Involute
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Involute 

Involute
 An involute of a circle can be obtained by rolling a line around the circle in a special way.
Contents 
Basic Description
Imagine you have a string attached to a point on a fixed curve. Then, tautly wind the string onto the curve. The trace of the end point on the string gives an involute of the original curve, and the original curve is called the evolute of its involute.The animation on the right gives an example of an involute of a circle. We can see that a straight line is rotating around the center circle. This is like unwinding a string from a pole and keeping it taut the whole time. The trace of the end point of the line is the involute of the circle. In the image, it is the red spiral.
The involute is also the roulette of a selected point on a line that rolls (as a tangent) along a given curve. Notice that while the rolling object can be anything (point, line, curve) for roulettes, it has to be a line to roll out involutes. For more information about roulettes, you can refer to the roulette image page.
History of Involute
The involute was first introduced by Huygens in 1673 in his Horologium Oscillatorium sive de motu pendulorum, in which he focused on theories about pendulum motion.
Christiaan Huygens was a Dutch mathematician, astronomer and physicist. In 1656, he invented and built the world's first pendulum clock, which was the basic design for more accurate clocks in the following 300 years. Huygens had long realized that the period of a simple pendulum's oscillations was not constant, but rather depended on the magnitude of the movements. In other words, if the releasing height of the pendulum changes, the time for one oscillation changes as well. In search of improvements to the pendulum clocks, Huygens showed that the cycloid was the solution to this issue in 1659. He found that for a particle (subject only to gravity) to slide down a cycloidal curve without friction, the time it took to reach the lowest point of the curve was independent of its starting point. But how can we make sure the pendulum oscillates along a cycloid, instead of a circular arc? This was the point where the involute came in.
In Horologium Oscillatorium sive de motu pendulorum, Huygens proved that the involute of a cycloid is another cycloid (you will see an example in the More Mathematical Explanation section). To force the pendulum bob to swing along a cycloid, the string needed to "unwrap" from the evolute of the cycloid. As shown in the image on the right, he suspended the pendulum from the cusp point between two cycloidal semiarcs. As a result, the pendulum bob traveled along a cycloidal path that was exactly the same as the cycloid to which the semiarcs belonged. Thus, the time needed for the pendulum to complete swings was the same regardless of the swing amplitude. ^{[1]}
To Draw an Involute
The involute of a given curve can be approximately drawn following the instructions below (there will be an example afterwards):
 Draw a number of tangent lines to the given curve.
 Pick a pair of neighboring tangent lines and set their intersection as the center. Then, with an endpoint at the point where one of those tangent lines meets the curve, draw an arc bounded by the two tangent lines. We call the line whose tangent point to the curve is also on the arc L_{1}, the other tangent line L_{2}, and the newly constructed arc Arc1.
 Pick the neighboring tangent of L_{2} and call it L_{3}. Set the intersection of L_{2} and L_{3} as the center. Then draw an arc bounded by these two tangents, using a radius that will make this arc join Arc1. In other words, the radius would be the distance between the point where L_{2} meets L_{3} and the point where L_{2} meets Arc1.
 Repeat the step above for the rest of the tangents: pick the neighboring tangent, draw the arc; pick the next neighboring tangent, draw another arc...
Here is an illustration of the construction procedure above:
This method does not produce the accurate involute curve because, for each of the line segments between the original curve and the involute (e.g. BA_{1}, CA_{2}, DA_{3}...), instead of using the length of arc of the original curve as its length, we use the sum of the segments of the tangents. To be more concrete, the length of segment BA_{1} should equal the length of arc AB; the length of segment CA_{2} should equal the length of arc BC. This is because the line segment represents the part of the string that has just been unwound. It was originally wrapped along the fixed curve. As a result, its length should equal the length of the part of the original curve that it "covered" before. However, with the construction process described above, the lengths of these line segments are in fact sums of tangent segments. For example, the length of BA_{1} is found by adding XA and BX, which is shorter than it is supposed to be. Therefore, the distance between the involute and the original curve is not perfectly accurate. Fortunately, this error gets smaller as we construct more tangent lines that are closer to each other.
A More Mathematical Explanation
In this section, we will derive the general formula for involutes, [...]
In this section, we will derive the general formula for involutes, and provide various examples of involutes and their equations. We will also introduce some properties of involute curves.
General Formula for Involutes
This section derives the general equation for involutes of curves:
As an example, the equation of the involute of a circle is going to be derived immediately in the examples of involutes section.
Recall that we can think of the involute as the path of the end point of a string that is unwound from a fixed curve. So the tangent line segment we see between a point on the original curve and its corresponding point on the involute can be considered the part of the string that has just been unwound from the fixed curve. Therefore, the length of this line segment equals the distance traveled by the contact point between the unwound string and the fixed curve (the tangent point).
If we are given a curve, we construct its involute by unwinding a string tautly from the curve. For the point on the original curve that has Cartesian coordinates (f(t ), g(t )), its corresponding point on the involute (the endpoint of the string) can be found with this formula:
where is the position vector for a point on the original curve (where the unwound part of the string touches the curve), is the unit tangent vector to the original curve at this point (the current direction of the string), s is the distance traveled by this point so far (how much of the string has been unwound), and is the position vector for the corresponding point on the involute curve (the position of the end of the string). The image on the right takes a circle as an example and explains these variables visually. How each part is calculated is going to be explained below.
Note: For the following lines, we will use f to represent f(t ) so that the formulas look neater. Similarly, g(t ), f′(t ), g′(t ) are shortened to be g, f′ and g′ respectively.
Generally, the tangent vector for a curve with a position vector is defined as . If you are not very familiar with calculus, you can check these webpages to learn more about tangent vectors, derivatives, and velocity vector.
In the case we have, the tangent vector is
Thus, the unit tangent vector is:
The distance traveled by the contact point can be calculated by taking the integral of its speed. In the horizontal direction, its velocity is the derivative f′; in the vertical direction, its velocity is g′. Therefore, the velocity of the point written as a vector is (f′, g′) (We can see that its velocity is exactly the tangent vector). Its speed is:
Thus, the point has traveled a distance of
where f(a ) is the point where the involute and the curve intersect (the starting point of the unwinding process).
Therefore, if we write the vectors using Cartesian coordinates and plug in the results we get above for each term in the equation, we have
Hence, the parametric equation for the involute is
In the examples of involutes section, you will see the derivation of the equation for the involute of a circle as an example.
Examples of Involutes
The image on the left shows the Involute of a Circle. It resembles an Archimedean spiral.
We know that the parametric equation for a circle with radius a is: For any point on the circle, its position vector is Then its tangent vector (also the velocity of the point on the circle) is The unit tangent vector is The speed of the point on the circle is The distance traveled by the point on the circle from its starting point is calculated by taking the integral of its speed: Plugging these numbers into the general formula, which is , we get the position vector of the corresponding point on the involute: Therefore, the parametric equation for the involute of a circle with radius a is: We can go through the same procedure to get the equations for all the involute curves below, but we are not going to derive all these equations in this page.
 
