The Equiangular Spiral

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.

radius = {?? ?}

The n array numbers your radius entries from 0 to 18.  If your total number of measurements is different from 19, adjust this array accordingly.  The Transpose command pairs the n entries and r entries to create a sequence of points {..., {n, (Subscript[r, n])}, ...} for plotting.

\(n = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}; 
    
rData = Transpose[{n, radius}]\)

Plot the data points.

p1 = ListPlot[rData]

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

p2 = ListPlot[?? ?]

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

Enter your fitted function here, and superimpose its graph on the data plot.

\(R[t_] := \(?? [??t];

        p3 = Plot[R[t], {t, 0, 20}, DisplayFunction -> Identity]

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

Part 3: Plotting a Spiral Curve

Use the fact that θ = t  (π/4) to build the function r(θ) from the function R(t).

r[theta_] := R[?? ]

Check your polar function by plotting it.

PolarPlot[r[theta], {theta, -2 Pi, 4.5  Pi}, 
      AspectRatio -> Automatic]

Enter your parametric formulas for x and y:

\(x[theta_] := \(?? theta;

        y[theta_] := \(?? theta;\)\)\)

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 = 60; 

    ParametricPlot[{x[theta], y[theta]}, {theta, 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[theta_] := r0\ Exp[k\ theta] Cos[theta]; 

    y[theta_] := r0\ Exp[k\ theta] Sin[theta]; \)

\(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.