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- This is a beautiful Lissajous Box. The Lissajous Curves on its sides have an angular frequency ratio of 10:7.
Lissajous Curves, or Lissajous Figures, are patterns formed when two harmonic vibrations along perpendicular lines are superimposed. For example, the following animation tells us how to generate a Lissajous Curve:
In the animation above, points X and Y are simple harmonic oscillators in x and y directions. They have the same magnitude of 10, but their angular frequencies are different. As we can see in the animation, the x - vibrator completes 3 cycles from the beginning to the end, while the y - vibrator completes only 2. In fact, these vibrators follow the equations of motion x = sin (3t ), and y = sin (2t ), respectively.
Now, we will try to get the superposition of these two vibrations, which is what we really care about. To get this superposition, we can draw from X a line perpendicular to x-axis, and from Y a line perpendicular to y-axis, and locate their intersection P. By simple geometry, P will have the same x-coordinate as X, and y-coordinate as Y, so it combines the motion of X and Y. As we can see in Figure 1, the trace of P turns out to be a complicated and beautiful curve, which we refer to as the "Lissajous Curve". More specifically, it's one Lissajous Curve in a big family, since we can easily generate more Lissajous Curves with other angular frequencies and phases using the same mechanism.
Mathematically speaking, since the motion of point P is consisted of two component vibrations, whose equations of motions are already known to us, we can easily get the parametric equations of P 's motion:
in which A and B are magnitudes of two harmonic vibrations, a and b are their angular frequencies, and φ is their phase difference. The term "phase difference" means that it's the difference between the two vibrations' initial phases.
The Lissajous Curve in Figure 1 has A = B = 10, a = 3, b = 2, and φ = 0. As we have stated before, we can get more Lissajous Curves by changing these parameters. The following images show some of these figures:
To see exactly how these parameters determine the appearance of Lissajous Curves, please go to the More Mathematical Explanation section.
A Dip Into the History
Lissajous Curves were named after French mathematician Jules Antoine Lissajous (1822–1880), who devised a simple optical method to study compound vibrations. Lissajous entered the Ecole Normale Superieure in 1841, and later became a professor of physics at the Lycee Saint-Louis in Paris, where he studied vibrations and sound.
During that age, people were enthusiastic about standardization in science. And the science of acoustics was no exception, since musicians and instrument makers were crying out for a standard in pitches. In response to their demand, Lissajous invented the Lissajous Tuning Forks, which turned out to be a great success since they not only allowed people to visualize and analyse sound vibrations, but also showed the beauty of math through interesting patterns.
The structure and usage of Lissajous Tuning Forks are shown in Figure 3-b. Each tuning fork is manufactured with a small piece of mirror attached to one prong, and a small metal ball attached to the other as counterweight. Two tuning forks like this are placed besides each other, oriented in perpendicular directions. A beam of light is bounced off the two mirrors in turn and directed to a screen. If we put a magnifying glass between the second tuning fork and the screen (to make the small deflections of light beam visible to human eyes), we can actually see Lissajous Curves forming on the screen.
The idea of visualizing sound vibrations may not be surprising nowadays, but it was ground-breakingly new in Lissajous' age. Moreover, as we are going to see in the More Mathematical Explanation section, the appearance of Lissajous Curves are extremely sensitive to the frequency ratio of tuning forks. The most stable and perfect patterns only appear when the two forks vibrate at frequencies of simple ratios, such as 2:1 or 3:2. These frequency ratios correspond to the musical intervals of the octave and perfect fifth, respectively. So, by observing the Lissajous Curve formed by an unadjusted fork and a standard fork of known frequency, people were able to make tuning adjustments far more accurately than tuning by ear.
Because of his contributions to acoustic science, Lissajous was honored as member of a musical science commission set up by the French Government in 1858, which also featured great composers such as Hector Berlioz (1803-1869) and Gioachino Rossini (1792-1868).
A More Mathematical Explanation
In previous sections, we have encountered this question for many times:
- What determines the appea [...]
