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The Equiangular Spiral - Why "Equiangular"?

Author(s): 
Lang Moore, David Smith, and Bill Mueller

The name "equiangular spiral" suggests that some collection of angles related to this curve are all the same -- equal -- constant. In this part of the module, we first explore what's constant about this type of spiral. Then we examine the relationship between the constant and certain angles.

  1. You should have found in Parts 2 and 3 that the nautilus seashell curve has a polar formula of the form 

     

    where r0 is the value of r at theta = 0 and k is a constant that incorporates both a multiple of pi and a growth constant. (Your formula may not look exactly like this, but it should be equivalent.) Explain why the ratio

    is constant. That is, the rate of growth of r is proportional to r itself.

Now we consider how that constant proportional growth rate is related to a constant angle. In the following figure, we show a segment of polar curve in red, along with three angles associated with a given point on the curve:

  • the polar angle theta to the point,

  • the angle alpha from horizontal to the tangent to the curve, and

  • the angle beta between the tangent and the radius through the point.

We see from the figure that alpha = theta + beta, so beta = alpha - theta. Now use a formula from trigonometry to relate tan(beta) to the tangents of the other two angles:

The tangents in this formula are easy to relate to the (x,y) coordinates of the point on the curve:

  1. We know from the polar to cartesian change-of-coordinate formulas (which you used in Part 3) that

    Find derivatives of x and y with respect to theta, and then combine the results to find dy/dx in terms of theta. You may want to use your helper application for this.

  2. In the formula for tan(beta), substitute the expression from the preceding step for tan(alpha). Simplify the resulting expression as much as possible. You definitely want assistance from your helper application here -- but you may have to help it see what steps to take. In particular, you may have to substitute sin(theta)/cos(theta) for tan(theta).

  3. Explain why the angle beta is constant -- that is, why beta is the same for every point on the spiral. That's the equiangle!

  4. Explain the relationship between beta and the constant ratio of growth rate of r to r itself.

Lang Moore, David Smith, and Bill Mueller, "The Equiangular Spiral - Why "Equiangular"?," Loci (December 2004)

JOMA

Journal of Online Mathematics and its Applications

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