# Involute

Involute
Field: Geometry
Image Created By: Chengying Wang

Involute

An involute of a circle can be obtained by rolling a line around the circle in a special way.

# Basic Description

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 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 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. Searching for improvements to the pendulum clocks, Huygens showed that the cycloid is the solution to this tautochrone problem in 1659. The time it takes for a particle (subject only to gravity) to slide down a cycloidal curve and reach the lowest point is independent of its starting point. But how can we make sure the pendulum oscillates along a cycloid, instead of a circular arc? This is 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). He found that to force the pendulum bob 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 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. [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. 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.

Here is an illustration of the construction procedure above:

 1. The black curve that point A, B, C, D, E, F lie on is the original curve we have. Draw the tangent line at each of the six points respectively. 2. 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 $\overset{\frown}{AA_1}$. 3. 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 $\overset{\frown}{A_1A_2}$ so that A2 lies on the tangent of point C. 4. 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. In the animation, the length of the rotating line segment is equal to the distance traveled by the point on the circle. This is easier to understand if we imagine we are unwinding a string. The line segment we see is the part of the string that has just been unwound from the circle. As a result, it should be as long as the distance traveled by the point of contact between the string and the circle. This property holds true for all other types of curves as well, as we shall see. The parametric equation for the involute of a circle with radius a is: $x = a(\cos t + t \cdot \sin t)$ $y = a(\sin t - t \cdot \cos t)$ The involute of a parabola looks like the image on the left. For example, if the the parabola is $x = t$ $y = t^2$ its involute curve is $x = \frac{1}{4} (2t - \frac{\operatorname{arcsinh}\ 2t}{\sqrt{1 + 4t^2}})$ $y = -\frac{t \operatorname{arcsinh}\ 2t}{2\sqrt{1+4t^2}}$ On the other hand, if the parabola is $x = t^2$ $y = t$ its involute curve is $x = -\frac{t \operatorname{arcsinh}\ 2t}{2\sqrt{1+4t^2}}$ $y = \frac{1}{4} (2t - \frac{\operatorname{arcsinh}\ 2t}{\sqrt{1 + 4t^2}})$ 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 $x = \cos ^3 t$ $y = \sin ^3 t$ The involute of the astorid is $x = -\frac{1}{4}\cos t (-2 + \cos 2t )$ $y = \frac{1}{4} (2 + \cos 2t ) \sin t$ The involute of a cycloid is a shifted copy of the original cycloid. If the cycloid is $x = t - \sin t$ $y = 1 - \cos t$ its involute curve is $x = t + \sin t$ $y = 3 + \cos t$ The involute of a cardioid is a mirrored, but bigger cardioid. For example, the cardioid is given as $x = (1 + \cos t) \cos t$ $y = (1 + \cos t) \sin t$ Its involute is $x = \frac{1}{2} (1 + 6\cos t - 3\cos 2t)$ $y = 6 \sin \left (\frac{t}{2} \right )^2 \sin t$

For more examples of involutes, you can visit WolframMathWorld -- Involutes and Evolutes

## Properties of Involutes

 The normal of the involute is tangent to the original curve. From the procedure of drawing an involute, we know that the involute curve is joined by arcs that are bounded by pairs of tangent lines of the original curve and centered at their intersections. Therefore, the pairs of neighboring tangents can be considered the radii of the circle which the arc between them belongs to. If we draw the involute curve accurately, we can know for sure that at each point on the involute, its normal is tangent to the original curve. Alternatively, we can imagine we are unwinding a string from a fixed curve. Then, we know that the string is always tangent to the fixed curve. Because the string is held taut, the string is perpendicular to the path of its end point. Therefore, we can conclude that the normal of the involute is tangent to the original curve. We can also think of involute as a curve orthogonal to all the tangents to a given curve. A parallel of an involute is also an involute and any two involutes are parallel. Every curve has many involutes because the initial point, where the involute intersects the original curve, can be chosen arbitrarily. The various involute curves obtained by choosing different initial points are parallel to each other. In other words, any two of them are a constant distance apart (this distance can be measured along their common normal). You might realize that not having a unique starting point for the line to roll along the original curve may result in involute curves to intersect each other (as found in the image on the right). However, this does not contradict the parallel property. For example, we have a circle and we unwind two strings from the circle. The string are attached to the circle at point A and B respectively. After some unwinding, the line that starts at point A will pass point B. From this point, the involute curves would be parallel as desired, because we can think of the case as a long string and a short string being unwound simultaneously from the circle. The starting point for both of them are point B . Involutes of a circle.

## 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. Therefore, the length of the part that has been unwound equals the distance traveled by the contact 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: $\vec r_{inv} = \vec r_{cur} - s\vec T$ where $\vec r_{cur}$ is the position vector for a point on the original curve (where the unwound part of the string touches the curve), $\vec T$ 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 $\vec r_{inv}$ 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 is: $s = \int_a^t\,ds = \int_a^t \frac{ds}{dt}\,dt = \int_a^t\sqrt{f'^2+g'^2}\,dt$ where f(a ) is the point where involute and the curve intersect. The unit tangent vector is: $\vec T = \frac{\frac{d\vec r_{cur}}{dt}}{\left\vert\frac{d\vec r_{cur}}{dt}\right\vert}=\frac{(f',g')}{\sqrt{f'^2+g'^2}}$ Therefore, if we write the vectors using Cartesian coordinates, we have $(x,y) = (f,g) - \int_a^t\sqrt{f'^2+g'^2}\,dt \cdot \frac{(f',g')}{\sqrt{f'^2+g'^2}} = (f,g)-\frac{s(f',g')}{\sqrt{f'^2+g'^2}}$ Hence, the parametric equation for the involute is $x = f - \frac{sf'}{\sqrt{f'^2+g'^2}}$ $y = g - \frac{sg'}{\sqrt{f'^2+g'^2}}$

# 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. To see why involute gear can revolve as if the pitch-circles are rolling against each other, consider two points, Q′ and R′, which will rotate to points Q and R in the same time interval. If Q′Y and R′Z are the tangent lines to the base-circles at Q′ and R′, respectively, then $Q'Y + ZR' = QP + PR$ Due to how involute is constructed (unwinding the string), we know that $Q'Y = \overset{\frown}{Q'Q} + QP$ and $ZR' = PR - \overset{\frown}{R'R}$ Therefore, $\overset{\frown}{Q'Q} = \overset{\frown}{R'R}$ 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[2] 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.

# References

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.