In previous sections, we have encountered this question for many times:
- What determines the appearance of Lissajous Curves?
In this section, we are going to answer this question in two ways. The first method is simple and direct, but is limited to several special cases. The second one applies to almost all Lissajous Curves, but as a result it's more subtle and complicated.
First Method: Direct Elimination of t
Since Lissajous Curves are defined by the following parametric equations:
In principle, one can use trigonometric formula to eliminate t from these equations, and get a relationship between x and y. See the following examples:
(Note: in all examples below, we are going to assume that A = B = 1, since changing these magnitudes will only make the curves dilate or contract in horizontal or vertical direction. They don't affect the structure of Lissajous curves.)
Example 1: line segment
If in addition to A = B = 1, we have a = b = 1, and = 0, then the parametric equations will become:
from which we can easily get:
Moreover, since the range of is from -1 to 1, we have:
Together, they give us the line segment shown in Figure 3-a.
Example 2: circle
In this case, we still have a = b = 1. But instead of = 0, we change it to :
Using the trigonometric identity , we will get:
Using the trigonometric identity , we will get:
Which gives us the circle shown in Figure 3-b.
Example 3: parabola
This time, we change the parameters into a = 1, b = 1, and , then the parametric equations will become:
from Eq.1 we can get:
Using the trigonometric identity , we can get:
Apply the formula , we can get:
Combine it with Eq.2: , we can get:
with x confined between -1 and 1. This gives us the parabola in Figure 3-c.
Conclusion: pros and cons
In the examples above, we can clearly see some advantages of the direct elimination method: it's clear, accurate, and easy to understand. However, these advantages are quickly shadowed by the complexity of calculation when we get to larger frequency ratios. For example, see the following parametric equations of a Lissajous Curve:
In principle, this could be solved by expanding the x- and y- function into powers of and :
These equations come from a general formula given in 16th century by French mathematician Vieta. For more information, please go to this page.
Notice that in these equations, if we consider and as unknowns, then we will have a set of two polynomial equations with two unknowns, and in principle we can solve and in terms of x and y. Then, the identity will give us a direct relationship between x and y. However, in practice, few people are willing to carry on with the algebra, because the calculations involved are just so cumbersome and annoying. To make things worse, as group theory tells us, not all polynomial equations of powers higher than 5 can be solved with exact expression of roots . So there is no guarantee that our effort will lead us to the answer. Even if they can, the relationship between x and y is going to be too complicated to tell us anything useful about the shape of the curve. So the method of elimination fails here, and we would like a new way to study these curves.
Second Method: Experiment and Observation
As shown in the previous discussion, the attempt to directly solve Lissajous Curves failed when we try to deal with large angular frequencies, so we have to find another way to study them. One such way is to study them through experiment and observation. That is, we can use computer software to draw some Lissajous Curves with different parameters, and see how they affect the appearance of Lissajous Curves.
When we study something with multiple variable parameters, it's much easier to study these parameters separately, rather than together. There are three variable parameters, a, b, and, in Lissajous Curves. So in the rest of this section we will fix the phase difference to study the angular frequencies a and b, and then fix the angular frequencies to study the phase difference.
Study a and b with fixed
The following table shows the Lissajous Curves:
with angular frequencies a and b varying from 1 to 5, and phase difference fixed at 0:
There are many interesting properties associated with this table:
- 1. All Lissajous Curves in the table are confined in a 2 * 2 square box. The curves can touch, but cannot go beyond, the lines x = 1, x = –1, y = 1, and y = –1, because the amplitudes of both horizontal and vertical vibrations are set to 1.
From these properties, we can see that many Lissajous Curves with different angular frequencies are actually the same thing, and we do not need to study all of them. In fact, we can use the following family to represent all Lissajous Curves with rational frequency ratios:
in which r is a rational number standing for angular frequency ratio. The argument for this goes as following: for any Lissajous Curve
in which a and b are integers, we can assume that , since if we can exchange their values, and according to property #3 the curve will only be flipped about line y = x. This doesn't affect the curve's structure, which is what we really care about.
