Difference between revisions of "Roulette"

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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 <balloon title="load:insideroulette">inside</balloon><span id="insideroulette" style="display:none">[[Image:Curtatecycloid.gif]]</span> or <balloon title="load:outsideroulette">outside,</balloon><span id="outsideroulette" style="display:none">[[Image:ProlateCycloid.gif]]</span> varying how the curve will look.  
 
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 <balloon title="load:insideroulette">inside</balloon><span id="insideroulette" style="display:none">[[Image:Curtatecycloid.gif]]</span> or <balloon title="load:outsideroulette">outside,</balloon><span id="outsideroulette" style="display:none">[[Image:ProlateCycloid.gif]]</span> varying how the curve will look.  
  
=Variations=
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{{hide|=Variations=
 
There are numerous cases of different roulettes; the most common ones have been named. A few of these are listed below:
 
There are numerous cases of different roulettes; the most common ones have been named. A few of these are listed below:
 
=== Trochoid ===
 
=== Trochoid ===
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[[Image:Involute.gif|left]]
 
[[Image:Involute.gif|left]]
 
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>}}
 
<br><br><br><br><br><br><br><br><br><br><br><br><br><br><br>}}
 
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== Interesting Application of the Concept ==
 
== Interesting Application of the Concept ==
 
{{Hide|1=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|catenary]] and the rolling curve does not need to be a circle but can be a polygon with sharp edges.  
 
{{Hide|1=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|catenary]] and the rolling curve does not need to be a circle but can be a polygon with sharp edges.  

Revision as of 10:53, 28 June 2012


Roulette
Roulette.jpg
Field: Geometry
Image Created By: Wolfram MathWorld
Website: Wolfram MathWorld

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:

Cycloid animated.gif

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.

{{{1}}}

Interesting Application of the Concept

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:

Click to stop animation.
Click to stop animation.



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

EmbedVideo does not recognize the video service "tubechop".






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References





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