spiral.html

The Equiangular Spiral

Consider the graph of the spiral passing through (x(t),y(t)) with the following properties:

The distance from the origin to the point (x(t),y(t)) is given by r(t)=exp(-k*t), where k is a parameter whose value is positive.
The angle from the horizontal x-axis to the point (x(t),y(t)) is t.

Here is a picture of this spiral:

Run the following movie to see why this is called the equiangular spiral .

You see a ray from the origin to a point P=(x(t),y(t)) on the spiral and a line tangent to the spiral at P as point P moves along the spiral. The angle to which the name equiangular spiral refers is the angle between these two lines.

Project: The Equiangular Spiral

a) Let j be be the number of letters in your last name. Let k = 0.1 j . You will investigate the spiral for which the distance from the origin to the point (x(t),y(t)) is r(t)=exp(-k*t) for this value of k.
b) Use the two defining properties of the equiangular spiral and the first picture displayed above to find both x(t) and y(t) as functions of t. It may help to notice that a vertical line segment dropped from point (x(t),y(t)) forms a right triangle with one angle t and hypotenuse r(t). Be sure to explain your reasoning clearly.
c) Plot an interesting portion (from t=0 to t=m for m on the order of 10*Pi will work well; you can experiment and choose the portion you find interesting) of the graph of this parametrically defined curve.
d) Calculate the slope of the line T that is tangent to the spiral at point P on the spiral and the slope of the ray R from the origin to point P.
e) Consider the following three angles: the angle between R and the x-axis, the angle between T and the x-axis, and one of the two angles between R and T. The sum of two of these is the third. Use your relationship to write the angle between R and T in terms of the other two. Suggestion: You will need to name these three angles. Be sure to define them clearly so that the reader can follow your work. You should include a hand-drawn diagram to make your work clear.
f) Use a trigonometric identity for the tangent of the sum or difference of two angles: and . Use the appropriate identity to find the tangent of the angle between R and T. Recall that the slope of a line is the tangent of its angle of inclination.
g) Explain why this spiral is called the equiangular spiral.
h)   Then use calculus to find the total length of this curve as t goes from 0 to 100*Pi.
Suggestion: You know a formula for finding the length of a portion of the graph of a function, when the function is given in terms of y=f(x).   That formula involves the derivative dy/dx.
    1. Use your knowledge of calculus to write dy/dx in terms of dx/dt and dy/dt.
2. Use the technique of substitution to write your integral in terms of the variable t.
3. Calculate this integral.
i) Finally, find the total length of this curve as the parameter t goes from 0 to infinity.
j) Write a concluding paragraph for your report.

Tools for this Project

To plot the spiral, define x(t) and y(t) then use the parametric form of the plot command:

> restart:

> x:=t-> ;

> y:=t-> ;

> with(plots):with(plottools):

> plot([x(t),y(t),t=0..6*Pi],scaling=constrained);

To create your movie:

> p2:=s->plot([x(t),y(t),t=0..24],scaling=constrained):

> q2:=s->plot([[(0,0)],[1.5*x(s),1.5*y(s)]],color=black):

> r2:=s->plot([[x(s)+D(x)(s),y(s)+D(y)(s)],[x(s)-D(x)(s),y(s)-D(y)(s)]], color=black):

> disp2:=s->display(q2(s),p2(s),r2(s),pointplot([x(s),y(s)],symbol=circle),labels=["",""],tickmarks=[0,0]):

> display(seq(disp2(.25*s),s=0..60),insequence=true,scaling=constrained);

To find derivatives and the length:

The command normal converts quotients of functions to a simpler form, in which common factors of numerator and denominator are assumed to be non-zero and canceled. This command may be useful in your calculation of the length of the spiral.

> normal(expression);

The Most Common Maple Commands

Academic Honesty Statement:

Place the following statement (by copying and pasting) at the end of your report and sign it in ink. Your instructor will not grade your report unless this signed statement appears at the end of your report.

I understand that I may work with others if I give them credit in this statement. I also understand that I am required to write my report--that to copy all or part of someone else's report or to allow someone else to copy all or part of my report constitutes plagiarism, which is a serious violation of academic honesty.

I worked with (replace this parenthetical remark with first and last names of those with whom you worked) on this project. I wrote my own report. I did not copy any of this report from anyone else and I did not allow anyone else to copy any of this report.

Signed: