Envelope
From Math Images
Blueaerialshell 

Blueaerialshell
 This is a beautiful blueaerialshell firework filling the sky. Each particle of the firework follows a parabolic trajectory, and together they sweep a parabolic area.
Contents 
Basic Description
In geometry, an envelope of a family of curves is the boundary of the area "swept" by the curve when we change the variable parameter in the curve function.
For example, suppose there is a ladder leaning on a wall. The ladder starts to slide down because someone steps on it. What will be the shape of the area swept by the ladder before he hits the ground?


Surprisingly, the area swept by a moving straight line does not have a straight boundary (see Figure 11). The envelope is the firstquadrant portion of an astroid . If we complete the envelope by letting the ladder sweep across the other 3 quadrants, we can see that it shapes like a star, which is why we call it an astroid (See Figure 12).
A gallery of beautiful envelopes
Envelope of lines
We can get more interesting envelopes if we play with the geometry:


In Figure 21, point A lies on line l, M is midpoint of OA, and line m is perpendicular to OA.
If we slide point A along line l, line m will sweep out a parabola, with point O as its focus and line l as its directrix.


Repeat the process in Figure 21,except point A now slides on a circle.
The result is an ellipse, with O and B as its focuses.


Repeat the process in Figure 23,except point B is outside the circle.
The result is a hyperbola, with O and B as its focuses.
So far we have got all of the three Conic Section Curves as envelopes of straight lines. However, the envelopes are in no ways restricted to straight lines. Circle, Ellipse, and other curves can make fantastic envelopes as well.
Envelope of circles
For circles, we can often make beautiful envelopes by sliding its center along a chosen curve.


In Figure 31, Points A and B are on circle O. The sweeping circle is centered at A, and passes B.
If we slide point A along the bold green circle O, we will get a cardioid as the envelope of the sweeping circle.
Cardioid got its name because it shapes like a heart. For more information, please go to this page.


In Figure 33, l is a vertical line that passes the center of circle O. Circle A has its center on Circle O and is tangent to l at B
If we slide point A along the bold green Circle O, we will get a nephroid as the envelope of the sweeping circle.
Nephroid is one of the Roulettes. The word "nephroid" means kidneyshaped.


In Figure 33, F1 and F2 are foci of the blue hyperbola. A is midpoint of segment F1F2. The sweeping circle O has its center on the hyperbola, and passes A.
If we slide O along the hyperbola, we will get a Lemniscate as the envelope of the sweeping circle.
Lemniscate is an eightshaped curve discovered by Jacob Bernoulli[1]. It has the polar equation of the form


Note that if in Figure 35, instead of having A as midpoint of foci segment, we move it to an arbitrary position inside the hyperbola, then we will get a variation of "lemniscate", which has a funny shape like a bunny's ears.
More complicated envelopes
The following envelopes have more complicated mechanism than previous ones. But they are also more interesting.
1.Astroid again, but this time using ellipses
Recall that in Figure 11, we showed how to construct an astroid using a line segment sliding on coordinate axes. Actually there is another way to generate the same astroid: using a family of ellipses.
For the math behind this, please go to the More Mathematical Explanation section.
2.Deltoid as envelope of WallaceSimson lines
The WallaceSimson line is related to a simple theorem in geometry proposed by William Wallace [2] in 1796. The theorem itself has nothing special, but with a little manipulation we can get a gorgeous envelope out of it.
Notice that deltoid, like cardioid and nephroid, belongs to the Roulette Family.
This envelope was firstly discovered and proved by Swiss mathematician Jakob Steiner. For the math behind this please go here [4].
A More Mathematical Explanation
 Note: understanding of this explanation requires: *Calculus
At the beginning of this page, we gave the following definition:
 An envelope of a family of curves is the boundary of the area swept by these curves when we change the variable parameter t.
However, if we want a more mathematical explanation of envelope, we have to redefine it in a more mathematical way, because two serious problems arise with the old definition when we dive into more math:
First, not all 2D curves can be represented as a single function y = f(x) with a variable parameter t. For example, a circle can not be expressed unless we use two functions, and . So we have to introduce new ways to describe curves.
Second, "boundary of the area swept by these curves" is a rough description in everyday language. It's not something that we can use to derive mathematical formula and equations. So we have to be clear about we mean by "sweep", "boundary", and so on.
In the rest of this section, we are going to deal with these two problems one by one, and show how can we get a good mathematical explanation of envelopes using the new definition.
Resolve the first problem: the power of level sets
The first problem is easily resolved if we describe 2D curves using level sets, rather than images of onevariable functions y = f(x).
In Multivariable Calculus, the level set of a twovariable function F(x,y) at height C is defined as the set of points (x,y) that satisfy the condition F(x,y) = C. For example, instead of writing y = 2x  1 for a line, we could write 2x  y = 1, in which F(x,y) = 2x  y and C = 1.
In general, level sets are more powerful than images of single functions when we need to describe 2D curves, since all single variable functions can be written in the level set form , but the converse is not true. Some of the level sets can not be rewritten as y = f(x) unless we use multiple functions (see the circle in Figure 61), others can't be reduced at all (see the level set curve in Figure 62).


Because of these advantages of level sets, in the rest of this section we will use , rather than , to describe a family of curves. At least for the purpose of envelopes, level set is sufficient to describe all 2D curves that we care about.
Resolve the second problem: the boundary condition
The next question is, given a family of level set curves F(x,y,t) = C with variable parameter t, how can we find its boundary and express it in mathematical language?
The answer is given by the boundary condition, which states that:
 For a family of level set curves F(x,y,t) = C with variable parameter t, it's envelope, or boundary of sweeping area, must satisfy the condition
To see why this is true, let's look more carefully at the ladder problem. The following two images are different phases of a ladder captured in the sliding process.


