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The Hypotrochoid
By Craig Anderson
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This paper explores the historic, asthetic and mathematical
qualities of the hypotrochoid curve. Its history is briefly presented through
the important figures of its conception. Its mathematics are explored
from the definition to a derivation of the parametric equations.
Its asthetic qualities are explored through visualisation and metaphor.
There are some areas of mathematics that require us to cast aside our compulsive preoccupation with practicality and allow ourselves the luxury of enjoying the pure aesthetic beauty without any need for further justification. Indeed, there must be an avenue where the free flowing lines of the art world find a crossroads with the analytical worlds of the equation and the computation. When such seemingly distant facets of academia arrive at such an intersection the results can be simply enigmatic. We are transported to a world that is at once purely logical in function yet pure beauty in form. The hypotrochoid is a prime example of an element of mathematics that lies on such an intersection.
Mathematicians first became fascinated with this
curve in the early 16th century. The initial interest seems
to be stemmed from a paper written in 1501 by Charles Bouvelles in an effort
to solve the problem of squaring the circle. Giles Persone de Roberval,
who played an integral role in finding the area for these curves, is given
credit for the name "trochoid." Blane Pascal, who referred to
these curves as "roulettes", actually offered cash prizes for
anyone able to solve the "problems" of the area and the center
of gravity of these complex shapes. Gallio Galilei, who referred to these
shaped as "cycloids", revered these curves for their graceful
beauty and their architectural potential. Although its origins are European,
the Japanese mathematician Wada Nei contributed greatly to our understanding
of the tenkyo kiseki, or "loci described by rolling." Ultimately,
for us 20th century dwellers, it was Hasbro's release of the
Spirograph that put the hypotrochoid into the awareness of the mainstream.
Hy"po- [Gr. under, beneath; akin to L. sub. See Sub-.]
Tro"choid n. [Gr. Like a wheel + -oid; cf. F. trocho\'8bde. See Troche.]
The hypotrochoid is actually a part of a subset of
a much larger family of curves called the roulettes. To put things in the
proper perspective, I will define the parents and grandparents of the hypotrochoid
first. A roulette is a curve generated by tracing the path of a point attached
to a curve that is rolling upon a another fixed curve without slippage.
These can be any two curves. For instance if we were to trace a point along
a parabola rolling along a fixed cardioid the resulting curve would be
a roulette. Note that the trace point does not need to lie directly on
the rolling curve. It needs only to be fixed about the rolling curve so
that its position relative to the rolling curve remains constant. A specific
subset of a roulette is known as the cycloidal curves. This is when the
rolling curve is a circle and the fixed curve is either a line or a circle.
Finally, the hypotrochoid is a specific case of the cycloidal curves in
which both the fixed curve and the rolling curve are circles and the rolling
circle is rolling against the inside of the fixed curve without
slippage.
Ok, this is where it starts getting fun! In order to
develop the parametric equations for this curve, we need to be able to
pinpoint the position of our trace point (P) at any given time. Our fixed
circle is circle C with radius a and center O. Our rolling circle is circle
S with radius b and center O'. The trace point P is distance h from the
center O' of the rolling circle. At t=0 P can be found at the coordinates
(a-b+h,0).
Now at time t>0 things begin to get far more complicated very quickly. As you can see the point of contact C travels around circle C in the counterclockwise direction through angle t. This is a result of circle S is rolling along circle C. This means that if you viewed circle S all by itself it would be rotating in a clockwise direction through angle b.
Let's take a moment to remind ourselves of our goal.
We want to track the position of point P at any given time. To do this
it will be convenient to implement the art of vector addition. Because
what we are interested in is position of P relative to the origin O, we
can focus on the way the vector 
changes
as time marches forward. A careful analysis of the diagram at the left
reveals that the vector will
always be equal to the sum of the vectors 
and 
.
Hence, the simplest form of the equation of the position
of P is:
Although this is very neat and quite tidy, it isn't
of much use to us at this point. Let's continue our analysis by examining
the vector . It can be seen
that the magnitude of does
not change with time and that point O' follows a counterclockwise circular
route of radius a-b. Thus,
Now we would be wise to devote our attention
to . This
vector's magnitude does not change with time either but it can be seen
that point P follows a clockwise circular path of radius h about
point O'. Thus, 
Ok, all we have left to do is to relate angle b to angle t. We can do this by taking note of the fact that arc(BC) must be the same length as arc(RC) due to the fact that both were created via the same rolling action. Now, given the relationship between arc length, angle, and radius tells us that arc(BC)=at and arc(RC)=b(b+t). Thus,
Whew, let's put all the pieces together,
Finally, this leads us to our ultimate goal
of finding the parametric equations: 
One last point of interest that I will address about
the mathematics of the hypotrochoid is the periodicity. If the ratio of
a to b is rational then the period can be found to be the numerator of
b if b is expressed as its own fraction. In other words if a=1 and b=3/4
then the ratio between them is rational and the numerator of b is three.
Thus the trace point will return to its original position after the rolling
circle completes three rotations. If you wish to express the period in
radians then simply multiply the value of the period by 2pi. The logic
behind this periodicity can be found by analyzing the coefficient of the
argument of the second sine and cosine functions of the parametric equations
(the ones that have an outer coefficient of h). If the ratio of a to b
is irrational then the curve is not periodic.
| Now that we've
labored our way through the mathematical rigor of the hypotrochoid let's
have a little fun with it. Single click on the "Animate" button
and draw a hypotrochoid. You can adjust the radii of both circles and the
trace point distance h by dragging the appropriate knob on the control
panel. You can clear the current drawing by clicking on the red "X"
in the lower left corner of the sketch. You may want to close some of the
toolbars on your browserin order to get a better veiw of the whole sketch.
If you experiment with the periodicity by manipulating the ratios between a and b you will find that it is quite difficult to create a perfectly periodic curve. This is due to some imprecision in the Geometer's Sketchpad measuring technique. Well, in all fairness, it is also quite difficult to manipulate something with a mouse to three decimal places of accuracy! You may notice that this applet window also has very poor resolution. This is because the software is plotting periodic discrete data points rather than a continuous plot. The results of this animation are quite different while executed in the GSP environment. To download this sketch and see the difference click here (you must have Geometer's Sketchpad to use this file, you can download a demo copy at the Keypress web site . . .see works cited). Well, I hope that this journey to the intersection between art and math has been an enjoyable one. If in fact it has, I would like to invite you to go to your local thrift store and scrounge the shelves for an old dusty Spirograph kit (you can still buy new ones too). Bring it home, put on a pot of coffee, measure each wheel, and build yourself a parameterized Spirograph kit! Just think of the possibilities . . . You can be the first kid on you block with the capability of constructing hypotrochoids with a mere pen. Consider the envy of your friends, neighbors, and peers. Well, just consider the fun you'll have. |
