Suppose that one of Ezekiel's wheels is fixed, while the other wheel rolls around its rim. Then a point attached to the rolling wheel traces a curve, called a trochoid. There are a great variety of trochoids. Some look like stars, some like flowers; some, maybe, like both. Along with beauty, they have interesting geometrical properties.
The rolling wheel may be inside or outside the static, or supporting, wheel. If it is inside, the curve is called a hypotrochoid; if outside, an epitrochoid.
The tracing point may be inside the rolling circle, outside it, or on the circle. If it is inside, then the curve is said to be curtate; if outside, it is prolate. If the point is on the circle, then the curve is an epicycloid or hypocycloid.
The word cycloid is usually applied to the curve generated by a point on the rim of a circle that rolls on a straight line. Following the precedent set by F. Morley, I shall use this word to include epicycloids and hypocycloids.
A trochoid is a closed curve, of finite length, precisely when the radius of the rolling circle is a rational multiple of the radius of the supporting circle. I will use the convention that this ratio, which I will call the wheel ratio, is positive if the two circles curve the same way at the point of contact. Thus the curve is a hypotrochoid if the wheel ratio is between 0 and 1; if it is negative or greater than 1, then the curve is an epitrochoid.
The wheel ratio is
where N and D are positive integers; usually we shall keep the fraction in lowest terms, so that the greatest common divisor of N and D is 1. Then, if the curve is a cycloid (epi- or hypo-), it will have D sharp points, or cusps, on the supporting circle. D is sometimes called the degree of the trochoid.
The distance from the center of the rolling circle to the tracing point I will call the arm, and the ratio of this to the radius of that circle I will denote by ρ, and call the arm ratio.
When N > 1, the trochoid can be generated by any one of N points equally spaced around the rolling circle. More generally, one can take any number, say M, of points equally spaced around the rolling circle, and ask what curves they generate. Let K be the greatest common divisor of N and M, with M = Km; then these points generate a rosette of m trochoids, each one traced by K points.
Here is an interactive display of trochoids, or rosettes of trochoids, along with the wheels that generate the curves. The top controls on the display enable you to select the sign, denominator, and numerator of the wheel ratio. Below these is a slider, which gives a range of possible values for the arm ratio, ρ; the selected value is shown below the slider. The interesting special case of cycloids can be chosen by clicking the button labeled "Cycloid." The control labeled "M" enables you to select the number of tracing points. Note that if M is equal to N, then all the points trace out the same trochoid.
There is a knob at the center of the rolling circle, represented by a black dot. If you move the cursor to that dot and press the mouse button, the dot will turn red, and it can be moved by dragging the mouse. This shows how the rolling wheel actually generates the trochoid. The dot will turn black, and revert to immobility, if you release the mouse button.
We can find an expression for the coordinates of an arbitrary point on a given trochoid. It is convenient to represent points in the plane by complex numbers. (Here is a refresher on complex numbers.) Let the supporting circle have radius 1, and center 0. Initially, let the roller be tangent to the supporting circle at 1; the center of the roller is at C0 = (1-w). Let the vector from C0 to the initial position T 0 of the tracing point be ρw. Let the arm C0T 0 (extended if necessary) intersect the roller at V0 = P0.
Now let us move the roller so that its point of contact with the supporting circle is P = eiθ = cosθ + i sinθ. Then the center of the roller moves to C = (1-w)eiθ. Let T be the position to which the tracing point moves, and let the intersection of the arm with the roller move to V. The arc length from C0 to C is θ; therefore the arc from V to P is also θ, so the angle from CV to CP must be (1/w)θ. Now θ specifies the direction of CP; so the direction of CV must be specified by (1 - 1/w)θ. Therefore the vector from C to T is
We have found
This is a "parametric representation" of the points on the trochoid. We will say that θ is the parameter of the point T. (For this particular trochoid, and this method of generating it.)
It is convenient to denote the value of T by T(w, ρ)(θ). Then T(w,ρ) is a function from real numbers θ to complex numbers denoting points in the plane. We will also start using "T(w, ρ)" as a name for a trochoid with wheel ratio w and arm ratio ρ. When ρ = 1 we will omit it; this "T(w)" is the name of a cycloid.
It is natural to multiply the expression T(w, ρ) by a constant, to change its scale or rotate it; and to add a constant, to shift its center from the origin.
This formula for T shows that a trochoid is generated by a combination of two circular motions. The first is associated with the motion of the roller around the stationary center; it has period 2π and amplitude (1-w). The second is associated with the rotation of the roller around its own center; it has period 2πw/(w-1) and amplitude ρw.
Or, we could define Q = ρei(1-1/w)θ and note that
Thus P and Q move around two circles of different radii, at different rates; and T divides the segment PQ in the constant ratio w : (1-w).
This suggests that we could think of Q as the contact point of a rolling circle on which is carried the vector going from wQ to T. The result is a second way of generating the same trochoid. The fact that each trochoid can be generated in two ways was discovered by Daniel Bernoulli in the seventeenth century. Here is how the two ways of specifying the trochoid are related:
Double Generation Theorem: If a trochoid is generated with a supporting circle of radius R, wheel ratio w, and arm ratio ρ, then the same trochoid can also be generated with a supporting circle of radius R′, wheel ratio w′, and arm ratio ρ′, where
In our algebraic notation, the trochoid R·T(w, ρ) is equivalent to Rρ ·T(1-w, 1/ρ).
As I said above, a trochoid is called "prolate" if ρ > 1 and "curtate" if ρ < 1. The double generation theorem seems to me to make this distinction somewhat unreal.
Here is an interactive display which demonstrates double generation. The controls are like those of the display farther up on this page; their values refer to the blue circles and arms. The values of R′, D′, N′, and ρ′ refer to the green circles and arms.
The lines normal to a trochoid are simply related to the supporting circles. There are also interesting groups of normal lines at a point on one of those circles. When the arm ratio is 1, there are similar patterns for the tangent lines.
An cycloid can be elegantly constructed as an envelope of its tangent lines.
The evolute of T(w) is a similar curve.
A trochoid whose supporting circle is rolling on another circle will envelop another trochoid.
The parametric representation of a cycloid can be extended to an interesting conformal map.
The MacTutor History of Mathematics archive has information about trochoids and many other curves.
Alexander Bogomolny's web site has very many interactive pages; one of them has an applet which demonstrates a particular case of the envelope of a rolling trochoid.