Epitrochoids
Epitrochoids |
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Epitrochoids
- An epitrochoid is a roulette made from a circle going around another circle. A roulette is a curve that is created by tracing a point attached to a rolling figure.
Contents
- 1 Basic Description
- 2 A More Mathematical Explanation
- 3 Why It's Interesting
- 4 How the Main Image Relates
- 5 Teaching Materials
- 6 About the Creator of this Image
- 7 Related Links
Basic Description
The epitrochoid can be imagined as 2 cut out circles. One is taped down to a paper, and the other has a stick that is taped to the circle, with a marker on that stick. The circle with a stick is pressed against the circle taped onto the paper, and the marker makes a pattern, or an epitrochoid on the paper. In a more detailed explanation, this page will go more in depth on the mathematical concept, the equations required for an epitrochoid, and its special forms.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Roulettes
Deriving the Parametric
Parametric Equations for an Epitrochoid:
http://www.math.hmc.edu/~gu/cu [...]Deriving the Parametric
Parametric Equations for an Epitrochoid:
1: As you can see, in the picture, there are 2 circles. We’ll call the radius of the bigger circle: a ; and the radius of the smaller circle: b. Since we are given these two radii, we can find their equations. So, the equation for Circle A(bigger one) would be x^2+y^2=a^2. The equation for the smaller circle will be different since the center of the circle is moved. So we know that the position of this center is the sum of the radius for Circle with radius a AND Circle with radius b. Thus, this equation for the smaller circle would be (x-m)^2 + y^2 = b^2.
From here, we have the two circles established:
Circle A-> x^2 + y^2 = a^2
Circle B->(x-m)^2 + y^2 = b^2
2: To find the coordinates for point P, we see that the y-value is 0 and the x-value is line h - (a+b). a+b is the sum of the 2 radii, and h is the line spinning. -> P0= (h-(a+b), 0)
3: On the picture to the right, you can see that beta is really the sum of the 2 mini arcs, t and t1 so by the Arc Addition Postulate, beta= t + t1 .
4: Since s = r(theta) -> a t = b t[1]
5: To solve this, solve for t1 first, thus you divide both sides by b to get: t1 = (a t)/b
6: Substitute this into the equation in step 3 to get: beta= t + ((a t)/b)
7: So if we look at the parametric equations for an epitrochoid, by definitions of a polar graph, sine is always the y-coordinate and cosine is always the x-coordinate. Variable h is the length of the line SP, while a and b represent their appropriate radii.
8: Now we can see that P= a+b [cos(t), sin(t)] - h [cos(beta), sin(beta)]. Sine and Cosine of (t) comes from theta; the first part of this determines where the outer circle is located. Sine and Cosine of beta (in which this case, we have found that beta= t + ((a t)/b) so…) we see in the second part of the parametric equations and because of the (h), this part determines the location of line segment h. By finding the place of h and the radii, we can get the position of point P.
Derivation from: http://msemac.redwoods.edu/~darnold/math50c/CalcProj/Sp99/Fran/epitrochoid.htm
Note:
For all of these above equations (Above and below): "h" is the distance from the point being traced to the center of the rotating/rolling circle.
The radius of the rolling circle is "b".
The radius of the stationary circle, that the smaller circle is rolling around, is "a".
Furthermore, the ratio for radii of the circles must be rational for the curve to be periodic. If it is irrational. then the epitrochoid will not repeat itself. The ratio is also important in determining the number of curves the epitrochoid will produce in one rotation around the circle.
Special Epitrochoids
The Epitrochoids are also the parent of many other mathematical functions:
Limaçon:
This is when the value of "a" is equivalent to the value of "b". "a" is the radius of the stationary circle, while "b" is the radius of the rotating circle. Once the two variables above are equal, then the epitrochoid becomes a , as shown below. As you can see, there are 3 different cases of this. They all review around the values of a and b.
Epicycloid:
An epicycloid occurs when the value of "h" is equivalent to the value of "b". Variable "h" is the distance from the point being traced to the center of the rotating. "b" as mentioned above, is the radius of the moving circle. When the two values are equal, it creates these:
From left to right, the values of a and b diet. The first one, a and b see equal. However, as you go to the right, you see that the values of a are increasing while the values of b are decreasing. Thus, out creates the differences in the bumps.
There is another case where the rotating circle is inside the stationary circle.
Explanation on the Position of the Point
The amount of times the point being traced goes back to where it started on the revolving circle is based on the two different circumferences. If the circumference of the revolving circle is a, and the stationary circle is 2a, that means the point on the revolving circle will end up where it started twice (2a/a=2). In other words, it takes the rotating circle two full rotations in order to return to the place where it first started.
Where the point is does not affect how many times arcs will be traced, as long as it is attached to the revolving circle. As demonstrated by this image, there are 2 curves no matter where the point is. If the point is on the edge of the circle, the curves will touch the edge of the stationary circle (red). If the point is inside the rotating circle, the curves will not touch the edge of the stationary circle (blue). If the point is outside of the rotating circle, the curves will loop into the stationary circle (purple).
However, the positioning of the point being traced does affect how curved the line is when the point goes back to its starting point. right
Relationship Between Radii and Circumference with the Number of Bumps(aka. curves)
There is an equation for the number of bumps the rotating circle takes as it rolls around the stationary circle once (from starting point back to starting point).
