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The Equiangular Spiral

(Alternate Version)

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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.

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

where **r**_{0} 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:

- 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.

- 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.

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

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