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
Basic Description
The basic concept of this is: a circle that can't move and a circle that can. The circle that can't move is fixed onto a surface (like paper). Within the circle that can move is a line that will create a line once this circle is moving around the fixed circle. The exterior circle will not slip away from the big circle because they are attached together by a point where both circles will always touch. Think of it as a rabbit that runs around its hole (that's hopefully on the ground) holding a marker while staying in the same position(i.e. a rabbit with its left hand on its head,would keep its left hand on its head), but twirling. The rabbit and whole are connected and cannot leave each other when this occurs. 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
1: First, it should be known that this is plotted in accordance with a po [...]Deriving the Parametric
1: First, it should be known that this is plotted in accordance with a polar grid. So, let's just say that "m" will be the sum of the 2 radii from the 2 circles -> m=A+B
2: The equation for a circle with its center at the origin is always x^2+y^2=r^2, where r is the radius. Thus, for the equation of the larger circle that has the radius "A" and its center at the origin of a graph, it will be this. -> x^2+y^2=A^2
3: For the equation of a circle whose center is not on the origin, then the formula becomes (x-h)^2 + (y-k)^2 =r^2. Thus, the equation of the smaller circle (with radius "B"), its "X" value will have to be(x-m) in this case because to find the center of the smaller circle on the graph, you would subtract "m" (the sum of the 2 radii) from x. -> (x-m)^2 + y^2= B^2
4: Define point P on its distance away from the origin in coordinate form when t=0 -> P[0] = ( m-h , 0 )
5: Since the small circle (radius "B")revolves counter-clockwise around the the big circle -> P= m [cos(t), sin(t)] - h [cos( B), sin( B)]
6: Angle B needs to be expressed in terms of "t", the angle of the big circle. Thus when the smaller circle rolls on the big circle, it produces the same arc length as the big circle.-> arc BC = arc RC
7: Now, because s = r(theta), "s" being the arc length, "r" being the radius and "theta" is the central angle in radians. -> A t = B t[1]
8: Solve the equation for t[1] -> t[1]= (At/B)
9:From the diagram, it's seen that -> b = t[1]+ t
10:Now substitute "B" into the equation from Step #8, to show Angle B in terms of "t"-> b = (At/B)+t
11:Since m=A+B, you simplify to get-> b = mt/B
Parametric Equations for an epitrochoid:
Derivation from: http://msemac.redwoods.edu/~darnold/math50c/CalcProj/Sp99/Fran/epitrochoid.htm
Note: Due to some technical issues... "b" represents "beta"; "A" represents "a" and "B" represents "b" in the photos; [] represents subscript.
For all of these above equations: "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 determing the number of curves the epitrochoid will produce in one rotation around the circle.
Other Significant Equations
(These equations were found at http://mathworld.wolfram.com/Epitrochoid.html)
This is the equation for the velocity of the rotating circle:
This is for the acceleration of the rotating circle:
The Equation for the Arc Length:
This is the equation for the amount of curvature:
Special Epitrochoids
Limaçon: When the value of "a" is equivalent to the value of "b".
Epicycloid: When the value of "h" is equivalent to the value of "b".
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
Why It's Interesting
It creates interesting shapes and can be applied to uses of art and engineering. Two examples of usages of an epitrochoids include the Wankel Rotary Engine and the Spirograph.
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. 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. It can produce epitrochoids. It can also produce hypocycloids, hypotrochoids, and epicycloid.
How the Main Image Relates
The concept of epitrochoids is explained.
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Related Links
Additional Resources
http://mathworld.wolfram.com/Epitrochoid.html* http://anna466-epitrochoid.freeservers.com/history_n_uses.html
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