- An involute of a circle can be obtained by rolling a line around the circle in a special way.
Imagine you have a string attached to a point on a fixed curve. Then, tautly wind the string onto (or unwind from) 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.
Involute is also considered as 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. He invented and built the world's first pendulum clock in 1956, which was the basic design for more accurate clocks in the following 300 years. Huygens had long realized that the period of 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 involute comes 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 semi-arcs. As a result, the pendulum bob traveled along a cycloidal path that was exactly the same as the cycloid to which the semi-arcs belonged. Thus, the time needed for the pendulum to complete swings was the same regardless of the swing amplitude. 
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 L1, the other tangent line L2, and the newly constructed arc Arc1.
- Pick the neighboring tangent of L2 and we call it L3. Set the intersection of L2 and L3 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 L2 meets the curve and the point where L2 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 segment between the original curve and involute (e.g. BA1, CA2, DA3...), 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 BA1 should equal the length of arc AB; the length of segment CA2 should equal the length of arc BC. This is because thee 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 BA1 is found by adding XA and BX, which is shorter than they are 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, provide various examples of involu [...]
In this section, we will derive the general formula for involutes, provide various examples of involutes and their equations. We will also introduce some properties of involute curves.
General Formula for Involutes
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. The tangent line segment between a point on the original curve, and its corresponding point on the involute can be considered as part of the string that has been unwound. This part was originally wrapped around the fixed curve. Therefore, the length of this line segment equals the distance traveled by the contact point (the tangent point) between the string and the fixed curve.
|If a point on the original curve has Cartesian coordinates (f(t ), g(t )), the corresponding point on the involute is:
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.
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.
The distance traveled by the 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′). Its speed is:
Thus, the point has traveled a distance of
where f(a ) is the point where involute and the curve intersect (the starting point of the unwinding process).
Generally, the tangent vector for a curve with a position vector is defined as . For more information about tangent vector, you can go [here].
In the case we have, the tangent vector is ,The unit tangent vector is:
Therefore, if we write the vectors using Cartesian coordinates, we have
Hence, the parametric equation for the involute is
Examples of Involutes
The image on the left shows the involute of a circle. It resembles an Archimedean spiral.
|The involute of a parabola looks like the image on the left.|
For example, if the 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 astorid 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 It's Interesting
|One of the most commonly used gearing system today is the involute gear. The image on the right is an example of such a gear.
In an involute gear system, it is desired that the two wheels should revolve as if the two pitch-circles were rolling against each other. This effect can be achieved if the teeth profiles are drawn as involutes of the base-circles and the tops of the teeth are arcs of circles that are concentric with, and bigger than the pitch circles.
Due to how involute is constructed (unwinding the string), we know that
Because Q′ and R′ move the same distance in the same time interval, we can conclude that Q and R move with equal velocities. Therefore, points on the pitch-circles will also move with same velocities.
|In the animation to the right, we can see that each pair of gear teeth has an instant contact point, called the pitch-point, and it moves along one single line as the gears rotate. This line is called the line of action and it is the common internal tangent of the two circles. In other words, the involutes of the two circles are always tangent to each other at a point on their common internal tangent.
Because of this design, a constant velocity ratio is transmitted and the fundamental law of gearing is satisfied. Also, having all the contact points on a single straight line results in a constant force and pressure, while for gear teeth of other shapes, the relative speeds and forces change as teeth engage, resulting in vibration, noise, and excessive wear. Lastly, involute gear has the advantage that it is easy to manufacture since all the teeth are uniform.
- There are currently no teaching materials for this page. Add teaching materials.
- Christiaan Huygens (n.d). Retrieved from http://www.robertnowlan.com/pdfs/Huygens,%20Christiaan.pdf
- 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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