# Parametrizing the Nephroid

Let's consider the path followed by a point on the boundary of a disk as it rolls around the boundary of a second disk whose radius is twice that of the first disk. (Curves formed by rolling disks of various radii in this manner are called epicycloids.) We will choose the particular disks of radius 1 and 1/2 as shown below, and follow the point on the boundary of the smaller disk labeled P in the picture.

### Figure 2: Following a point on a rolling circle.

To establish some notation, let's say

• R = 1 is the radius of the larger disk
• r = 1/2 is the radius of the smaller disk
• t is the angle of elevation of the center C of the smaller disk with respect to a horizontal line going through the center of the larger disk.
• s is the angle of elevation of P with respect to a horizontal line going through C.
How should s and t be related in general? Let's collect some data to figure this out.

## Question 1

The Epicycloid Generator allows us to simulate the rolling of one circle around another. The point P that we will trace begins at 180 degrees. In other words, when t=0, then s=180 degrees. Make a table of t versus s values for values of t in the interval [0, 360] and use this table to help you answer the following questions:
• How many times does s=0 as the outer circle rolls around the inner circle? For what values of t does this occur?
• If we stay away from the t values for which s becomes zero, how does s change for a given change in t? In other words, what is ds/dt?
• Use your data and the answers to the previous questions to write down a simple formula for s as a function of t. (Hint: even though the data indicates that the underlying function is discontinuous, use the fact that 0 and 360 degrees represent the same point on a circle to write down a continuous function.)

You might be surprised by the answer to the previous question, if you had reasoned as follows: "Since the larger disk has twice the diameter, it has twice the circumference. Therefore the smaller disk will have gone through twice the angular rotation as the larger one during the same period of time, and s = 2t."

Although the above reasoning is slightly wrong, it is nevertheless true that the small circle has half the arclength of the large circle, and therefore as it rolls, it rotates twice due to the rolling. If the above statement is true, then there must be another factor that causes the small circle to rotate faster than our intuition suggests.

## Question 2

Suppose the small circle does not roll around the edge of the big circle, but rather is rigidly connected to the larger circle so that it moves around the circle as in the figure below.

How many rotations will a dot on a non-rolling circle make as the circle moves once around the larger circle? Use this write down a parametrization for the nephroid as x = x(t), y=y(t), where x(t), y(t) are some trigonometric functions of t. Check that your parametrization gives a nephroid using the Parametric Curve Plotter.

### Figure 3: Following a point on a circle that rotates, but does not roll.

Next: Transforming a Parametrization into an Implicit Algebraic Equation
Up: Introduction
Vic Reiner <reiner@math.umn.edu>
Frederick J. Wicklin <fjw@geom.umn.edu>
Last modified: Thu Apr 4 14:50:09 1996