The next step is to divide both angular frequencies by b. According to property #2, the curve will not change, and we will get:
The last set of parametric equations belongs to the family we mentioned above. So we only need to study this family of Lissajous Curves, since others can all be reduced to this case.
The following animation shows some of the Lissajous Curves in this family, with the frequency ratio a / b varying continuously from 0 to 1:
Surprisingly, as we can see in the animation, most of these Lissajous Curves are rather convoluted. But there are some simple and beautiful ones scattered in them. A more careful examination shows that, when these simple patterns occur, the frequency ratio must be equal to a simple fraction. This phenomenon is not hard to understand if we look at the generation of Lissajous Curves once again. Suppose the two component vibrations start at t = t0. As long as the frequency ratio is rational, the moving point will eventually return to its starting place, and make a closed Lissajous Curve. Suppose this happens at t = t1. So the time period between t0 and t1 is a complete cycle of this Lissajous Curve. Moreover, since the starting point and ending point overlap, we must have:
Substitute into the parametric equations of Lissajous Curve with rational frequency ratio, we can get:
which leads to:
in which k1 and k2 are integers. The other possibility is omitted because they represent the intersections inside one cycle. At these intersections, although the positions overlap, the velocities don't. So the Lissajous Curve is not closed at these points.
Substitute Eq.2 into Eq.1, we get:
since a / b is assumed to be an irreducible fraction (if not, we can divide them by their common factor without changing the Lissajous Curve), the smallest k1 and k2 that satisfy this equation are k1 = a and k2 = b. Substitute back into Eq.2, we can get:
So the larger b is, the longer it's going to take before the Lissajous Curve closes and repeats itself, and the more convoluted it's going to be. For a simple angular frequency ratio like 1/2, the vibrations soon start to repeat, and the Lissajous Curve is simple, as shown in the previous table. However, a ratio like 37/335 will make the curve much more complicated. In an extreme case, if the ratio is irrational, then both a and b will be infinitely large, and the Curve is no longer closed. This special case is treated later in this section. Click here to see.
In conclusion, the angular frequency ratio a / b, reduced to simplest fraction, determines the complexity of Lissajous Curves. Large a and b lead to complicated Lissajous Curves; small a and b give us simple ones. This is why Lissajous Tuning Forks are so suitable for tuning notes. In music theory, most of the important intervals are simple fractions. For example, the interval of a perfect octave is 1:2, perfect fifth is 2:3, perfect fourth is 3:4, and so on. These intervals all correspond to simple Lissajous Curves with distinctive features.
Study with a and b fixed
In the previous subsection we have figured out how angular frequencies of a Lissajous Curve affect its appearance. Now we are going to fix the angular frequencies to study the third, and last, variable parameter: the phase difference .
The following animation shows the Lissajous Curve
with varying continuously from to :
An interesting fact to notice is that, the animation above looks more like a rotating 3-D curve, rather than a changing 2-D one. The reason for this illusion is related to another way to define Lissajous Curves. In the beginning of this page, we introduced the following definition:
- Lissajous Curve is the superposition of two harmonic vibrations.
However, this is not the only definition for Lissajous Curves. These curves can also be viewed as the projection of a 3-D harmonic height function over a circular base. The following set of images explain this definition in more detail:
The first step to generate this harmonic height function is to draw a circular base in x-y plane, as shown in Figure 5-b. The parametric equation of this circular base is:
The variable parameter here doesn't change the shape of the circle, as we still have the relationship . But if we change the value of , then the circle will rotate about the origin O. Of course we can't see the motion here, because O is also the center of the circle. However, this rotation is going to make a difference later.
In the next step, we raise (or lower) each point in the circular base to a certain height. This height is determined by the function:
The raising process is shown in Figure 5-c. Note that if we change now, the rotation is visible, since the curve's rotational symmetry is broken in the raising process.
