A Rolling Trochoid Envelops Another Trochoid

What happens when a trochoid's supporting circle is rolling around another circle? We will call this a rolling trochoid, and look for curves which it envelops.

Rolling Trochoid Theorem: Let T(w) be an epicycloid or hypocycloid, generated with rolling circle C; let C serve as the supporting circle of a trochoid similar to T(w′, ρ). Then the rolling trochoid envelops a trochoid T(ww′, ρ).

This theorem was first proved (so far as I know) in a paper by Frank Morley, "On adjustable cycloidal and trochoidal curves", American J. of Math. 16(1894) 188-204. I have seen at least one recent reference to it in print, which I have not been able to track down.

Informal Proof. Let CS be the stationary circle on which C rolls. Then w is the radius of C. Let CR be the circle which rolls on C to generate the "rolling" trochoid similar to T(w′, ρ). Then ww′ is the radius of CR. The tracing point T of that trochoid is distant ρww′ from the center of CR. We may move CR , if necessary, so that its contact point with C is the point P of contact between C and CS . Now let C and CR move so that they retain this common point of contact with CS. Relatively to CS, T traces out T(ww′, ρ), and relatively to C it traces out the rolling trochoid. Moreover, the line PT is normal to both trochoids at T. Therefore the two trochoids are tangent to each other at T, QED.

If ρ = 1, then the rolling trochoid has cusps on C, which trace out cycloids as C rotates. Even when ρ ≠ 1, this set of points, which we may call a chief group following Morley, move on a rosette of cycloids. In the case where ρ = 1, the theorem is sometimes stated in terms of an epicycloid or hypocycloid whose cusps are guided by cycloids.

The trochoid T(ww′, ρ) is not the entire envelope of the rolling trochoid. Let w′ = ±N′ /D′. Then there are N′ points attached to CR each of which traces out the rolling trochoid, relatively to C. Relatively to CS , these points trace out a rosette of trochoids similar to T(ww′, ρ).

How many different trochoids are there in this rosette? To deal with this question we must go into rather fussy detail about the wheel ratios. The wheel ratio of the envelope trochoid is we = ww′ = ±(NN′)/(DD′). This expression may not be in lowest terms. Let f = gcd(N, D′) , that is, the greatest common divisor of N and D′, and let f′ = gcd(N′, D) . Then the lowest-term expression for we is

we = ±Ne/De , where
Ne = (N/f)(N′/f′) and
De = (D/f′)(D′/f) .
Now the rosette is traced by N′ points equally spaced around CR , whereas there are Ne points, equally spaced, which would trace one of the trochoids. The number K of points in both of these groups is gcd(Ne , N′). What's that? Well, Ne and N′ have at least a common divisor of (N′/f′); we can take this out of the "gcd" expression and say K = (N′/f′)gcd((N/f), f′). But this new "gcd" is 1. Why? Because the first argument, (N/f), is a divisor of N and the second argument, f′, is a divisor of D; but gcd(N, D) = 1. (Whew.) To summarize: of the N′ points that trace the rosette, a group of (N′/f′) traces one trochoid, and so there are f′ trochoids in the rosette.

For an example, try w = 1/6 and w′ = 4/3. Then each trochoid is enveloped at two points, and there are two trochoids in the rosette. Set the arm ratio according to taste; I recommend 1.40.

That rosette is not the whole story of the envelope of the rolling trochoid. Consider the case where ρ = 1. The double generation theorem tells us that the rolling trochoid is also T(1-w′). Therefore there is also a rosette of cycloids congruent to T(w(1-w′)).

If ρ ≠ 1 then I am not sure what is implied by the alternate generation of the rolling trochoid. The difficulty is that the alternate generation requires a supporting circle of radius ρw. This does not roll, without slipping, on any stationary circle. The envelope will include a rosette of curves, but are they trochoids?

Here is a display which demonstrates the rolling trochoid theorem. The upper set of controls determine the value of w; the lower set determine w′ and ρ. The "knob" is at the point of contact of CS, C, and CR. There are checkboxes to control how much detail appears. The "arm," that is, the radius of C ehich carries the tracing point of the rolling trochoid and the envelope, may be shown or hidden. The trochoid T(ww′,ρ) is always shown; when it is part of a rosette, the other trochoids in the rosette may be shown or hidden. So may the alternate rosette, when ρ = 1. Finally, two check boxes control the visibility of the guide cycloids and the common normals to the rolling trochoid and the envelopes.

Some special cases

Some of the simpler cases of the rolling trochoid theorem are widely known, and were discovered independently. In all of these, only cycloids are considered, so ρ = 1.

The hypocycloid with three cusps is often called the deltoid. If it is inscribed in a circle of radius 1, then every tangent to the deltoid meets the curve in two points, which are a distance of 4/3 from each other. Thus the segment of that length, moving with its endpoints guided by the curve, also envelops the curve. This is the case of the rolling trochoid theorem with w = 2/3 and w′ = 1/2. Thus, the rolling cycloid is a line segment, and the guiding cycloid is just an alternate generation of the envelope.

Let two perpendicular lines be drawn, and let a segment move so that one end is on each of the lines. Then the envelope of the segment is a four-cusped hypocycloid, or astroid. This is the rolling trochoid theorem with w = 1/2 and w′ = 1/2. The two lines are the "rosette" of guide cycloids.

The cardioid is the epicycloid with w = -1. Every line through its single cusp intersects it in two other points, whose distance is twice the diameter of the stationary circle. To bring this theorem into the framework of the rolling trochoid theorem, we use the alternate generation of the cardioid, with w = 2, and let w′ = 1/2. But this implies that the envelope has wheel ratio ww′ = 1. What can that mean? What it means, at least for this case of this theorem, is that the envelope degenerates to a single point, which happens to be the cusp of the cardioid.

In the above examples, the rolling trochoid is a line segment. Morley, in the paper mentioned above, generalized the deltoid example to have other hypocycloids as the rollers. He took w = (n-1)/n, so that the guide cycloid is the n-cusped hypocycloid, generated in the alternate way; and w′ = 1/(n-1). As before, the envelope coincides with the guide. The rolling hypocycloid has all of its n-1 cusps on the guide, and is tangent to it at another point. Morley built up some elaborate extensions of this construction, and a theory of "penosculant" curves to provide a framework. Alex Bogomolny has an introduction to it, with interactive applets.

Last modified on $Date: 2015-05-02 12:51:16 -0400 (Sat, 02 May 2015) $

Christopher J. Henrich