Process of testing of the relationship of "normal" epitrochoids:
We called normal epitrochoids the epitrochoids that has bumps that do not cross, and the epitrochoid reaches it's starting point after one revolution of the rotating circle.
Note: a is the radius of the stationary circle, and b is the radius of the rotating circle
Normal epitrochoid test 1: a=3 b=1 Here, we see that 3 curves were created. We see that since 3/1 is a whole number, the curves stay at the same place. 3/1=3
Normal epitrochoid test 2: a=6 b=3 Here, we see that 2 curves are created. 6/3=2.
From the tests, above we discovered:
If a/b produces a full number, then it will produce an epitrochoid where the curves all fit completely in one revolution.
There is a similar relationship between the circumferences.
If N= number of "bumps" in one rotation and C1= Circumference of stationary circle, C2= Circumference of the rotating circle,
Then N=C1/C2
This relationship is exactly the same as the relationship between the radii because:
In order to get the circumferences, you multiply both radii by the same thing (2 pi). Since you are using a ratio (R1/R2), you are basically multiplying the denominator and the numerator with the same number. This means that the ratio stays the same. Therefore, the ratio C1/C2 works.
Process of testing of the relationship of complicated epitrochoids:
After we tested the epitrochoids that go back to their starting point in one revolution, we tested on the more complicated epitrochoids
Test 1: a= 8 b= 7 Though counting the number of curves may be a little overwhelming, we should also acknowledge that the ration between a and b also have influence on whether the epitrochoid will go back to its original starting point. 8/7=1.142857... Though the epitrochoid doesn't make it all the way back in one full revolution, it does eventually after 8 bumps. Is there a relationship between the number of bumps and the circumfrences for numbers that don't add up perfectly? Let us look at more examples
Test 2: a=13 b=5 Here, we also see this fancy, interesting epitrochoid, but it does not return back to the starting point after one revolution, but it does eventually makes its way back after 13 loop. 13/5=2.6
Test 3: a=8 b=6 Here, we see another fancy, interesting epitrochoid. It makes it's way back to the starting point after 4 bumps
There is a relationship between all three of these tests. The first test had a=8 and b=7, with a total of 8 bumps. The second test had a=13 and b=5, with 13 bumps. The third test had a=8 and b=6, with 4 total bumps. In both the first and second test, the number of bumps were both equivalent to the value of a. If the first 2 tests had a and b written as a fraction, the number of bumps would be equal to the numerator. The third test was different from the first two, as it did not fit the pattern. However, if it was written as a fraction, it would be 8/6, it could be simplified into 4/3, and the number of bumps is still equivalent to numerator.
Final Equation
number of bumps = the numerator in the simplified version of a/b
This new equation also applies to epitrochoids where a and b divide into full numbers. For example, test number 2 of normal epitrochoids have a=6 and b=3. 6/3 can be simplified into 2/1, and the number of bumps is equivalent to the numerator.
The only difference between "normal" epitrochoids (when the numbers divide perfectly)and complicated epitrochoids (when there is a remainder after a/b), is that complicated epitrochoids don't meet back after one revolution.
NOte: the graphs above were made using Graph Sketch, which can be accessed here: http://graphsketch.com/parametric
Why It's Interesting
It creates interesting shapes and can be applied to uses of art and engineering. Three examples of usages of epitrochoids include the Wankel Rotary Engine, the Spirograph, and the Ptolemaic System.
Wankel Rotary
The Wankel Rotary Engine was created by Felix Wankel in the 1920s that is used in cars and vehicles as engines. At the time, it was a new type of gasoline engine. The rotor itself is an equilateral triangle while the bore is made of an eptitrochoid curve. This machine requires 2 pivots; the center one will go at a 270°turn while the other one will go at a negative 180° turn, this creating the path of motion. By following the path that the rotor tip creates, you can see the traces produced. For another explanation, you can go on: Involute
Spirograph
The Spirograph is a geometric tool used in manually drawing roulettes. Since a spirograph utilizes 2 different sized circles that can go whatever way you want, it can produce epitrochoids. It can also produce hypocycloids, hypotrochoids, and epicycloid.
Ptolemaic System
. In Ptolemy's concept for his view of the universe, the Earth was the center of the universe and everything else revolved around it. It was not until the Heliocentric theory by Copernicus that this theory became challenged. Today, modern technology has shown that the Heliocentric theory is correct. However, Ptolemy's astronomical system is still acknowledged today. The orbits in this system are epicycloids, a special form of epitrochoids. As you can see, the method above known as equant, by moving Earth off of the center of the circle and at the same time, having a planet rotating on its own orbit around the circumference of the circle.
How the Main Image Relates
The concept of epitrochoids is explained.
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
About the Creator of this Image
Albrecht Duerer is credited to being the first person to describe the epitrochoids in his book "Instruction in Measurement with Compass and Straight Edge" in 1525. He said they were spider lines. Albrecht Duerer himself is a painter, engraver, printmaker, mathematician, and theorist from Nuremberg, Germany who lived during the time of the Northern Renaissance.
This page was written by Constance Lee, Kevin Liu, and Kevin Yang. They are students at J. R. Masterman high school.
Related Links
Additional Resources
- http://mathworld.wolfram.com/Epitrochoid.html
- http://anna466-epitrochoid.freeservers.com/history_n_uses.html
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