Roulette
- Four different roulettes formed by rolling four different shapes and tracing a fixed point on each of these shapes.
Basic Description
Suppose you see a nickel rolling on the sidewalk. Imagine a pen traced the path of one fixed point on the coin as it rolled. A curve would be created. This curve is called a roulette. The example is depicted below:
However, a roulette is not restricted to straight lines and circles. The rolling object can range from a point on a line to a parabola to a decagon to anything. Similarly, the surface on which this curve rolls does not have to be a line. It can be a parabola as well, or a circle, among many others. There are a few restrictions that apply:
- The curve that is not rolling must remain fixed.
- The point on the rolling curve must remain fixed.
- Both curves must be differentiable.
- The curves must be tangent at all times.
In the example of the rolling nickel, we imagine that the point of the pen is somewhere on the edge of the nickel. However, this point does not have to be on the edge of the rolling object. It can be also be inside
or outside,
varying how the curve will look.
Variations
There are numerous cases of different roulettes; the most common ones have been named. A few of these are listed below:
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Trochoid
A trochoid is any roulette in which the fixed curve is a line and the rolling curve is a circle.When the fixed point is on the edge of the rolling circle, then the roulette is called a cycloid. If the point is inside the rolling circle, then the curve produced is called a curtate cycloid. If the point is on the outside of the rolling circle, then the curve produced is called a prolate cycloid.
Example of a curtate cycloid
Hypotrochoid
A hypotrochoid is any roulette in which both the rolling curve and the fixed curve are circles and the rolling circle is on the INSIDE of the fixed circle. The fixed point can be on the
edge
,
inside,
or
outside
of the rolling circle.
Example of a hypocycloid
Epitrochoid
An epitrochoid is any roulette in which both the rolling and the fixed curve are circles and the rolling circle is on the OUTSIDE of the fixed circle. Like in the previous cases, the fixed point can be on the
edge
,
inside
, or
outside
of the rolling circle.
Example of an epicycloid
Involute
An involute is a more complicated roulette in which the rolling object is actually a line, and the fixed curve can be any other curve. In the case of the involute, the line acts as an imaginary string and as the line rolls, the string winds in around the curve. The pattern traced by the endpoint of the string is the roulette.
Example of an involute with a circle as the fixed curve
Interesting Application of the Concept
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The above roulettes are only a few of many different types of this curve. The main image of the page demonstrates that the fixed curve can be a catenary and the rolling curve does not need to be a circle but can be a polygon with sharp edges.
Below are a few examples of this concept:
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It is easy to imagine a nickel rolling on the floor, but how can we imagine a square rolling a on a bumpy road? Professor Stan Wagon of Macalester College created a square-wheeled tricycle and demonstrated that it is possible for square wheels to work. Below is a short video that shows how this tricycle works. For more information go to Macalester Math and Science
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References
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