by: Marcia Bailey
Linda Crawford
Cynthia Dozier
The idea for this essay came from the following problem:
Take two quarters and place them face side up on a table. Keep one stationary and rotate the other around it counterclockwise, keeping the edges in contact. Be careful not to let the rolling quarter slip. Something strange appears to happen--the rolling quarter makes one complete revolution in only half the distance around the fixed quarter. What’s going on here? The circumferences of the two quarters are the same so why does the rolling quarter make 2 complete revolutions in one roll around the fixed circle?
The above problem will be investigated in this essay. Other problems related to rolling circles will also be considered.
We can use GSP to simulate the rolling of the quarter around the quarter. Click here for the simulation. Observe that the rolling quarter does make two revolutions in one roll about the fixed quarter. (Hint: Keep your eye focused on point P.)
Let’s now consider what happens if the ratio of the two radii is not 1 to 1. Let C be the fixed circle with a radius of f; let C’ be the rolling circle with a radius of r. Suppose the ratio of f to r is 2 to 1. C’ makes 3 revolutions in one roll about C. Click here for the GSP simulation.
It can be shown that C’ makes (f+r)/r revolutions as it rolls around the outside of C once.
Investigate the problem for various ratios of f to r. When you open the GSP sketch, drag on points B and/or D to change the ratio of f to r. When you reach the ratio you want to investigate, animate the sketch. Verify that C’ does make (f+r)/r revolutions. Click here.
To prove that the number of revolutions is (f+r)/r, observe that the locus of the center of C’ as it rolls about C is a circle concentric with C and having a radius of f + r. Hence, the center of C’ travels a distance of 2 (f + r) as C’ rolls around C once. But the center of C’ travels a distance of 2 r every time C’ makes one revolution. Therefore, the number of revolutions that C’ must make in one roll about C is
2 (f + r) or ( f + r)/r.
2 r
Let P be a point on C’. What would be the locus of P as C’ rolls about C? Use GSP to investigate this situation for various f / r. Click here and again drag on points B and/or D to obtain the desired ratio.
The curve defined by the locus of point P as C’ rolls around C is called an epicycloid. There is a cusp of the first kind at every point at which P touches the fixed circle. Notice that the number of arches is equal to f /r. In other words, when f = r, the curve has only one arch; there are two arches when f/r is 2 to 1, and n arches when f /r is n to 1. For f = r, the special curve called the cardioid is obtained. The parametric equations of the epicycloid are:
x = (f + r) cos t - r cos [((f+r)/r) * t],
y = (f + r) sin t - r sin[((f+r)/r) * t].
TRY THIS: Use the GSP sketch to trace the locus of point P for f/r = 3/1. (You may find it instructive to turn the grid on in the GSP program and then drag the center of C to the origin.)
Now use the TI-82 to graph the above parametric equations for f = 3 and r = 1. Are the sketches on GSP and the TI-82 the same?
Click to trace the locusof point M, the midpoint of the radius of the rolling circle. The locus of this point is called an epitrochoid.
Epitrochoids are generated when the locus of any fixed point Q on the radius of the rolling circle, or on the extension of the radius, is traced as the circle C’ rolls about C. If h denotes the distance from the center of the rolling circle to point Q, the parametric equations of the epitrochoid are:
x = (f + r) cos t - h cos [((f+r)/r) * t],
y = (f + r) sin t - h sin[((f+r)/r) * t].
Notice that an epicycloid is a special epitrochoid.
In the GSP sketch that follows point Q has been constructed on the line which contains the radius of circle C’. Trace the locus of Q for various distances h. Click here.
When h > r, the locus contains loops (where the number of loops is equal to the ratio of f to r). When h < r, the locus does not contain any loops.
Now let’s consider the situation where C’ rolls along the inside of C; that is, C’ is internally tangent to C. Let f to r be 2 to 1. Use GSP to simulate the motion. Click here. C’ makes one revolution as it rolls once around the inside of C?
Examine the situation for various ratios of f to r. Drag on points B and/or D to obtain the ratio desired. Then animate the sketch. For example, if the ratio of f to r is 4 to 1, C’ makes 3 revolutions as it rolls around C once. Click here.
The locus of the center of C’as C’ rolls around C once is a circle concentric with C which has a radius of (f - r). Thus, the center of C’ must travel a distance of 2 (f - r) in order for C’ to roll around C once. Since the circumference of C’ is 2 r, then the number of revolutions C’ must make in one roll about C is
2 (f - r) or (f - r)/r
2 r
Now let’s consider the locus of a point P which is located on C’. The locus of P as C’ (in a clockwise rotation) rolls around the inside of C is called a hypocycloid. The hypocycloid has a cusp of the first kind at every point at which it touches C. If f to r is 2 to 1, there are 2 cusps; for f to r of 4 to 1 there are 4 cusps; in general, the number of cusps is f /r. Click here to investigate.
The parametric equations of the hypocyloid are:
x = (f - r) cos t +r cos [((f- r)/r) * t],
y = (f - r) sin t - r sin[((f - r)/r) * t].
Let f = 4 and r = 1. Compare the GSP sketch of the locus of P with the graphs of the parametric equations.
As a last investigation, consider the locus of a point P on a circle C’ as C’ rolls along a line. Let the radius of C’ be r.
Click here.
The curve determined by this locus is called a cycloid. The parametric equations of the cycloid are:
x = r(t - sin t) and
y = r(1 - cos t)
Let T be a point located a distance of d from the center of C’. Trace the locus of point T. The curve traced out by T as C’ rolls along the line is called a trochoid. Look at the locus for various values of d by dragging on point T; consider cases where d < r and where d > r. Click here for the GSP sketch. (The cycloid is the special case where d = r.)
Below are the steps to do the constructions which simulate the rolling of a circle C’ both around the outside and along the inside of a fixed circle C.
A. To construct C’ to roll along the inside of C:
1. Construct a segment AB whose length F will be the radius of C.
2. Now construct the segment with a length of R. Proceed as follows: Measure AB; then use the calculator to determine the measure (R/F)*(length of AB). Select this measure and choose MARK DISTANCE under the TRANSFORM menu. Construct a point X; with X selected, choose REFLECT (by a marked distance). Draw the segment connecting X and its reflection. The length of this segment is R.
3. Now construct a segment whose length is (F - R). This length is the radius of the concentric circle.
4. Construct a point to be the center of circle C. Construct C.
5. Construct the circle CC which is to be concentric with C and having radius (F - R).
6. Construct a point Y on CC. Now construct the circle with center Y and radius R. This circle is C’ and should be internally tangent to C.
7. Now construct the “helper” circle. Construct a point Z away from the existing constructions. Now construct the circle with center Z and radius R. Construct a point W on this circle and draw the segment WZ.
8. Construct a line through Y and parallel to WZ. Determine one of the points of intersection (call it T) of this parallel line and the circle C’. Construct segment YT and then hide line YT.
9. Construct an animation button. Let Y move around circle CC and W move around the “helper” circle.
10. C’ should roll around C.
11. Hide objects as desired.
B. To construct C’ to travel around the outside of C:
Perform the same steps as above except that the radius of the concentric circle will be (F + R). Construct a segment with this length to use.
REFERENCES
Cofman, Judita. What to Solve? New York: Oxford University Press. 1991. p. 97.
Fremont, Herbert. Teaching Secondary Mathematics through Applications, 2nd edition. Boston: Prindle, Weber & Schmidt. 1979. pp.276-282.