Hypotrochoid

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Hypotrochoid
Three Hypotrochoid curves combined, each represented by a different color: green, yellow, and orange.

Contents

Basic Description



If the blue circle rolled (like a wheel) along the edge of the larger circle and the red point acted as a pen, leaving a trace, then the red drawing created would be a hypotrochoid. We will refer to the red point as P.

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Variations

There are three subgroups within the Hypotrochoid family. The point P can be in three different general locations in relation to the rolling circle: inside, outside, or on the edge.

  • Inside the rolling circle

     If P is inside the rolling circle, then the distance of the line from the center of the inner circle to P is less than the radius of the inner circle. We call this a curtate hypocycloid.

  • Outside the rolling circle

     If P is outside the rolling circle, then the distance of the line from the center of the inner circle to P is greater than the radius of the inner circle. We call this a prolate hypocycloid.

  • On the edge of the circle

     If P is on the rolling circle, then the distance from the center of the inner circle to P is equal to the radius of the inner circle. We call this simply a hypocycloid.

Demonstration

If you can see this message, you do not have the Java software required to view the applet.


The interactive applet on left is an illustration of a Hypotrochoid.
You can change the radius of the inner and outer circles as well
as the distance of the point, P to the center of the circles.



A More Mathematical Explanation

Note: understanding of this explanation requires: *Trigonometry and Basic Parametric Equations
Image:HypotrochoidDiagram2_1000.gif
A hypotrochoid is a roulette traced by a point P attached to a circle of radius b rolling around the inside of a fixed circle of radius a, where P is a distance h from the center of the interior circle. Note that the point P can be anywhere in relation to the interior circle.








Hypotrochoids are graphed using the following parametric equations


  • x(t)=(a-b)cos(t)+hcos(\frac{a-b}{b}t)
  • y(t)=(a-b)sin(t)-hsin(\frac{a-b}{b}t)

All hypotrochoids are defined by these parametric equations. They all belong to a family of curves, which includes other types of trochoids.

The domain of t has a large effect on what the hypotrochoid will look like. If we allow t to vary from 0 to 2\pi, then the interior circle will only make one revolution around the fixed circle. In some cases this will not result in a full pattern, meaning that the point P will not always return to its starting place. For example, let the radius of the interior circle b be 3, the radius of the fixed circle a be 5, and the distance from P to the center of the interior circle h be 5. By revolving the interior circle once, the resulting image looks as follows:

In order to obtain a pattern that has the same symmetric characteristics as the one at the top of the page, the interior circle must make at least three complete revolutions, therefore ranging from 0 to 6\pi. The following animation depicts this idea where a is 5, b is 3, and h is 5:
Image:Hypotrochoid35.gif

Setting a Domain

There is a method to determine the domain of t in order to complete a full pattern where the curve ends at the point where it started. To find the maximum value of the domain, the values of the outer and inner radius, a and b respectively, are used.

The inner circle generally starts at t=0, so the x coordinate will be a-b+h and the y coordinate will be 0. Similarly, when t is zero, cos(t)=1 and sin(t)=0. A full patern will occur when both x and y coordinates have the values of the same initial coordinates at t=0. This occurs at 2\pi, 4\pi, 6\pi: essentially, any multiple of 2\pi.

There is a general value for the maximum endpoint that will result in one complete patern ending at the starting point. This value of t is
2\pi\frac{b}{gcd(a,b)}

where gcd(a,b) stands for the greatest common factor of a and b.

How The Main Image Relates to Hypotrochoids

To recreate the main image, a graphing program such as Mathematica is required. Though there appear to be three different curves, each being a different color, they all have the same a, b, and h values of 100, 2, and 70 respectively. What makes them look like three different curves is the fact that only a limited number of sample points are plotted for each graph. Rather than plotting infinitely many points, there are three curves of the same graph with 75, 125, and 175 sample points.

Notice how the plot of the curve with 125 sample points does not overwrite the plot of the curve with 75 sample points. This is because the amount of sample points selected are evenly spread out on the curve separated by the same distance between each. When the number of sample points increases the distance between each decreases, so the points will generally not overwrite a curve with less sample points which are further apart.




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Related Links

Additional Resources

Reference used - http://mathworld.wolfram.com/Hypotrochoid.html
Reference used - http://en.wikipedia.org/wiki/Hypotrochoid
Reference used - http://online.redwoods.cc.ca.us/instruct/darnold/calcproj/Fall98/CraigA/project3.htm






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