- 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 a point on the string gives an involute of the original curve, and the original curve is called the evolute of its involute. The image on the right shows an involute of a circle.
In other words, an involute is the roulette of a selected point on a line that rolls (as a tangent) along a given curve.
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 base design for more accurate clocks in the following 300 years. Huygens realized that the oscillations of simple pendulum did not have a constant period. If the magnitude of the pendulum's swing changes, the time for one oscillation changes as well. In 1659, Huygens showed that the cycloid is the solution to this problem. The time it takes for a particle (subject only to gravity) to slide down a cycloid curve and reach the lowest point is independent of its starting point. But how can we make sure the pendulum oscillate along a cycloid, instead of a circular arc? This is the point that 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). He found that to make the pendulum bob swing along a cycloid, the string needed to "unwrap" from the evolute of the cycloid. Therefore, he suspended the pendulum from the cusp point between two cycloidal semi-arcs. (add image here) As a result, the pendulum bob travels along a cycloidal path that's exactly the same as the cycloid to which the semi-arcs belong. Thus, the time needed for the pendulum to complete swings is 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. With the center at their intersection, draw an arc bounded by the two tangent lines, passing through one of the points of contact between the tangent line and the curve.
- Pick the next pair of neighboring tangents. Set their intersection as the center. Then draw an arc bounded by the two tangents, using a radius that will make the arcs join. Repeat this step for all pairs of neighboring tangents.
|The image on the right is an illustration of the construction procedure above. The black curve that point A, B, C, D, E, F, G lie on is the original curve we have.
First, draw the tangent line at each of the seven points respectively.
We pick the tangents at point A and B as the first pair of neighbors. They intersect at point X. Then, set point as the center, draw an arc that passes point A and is bounded by the two tangents. We now have .
Pick the next pair of neighboring tangents, namely the tangent lines at point B and C. Their intersection is point Y. Set Y as the center, draw so that A2 lies on the tangent of point C.
Repeat the steps above for the remaining pairs of tangents. Then we get the red curve, which is an involute of the black curve.
This method does not produce the accurate involute curve because, instead of using the the length of arc of the original curve, we use the sum of the segments of the tangents. Therefore, the distance between the involute and the original curve is not perfectly accurate.However, this error gets smaller as we construct more tangent lines that are near enough together.
A More Mathematical Explanation
In this section, we provide various examples of involutes and their equations. We will also introduce [...]
In this section, we provide various examples of involutes and their equations. We will also introduce some properties of involute curves and derive the general formula.
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 images on the left.|
For example, if the the parabola is
its involute curve is
|On the other hand, if the 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
General Formula for 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.
Things I Plan to Change
- Currently searching for more history so that there could be a section about the discovery/development of involutes.
- Make the illustration of how to draw an involute into an animation
- I posted some questions on the discussion page. Team, please take a look.
- 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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