The involute of a parabola looks like the images on the left. For example, if the parabola is its involute curve is  
On the other hand, if the parabola is
its involute curve is  
The involute of an astroid is another astroid that is half of its original size and rotated 1/8 of a turn. For example, if the parametric equation for the astroid is The involute of the astroid is  
The involute of a cycloid is a shifted copy of the original cycloid. If the cycloid is its involute curve is  
The involute of a cardioid is a mirrored, but bigger cardioid.
For example, the cardioid is given as Its involute is 
For more examples of involutes, you can visit WolframMathWorld  Involutes and Evolutes
Properties of Involutes

 Why are the various involutes of a given circle parallel to each other?
We can approach this question through string unwinding again.
 Since each involute of the circle is symmetrical, we can look at one side of the cusp first. That is, we unwind two strings from a circle following the same direction. The endpoints of the strings are initially at points A and B (see the image below on the left). After some unwinding, the line that starts at point A will pass point B. Then, it is easy to see that the involute curves are parallel as desired. Starting at point B, we can also think of the case as a long string (the one drawing out the red involute) and a short string (the one drawing out the blue involute) being unwound simultaneously from the circle, and the starting point for both of them is point B (see the image below on the right). Therefore, they are always a constant distance apart.
 For similar reasons, curves on the other side of the cusps (formed by unwinding the string in an opposite direction) are a constant distance apart as well. If we put the two parts of the involute curves together, we can see that all the involutes of a circle are parallel to each other.
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References
 ↑ Christiaan Huygens (n.d). Retrieved from http://www.robertnowlan.com/pdfs/Huygens,%20Christiaan.pdf
 ↑ ^{2.0} ^{2.1} Wikipedia (List of gear nomenclature). (n.d). List of gear nomenclature. Retrieved from http://en.wikipedia.org/wiki/List_of_gear_nomenclature
 ↑ Wikiversity (Gears). (n.d.). Gears. Retrieved from http://en.wikiversity.org/wiki/Gears
LockWood E.H.(1967). A Book of Curves. The Syndics of the Cambridge University Press.
Yates, Robert C.(1952). A Handbook on Curves and Their Properties. Edwards Brothers, Inc.
Wikipedia (Involute gear). (n.d.). Involute gear. Retrieved from http://en.wikipedia.org/wiki/Involute_gear
Wikipedia (Involute). (n.d.). Involute. Retrieved from http://en.wikipedia.org/wiki/Involute
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