In Figures 63 and 64, different phases of a sliding ladder are indicated using different colors. One may immediately notice that if we connect the intersections of different phases, we will get a curve close to the astroid envelope. It is not exactly the envelope because the time intervals between different captured phases are finite. However, if we make the time intervals shorter, the curve connecting them will get closer and closer to the envelope.
This is not mere coincidence. Same thing happens in the elliptic envelope of astroid (see Figure 65) . Actually it can be shown that if we choose two phases with infinitely small time interval, then their intersection must lie on the envelope. The argument goes as following:
In Figure 65, it's obvious that in one phase, the only segment that contributes to the envelope is between its intersections with the previous and next phase. For example, in the green ellipse, which has variable parameter t = 0.40, the segment above point A is covered by the orange envelope, thus not contributing to the envelope. Similarly the segment below point B does not contribute since it's covered by the blue ellipse. So the envelope is actually consisted of many "chunks" of segments that's not covered by other phases.
Now consider what happens if we increase the number of captures, and shorten each time interval:
 Two neighboring phases will approximate each other.
 Two intersections A and B will be closer.
 Segment AB will be shorter and smoother.
Eventually, when the number of captures goes to infinity, points A and B will meet together, and segment AB will be "squeezed" into one point on the envelope (call this point P). Recall that P is the intersection of two neighboring phases:
 and
which gives us:
This is the boundary condition we are trying to prove. The same argument goes for every other envelope.
One may also notice that, since the envelope only touches each curve at this single "squeezed" point, it is tangent to every curve in the family. In fact, envelope can also be defined as a curve that is tangent to every one of a family of curves.
Conclusion and Application
Now we have the family of level set curves:
and the boundary condition:
Since every point on the envelope must satisfy both equations, we can combine them to solve for a 2D envelope curve. However, the calculation involved is rather long and complicated, so here I will only prove the envelope of a sliding ladder is an Astroid.
Proof for the Astroid envelope
See Figure 66, the length of the ladder is a. For simplicity we will only consider the envelope in first quadrant.
Choose xcoordinate of point A as the variable parameter t. So ycoordinate of point B is
Thus the equation of line AB, our sweeping curve, is:
Differentiate Eq.1 to get the boundary condition:
From which we can get:
Substitute Eq.1 into Eq.2 twice, first substitute x, and then y, we get:
Substitute back into Eq.1, we get:
Which leads to:
 , the equation of an Astroid.
Other proofs are similar.
Why It's Interesting
Although envelope looks like a pure math concept, it does have some interesting applications in various areas, such as Microeconomics, Applied Physics, and String Arts:
Application in Microeconomics: the Envelope Theorem
Economists often deal with maximization or minimization problems: to maximize benefit, minimize cost, maximize social revenue, and so on. However, the problem is that there are so many variable parameters in economics. How many men should I hire? How much land should I buy or rent? Should I invest more money to buy new machines, or should I just do with old ones? Because of all these variable parameters, economists often end up doing maximization or minimization of a family of curves, rather than a single curve (see Figure 71)
So here comes the Envelope Theorem. It allows economists to find the envelope of a family of curves first, and then determine the maximum or minimum value on the envelope. Since no points go beyond the envelope, this point must be the absolute extrumum among the whole family of curves
For more about the envelope theorem, please go here.
Application in Physics: Envelope of Waves


In physics, if we combine two wave of almost the same wavelength and frequency, we will get a beating wave (see Figure 72). For such a wave, physicists usually care more about its envelope, rather than the wave itself, since the envelope is what people will actually hear, or see. For example, the two branches of a tuning fork are almost, but not exactly identical. So if a tuning fork starts to vibrate, its two branches will produce two slightly different sound waves. The superposition of these two waves is a beating sound wave with varying amplitude. This is why people can hear "beats" when they strike a tuning fork.
A similar mechanism is used in Amplitude Modulation broadcasting. Different waves are superposed with each other to form a sinusoidal carrier wave with changing amplitude, which can be used to carry audio signals (see Figure 73). For more about broadcasting, please go here.
Application in String Arts


String Arts is a material representation of envelope, in which people arrange colored straight strings to form complicated geometric figures.
For more about String Art please go here.
How the Main Image Relates
As pointed out in the main image, the envelope of all particles' trajectories in an exploding firework is a parabola. Here comes the explanation:
Figure 81 shows a simulation of the exploding process. Blue parabolas are trajectories of particles, and the red parabola is their envelope.
For the envelope to be parabolic, we have to make several assumptions:
 The firework is consisted of many shinning particles, each projected from the origin at the same time, with same velocity v.
 Particles are subject to constant gravity, with gravitational acceleration g.
 Air friction can be neglected. In fact, this turns out to be an arguable assumption. Most firework particles are relatively small and light, so they could be significantly deflected by air friction. However, the case with air friction is way too complicated for this page. Besides, air friction can be neglected, at least on some big and heavy firework particles. So we can still accept this assumption and see what happens.
With the assumptions above, we can write out the trajectory of one particular particle:
 ,
in which is the angle of projection. This is a result from simple mechanics. For more about projectile trajectory, please go here.
For now, let's leave Physics behind and focus on the curves themselves. In this trajectory, is the variable parameter. If we denote by , we can write out a family of curves (in level set form):
Differentiate to get the boundary condition (see the More Mathematical Explanation section):
Substitute Eq 2 into Eq 1, after doing some algebra we can get:
 ,
which gives us a parabolic envelope of the particles' trajectories.
As we have discussed before, this is not true for all fireworks. Because of air friction, most fireworks have an envelope more like a sphere. Nonetheless, this parabolic pattern can be seen somewhere else, such as fountains or explosions. This analysis is also useful in the study of "safe domains" in projectile motion.
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