by
Franziska von Herrath
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| 'a'=1,'b'=(2/5),h=1.2 | 'a'=1,'b'=(3/5),h=1.2 | 'a'=1,'b'=(3/5), h=.6 |
| Abstract: This report will provide an illustrated study of the epitrochoid, including a description of the special plane curve, the derivation of its parametric equations, and a brief history. |
Description:
The epitrochoid is a
special case of a roulette. The curve is the locus of a point 'P'
that is rigidly attached to a small circle of radius 'b' which rolls
without slippage on the outside of a bigger circle with radius 'a'.
If the ratio of the curves' radii is rational, the curve is periodic.
In the case that the ratio is irrational, the epitrochoid will not repeat
itself. The ratio of the circles' radii also determines the number
of cusps the epitrochoid has. The above pictured curves all have
a ratio of 1/5, so note the five cusps on the curves' inside and outside.
Since the distance from the center of the smaller circle to the point 'P'
is variable, the epitrochoid is a parent to many other more specialized
legendary plane curves like the epicycloid, whose point 'P' is located
on the circumference of the smaller circle with radius 'b,' or,
for example, the limacon whose two circles are of equal radius.
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Deriving the Parametric Equations:
Define 'm' to
be the sum of the two circles' radii:
[1]
m= a + b
The equation for the big circle of radius
'a' is, since it is centered at the origin, as follows:
[2]
x^2 + y^2 = a^2.
The equation of the small circle is consequently:
[3]
(x-m)^2 + y^2 = b^2.
The point 'P' at t = 0 can be represented
in coordinate form by its distance from the origin :
[4]
P0 = ( m-h , 0 ),
where h is the distance from the small circle's
center to the point 'P'.
As the small circle with radius 'b'
revolves counterclockwise around the big circle with radius 'a',
the coordinates of the point 'P' can be described with the equation:
[5]
P= m [cos(t), sin(t)] - h [cos( b),
sin( b)].
Now the angle b
needs
to be expressed in terms of the bigger circle's angle 't' in order
to arrive at a parametrization with respect to one variable.
As the small circle rolls on the big one, it 'unwinds' the same arc length
as the big one:
[6]
arc BC = arc RC.
Since s = rq,
thus:
[7]
a t = b t1,
From the picture one can learn that:
[8]
b = t1+
t .
Solving equation [7] for t1
, one obtains:
[9]
t1= at/b.
Substituting this result back into equation
[8],
one
can express bin
terms of t alone:
[10]
b = at/b
+ t.
Combining by remembering that m=a+b,
one attains:
[11] b = mt/b.
Consequently, the parametric equations for the epitrochoid are:
y = m sin (t) - h sin (mt/b)
If, after all that math, you still don't believe these equations will work, witness with your own eyes how an epitrochoid revolves!
Have a little fun with an animation! You can manipulate the length of the revolving arm and change the sizes of the circles. Remember that the circles are in proportion to each other, so changing the size of one will automatically change the size of the other. All you need is a copy of Geometer's Sketchpad and you will be able to see this animation. . . Click here!
A Brief History:
The epitrochoid
was first described by Albrecht Duerer in 1525, who called the curve "spider
lines" because he thought the curve bears resemblance to an arachnid. He
mentioned it in his book " Instruction in Measurement with Compass and
Straight Edge." Ole Roemer studied epitrochoids in 1674 in connection
with his research concerning gear teeth. The curve was also examined
by a variety of mathematicians, such as Leibniz, Newton, the Bernouills,
La Hire and Desargues, who did extensive work in descriptive geometry,
and L'Hopital. Today epitrochoids can be found in rotary combustion
engines by noting the path that the rotor tip of the eccentric shaft traces
out upon revolving.
Works Cited:
Lockwood, E.H, A Book of Curves,
Cambridge UP, 1963.
Lawrence, J. Dennis, A Catalog of
Special Plane Curves, Dover Books, 1972.
Web site: Xah:
Special Plane Curves: Epitrochoid