The Equiangular Spiral (Alternate Version)

Enter the plotting package:

<< Graphics`

Part 1. Background

(No response is required in Part 1.)

Part 2. Measuring a Spiral Seashell

Enter your radial measurements in the following array, separated by commas.

Enter your x, y, r, and t measurements here.  Adjust your values of t (= θ) as necessary, by adding multiples of 2π.

Construct the (θ, r) pairs by entering the next line.

rData = Thread[t, r];

Plot the data points.

p1 = ListPlot[rData];

Construct an appropriate plot of the data to determine the growth type.

p2 = ?? ;

Do your calculations here to determine a function that will reasonably fit the radial measurment data.

Enter your fitted function here, and superimpose its graph on the data plot.  Note that we are using R for the dependent variable, since r is already in use for data values.

\(R[t_] := \(?? ;
p3 = 
          Plot[R[t], {t, 0, 6*Pi}, PlotRange -> {0, r[\([n]\)]}, 
            DisplayFunction -> Identity]; 

        Show[p1, p3, DisplayFunction -> $DisplayFunction];\)\)

Part 3: Plotting a Spiral Curve

Check your polar function by plotting it.

PolarPlot[R[t], {t, -2*Pi, 4.5*Pi}, 
        AspectRatio -> Automatic];

Enter your parametric formulas for x and y:

\(x[t_] := \(?? ;
y[t_] := \(?? ;\)\)\)

Check your parametric equations by plotting the curve.  You may change the θ interval by changing a and/or b.  You may change the x-range and y-range by changing c.  In particular, you may use c to zoom in on the graph.  For very small values of c, you will need to make a and b smaller.

a = -2*Pi;  b = 4.5*Pi; 
c = 300;
p = 1000;
\(\(ParametricPlot[{x[t], y[t]}, {t, a, b}, 
	
        PlotRange -> {{\(-c\), c}, {\(-c\), c}}, 
        AspectRatio -> Automatic];\)\)

Part 4: Why "Equiangular?"

Answer question 1 here.

Enter the parametric formulas for the equiangular spiral:

x[t_] := r0 Exp[k t] Cos[t];
y[t_] := r0 Exp[k t] Sin[t];

\(r0 = \(?? ;
k = \(?? ;\)\)\)

Do the rest of your calculations here, and answer the rest of the questions in this part.

Part 5: Summary

Answer the summary questions here.