### The Equiangular Spiral (Alternate Version)

#### Part 4: Why "Equiangular"?

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 will explore first what's constant about this type of spiral. Then we will look at what that has to do with 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), and substitute sin(theta)/cos(theta) for tan(theta). 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.

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.