Finally, if we make the projection of that rotating height curve onto the y-z plane, as shown in Figure 5-d, we can see that it's exactly same as the animation in Figure 5-a. In other words, this Lissajous Curve can be viewed as the projection of this 3-D height function. Changing the value of makes the 3-D curve to rotate, and in turn changes the 2-D curve. In fact, this is why we had the 3-D illusion in Figure 5-a.
Algebraic analysis agrees with this result. As we have seen, the parametric equations of this harmonic height function are:
To project it onto the y-z plane, we can fix its x component to be 0:
Compare this projection to the Lissajous Curve we had before:
we can see that they are indeed the same thing.
Although we used a special case a = 1, b = 3 in our discussion, the result applies to all Lissajous Curves with rational frequency ratios. The following images show the 3-D height function of some other Lissajous Curves:
To put it all together: a java applet
So far we have talked much about the appearance of Lissajous Curves. We know that some simple cases can be solve by direct elimination of t in the parametric equations, that the frequency ratio of a Lissajous Curve determines its complexity, and that the phase difference affects a Lissajous Curve by rotating its corresponding 3-D height function. Here is an interactive java applet that puts these all together. It allows the user the change both angular frequencies from 1 to 9, and animate the curve by changing :
What happens when things get irrational?
We have limited previous discussions to Lissajous Curves with rational frequency ratios. So, one may naturally wonder, what happened to all those with irrational frequency ratios? Well, they have all died painfully because of their irrationality ...
Just kidding. They are still there, waiting for us to study. For example, see the following Lissajous Curve:
It's a known fact that is irrational. So the frequency ratio here is also irrational, and this curve is going to be radically different from any one we have encountered so far. See the animation below to get a sense of what it looks like:
Figure 6 shows the trace this Lissajous Curve in accelerating motion. In the beginning, it looks just like an ordinary Lissajous Curve. However, soon we can see the difference: this curve is never closed! It kept going on and on, and eventually fills the whole 2*2 box. In fact, it can be shown that all Lissajous Curves with irrational frequency ratios cannot close. Without loss of generality, let the parametric equation of such a Lissajous Curve be:
in which r is an irrational number.
Now we suppose that the curve is closed, and try to derive a contradiction. Let the starting time be 't0 and the closing time be t1 as before, so we must have:
which gives us:
which leads to:
in which p and q are integers. The other possibility is omitted due to the same reason discussed before.
Substitute Eq.2 into Eq.1, we can get:
However, recall that we assumed r to be irrational, which means it cannot be written as an integer fraction . So the equation above cannot be true, and this Lissajous Curve is never closed.
Why It's Interesting
As a family of beautiful figures, Lissajous Curves are themselves an interesting subject to study. Moreover, they also have some practical applications, including oscilloscopes and harmonographs.
Application to Oscilloscopes
Oscilloscope is a type of electronic instrument in physics that allows observation of constantly varying signal voltages. The following image shows the simplified structure of a typical Cathode Ray Oscilloscope:
In Figure 7-1, the election gun at left generates a beam of electrons when heated, which is then directed through a deflecting system. The deflecting system is made of two sets of parallel metal plates, one for deflection in x - direction, and the other for y - direction. A signal voltage applied to the X-plates gives them an electronic potential difference, generates a uniform electronic field between them, and makes the electron beam to deflect in x - direction. Same for the y - plates. The angle of deflection is proportional to the voltage applied.
After passing the deflecting system, the electron beam is then directed to a screen, which is covered by fluorescent material so that we can see green light on the places hit by electrons. If there is no voltage applied to the deflecting system, then the electron beam hits the screen right at the center. If there is voltage applied, then the electrons will hit somewhere else. So the oscilloscope makes signal voltages visible to us.
Now if we apply a sinusoidal signal on each set of the plates, then both the X-plates and the Y-plates will have varying electronic fields between them, and the electron beam will oscillate in both directions. As a result, the trace on the screen should be the superposition of these two oscillations. As we have discussed before, this is a Lissajous Curve. The following images show some of the Lissajous Curves achieved on oscilloscopes:
Similar to Lissajous Tuning Forks, Lissajous Figures on the oscilloscope can give us some information about the two component vibrations. For example, just by looking at the Lissajous Curve in Figure 7-b, experienced observers can tell that the frequency ratio between its two component vibrations is 1:3, and the phase difference is . Engineers and physicists often use this method to analyze signals and waves.
