqrt[2]))}]

length[{Cos[(π/3)],Sin[(π/3)]}]

Of course the vector from the origin to any  point of the form {Cos[t],Sin[t]} has length 1.  (Why?)  

In other words any  point of the form {Cos[t],Sin[t]} lies on the unit circle.

ParametricPlot[{Cos[t],Sin[t]},{t,0,(π/2)},PlotRange->{{-1.2`,1.2`},{-1.2`,1.2`}},AspectRatio->1]

Sinusoidal cogs

The latter length fluctuates between .9 and 1.1.  (Why?)   Consequently we should be able to anticipate what the next picture will be like.  We call it a sinusoidal cog.

ParametricPlot[{(1+0.1` Sin[8 t]) Cos[t],(1+0.1` Sin[8 t]) Sin[t]},{t,0,2 π},PlotRange->{{-1.2`,1.2`},{-1.2`,1.2`}},AspectRatio->1]

ParametricPlot[{(1+0.1` Sin[8 t]) Cos[t],(1+0.1` Sin[8 t]) Sin[t]},{t,0,2 π},PlotRange->{{-1.2`,1.2`},{-1.2`,1.2`}},AspectRatio->1]

A daisy

A deeply indented sinusoidal cog is a daisy:

ParametricPlot[{(1+0.7` Sin[16 t]) Cos[t],(1+0.7` Sin[16 t]) Sin[t]},{t,0,2 π},PlotRange->{{-2,2},{-2,2}},PlotPoints->50,AspectRatio->1,Axes->None]

ParametricPlot[{(1+0.7` Sin[16 t]) Cos[t],(1+0.7` Sin[16 t]) Sin[t]},{t,0,4 π},PlotRange->{{-2,2},{-2,2}},PlotPoints->60,AspectRatio->1,Axes->None]

ParametricPlot[{(1+0.7` Sin[16 t]) Cos[t],(1+0.7` Sin[16 t]) Sin[t]},{t,0,6 π},PlotRange->{{-2,2},{-2,2}},PlotPoints->50,AspectRatio->1,Axes->None]

You can get different styles of daisy by modifying the following 
parametric description to get longer and/or overlapping petals:

ParametricPlot[{(1+Abs[Sin[8 t]]) Cos[t],(1+Abs[Sin[8 t]]) Sin[t]},{t,0,2 π},PlotRange->{{-4,4},{-4,4}},PlotPoints->50,AspectRatio->1,Axes->None]