Application to Harmonographs
A harmonograph is a mechanical apparatus that employs pendulums to create geometric images. The drawings created are typically Lissajous curves, or related drawings of greater complexity. See the following video the get a sense of how it works:
As we can see in the video, a typical three pendulum rotary harmonograph is consisted of a table, a drawing board, a pen, and 3 pendulums. Two of them are linear pendulums oriented in perpendicular directions, and they control the motion of the pen. The third pendulum is free to swing in both directions, and it's connected to the drawing board.
Harmonographs can be used to draw Lissajous Curves. We only need to fix the pendulum connecting to the drawing board, and assume that there is no friction in the other two pendulums. In mechanics, it is a known fact that motion of a frictionless pendulum can be viewed as simple harmonic motion, provided that the swinging angle is small. So the pen's motion is the superposition of two perpendicular harmonic vibrations, which is a Lissajous Curve by definition.
However, in practice, friction cannot be completely eliminated. So the two linear pendulums are actually doing damped oscillations, rather than simple harmonic motion. Physicists give us the following equation of motion for damped harmonic oscillations:
in which is called the damping constant. The larger is, the more heavily this oscillator is damped, and the faster its magnitude decreases.
Since both linear pendulums are doing damped harmonic oscillations, the pen should have the following equation of motion:
If , then the common factor can be extracted from the equations above, and we are left with the parametric equations of a Lissajous Curve:
which gives us a Lissajous Curve with exponentially decreasing magnitudes. For example, see the following computer simulations of the harmonograph with A = B =10, = 3, = 2, and:
Figure 8-a shows the damped case with = 0.04, and Figure 8-b shows the corresponding frictionless motion with = 0. One can clearly see that they have similar shapes, except the magnitude of the curve in Figure 8-a decreases a little bit after each cycle, which is exactly what we mean by damping.
If , then things get more complicated, because the shape of the curve is distorted during the damping process. For example, see the following images:
The three curves above all have A = B =10, = = 1, and . Figure 8-c shows the frictionless case, which is a Lissajous Curve we have discussed before. Figure 8-d shows the damped case with equal damping constants, and one can see that the motion decreases uniformly in both directions. Figure 5-e shows the damped case with unequal damping constants. Since the motion in y - direction is damped much more heavily than in x - direction, the shape of this curve is distorted towards x - axis during the damping process.
If we add more complexity by releasing the free pendulum connecting to the drawing board, then the curve will be the superposition of all these motions. We are not going to study the math behind, since it's way too complicated with more than 10 variable parameters. However, as we have seen in the video, more complexity also gives us more beautiful and interesting images. The following images are works created by harmonographs. Some of them are computer simulations, others are real pictures from harmonograph makers:
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- ↑ Jules Antoine Lissajous, from Wikipedia. This is a biography of Jules Lissajous, discoverer of Lissajous Curves.
- ↑ Lissajous tuning forks: the standardization of musical sound from Wipple Collections. This is a brief introduction to Lissajous' Tuning Forks and his contribution in acoustic science.
- ↑ List of Trigonometric identities, from Wikipedia. This page lists some of the trigonometric formula we used the derive the shape of Lissajous curves.
- ↑ Polynomial, from Wikipedia. This briefly explains why we can't find a general solution for equations of powers higher than 5.
- ↑ Interval (music), from Wikipedia. This article explains more about musical notes and their frequency intervals.
- ↑ Animated Lissajous figures.This is the source of the embedded java applet.
- ↑ Three Pendulum Harmonograph, from youtube. This is the source of the embedded video.
- ↑ Pendulum, from Wikipedia. This article explains the physics behind pendulums.
- ↑ Damping, from Wikipedia. This article explains how we derive the equation of motion for damped oscillators.
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