Envelope
From Math Images
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<br>  <br>  
  In geometry, an '''envelope''' of a [[#Familyfamily of curves]] is the boundary of these curves' "sweeping area". In most cases,  +  In geometry, an '''envelope''' of a [[#Familyfamily of curves]] is the boundary of these curves' "sweeping area". In most cases, the envelope is tangent to each member of the family at some point. 
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  +  <br>  
This envelope was firstly discovered and proved by Swiss mathematician Jakob Steiner. In 1856 he published a paper, giving a lengthy proof of why we get a Deltoid when moving the WallaceSimson Line. A simplified version of this proof can be found [http://www.rac.es/ficheros/doc/00045.pdf here]<ref name = ref3>[http://www.rac.es/ficheros/doc/00045.pdf M. de Guzman, 2001, ''A simple proof of the Steiner theorem on the deltoid'']. This is a simplified version of Jakob Steiner's proof.</ref>.  This envelope was firstly discovered and proved by Swiss mathematician Jakob Steiner. In 1856 he published a paper, giving a lengthy proof of why we get a Deltoid when moving the WallaceSimson Line. A simplified version of this proof can be found [http://www.rac.es/ficheros/doc/00045.pdf here]<ref name = ref3>[http://www.rac.es/ficheros/doc/00045.pdf M. de Guzman, 2001, ''A simple proof of the Steiner theorem on the deltoid'']. This is a simplified version of Jakob Steiner's proof.</ref>.  
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  In general, level sets are more powerful than graph of single functions when we need to describe 2D curves, since all single variable functions <math>y = f(x)</math> can be written in the level set form <math>F(x,y) = f(x)  y = 0</math>, but the converse is not true.  +  In general, level sets are more powerful than graph of single functions when we need to describe 2D curves, since all single variable functions <math>y = f(x)</math> can be written in the level set form <math>F(x,y) = f(x)  y = 0</math>, but the converse is not true. For example, like the circle in [[#Figure61Figure 61]], can not be rewritten as y = f(x) unless we use multiple functions. In a more extreme case, the level set<math>x^5 + y +\cos y = 1</math> in [[#Figure62Figure 62]] is not even possible to be reduced to y = f(x) form, because <math>\cos y</math> is a [[#Transcendentaltranscendental function]]. 
{{{!}}border="0" cellpadding=20 cellspacing=20  {{{!}}border="0" cellpadding=20 cellspacing=20  
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  ==  +  ==Resolving the second problem: the boundary condition== 
<div id = "Boundary"></div>  <div id = "Boundary"></div>  
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  ::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<br><math>{\partial F(x,y,t) \over \partial t} = 0</math> .  +  ::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:<br><math>{\partial F(x,y,t) \over \partial t} = 0</math> . 
  +  This condition is easy to prove using '''the implicit function theorem''', which states that, if we have a level set <math>F(x,y) = 0</math>, then ''y'' can be viewed as an ''implicit function'' of ''x''. If the value of ''y'' changes, the value of ''x'' also has to change, since the condition <math>F(x,y) = 0</math> must always be satisfied. Sometimes we can derive an explicit function <math>y = f(x)</math> out of it, sometimes we can't, as shown in the [[#Transcendentaltranscendental function example]]. But the failure of derivation doesn't mean ''x'' and ''y'' are unrelated. They are still related through this "implicit function".  
  
  
  
  +  This theorem can be generalized to functions of three or more variables. For example, look at a function <math>F(x,y,z) = 0</math> of three variables. If we fix the value of one variable, say ''x'', then we will have ''y'' as an implicit function of ''z''.  
  +  "Well", one may ask, "this is an interesting theorem, but how does it relate to envelopes?" The trick is to do the same thing to the level set that's going to sweep our envelope. In the level set <math>F(x,y,t) = C</math>, if we fix the value of x, then y will be an implicit function of t. More importantly, the maximum and minimum value of that implicit function must lie on the envelope, since the envelope should also be the boundary of this implicit function by definition.  
  +  For example, let's revisit the [[#Beginningladder problem and Astroid envelope]]. First, we can fix an ''x'' value by drawing a vertical line, as shown in [[#Figure63Figure 63]]. Each phase of the ladder intersects this line at a different point, and the ycoordinate of the intersection is an implicit function of the ladder's position. What we want, however, is the highest and lowest among these intersections, because they must lie on the envelope. If the position of these two points can be determined, then we can locate two points like this for each fixed x value, and the envelope is just a collection of these highest and lowest points (see [[#Figure64Figure 64]]).  
  +  {{{!}}border="0" cellpadding=20 cellspacing=20  
+  {{!}}{{AnchorReference=Figure63Link=[[Image:Astroid_x.pngcenterthumb350pxFigure 63<br>y as an implicit function of t]]}}{{!}}{{!}}{{AnchorReference=Figure64Link=[[Image:Astroid_x_2.pngcenterthumb350pxFigure 64<br>Highest and lowest points lying on envelope]]}}  
+  {{!}}}  
  Now  +  Now, the only problem remains is to determine the maximum and minimum value of this implicit function. This could be solved by using the '''chain rule''' in multivariable calculus. The chain rule is a formula for computing the derivative of the composition of two or more functions. For example, if we have a function 
+  <br><br>  
+  :<math>F(x(t),y(t))</math>  
+  <br>  
+  in which <math>x(t)</math> and <math>y(t)</math>are differentiable functions of t. Then the chain rule claims that:  
+  <br><br>  
+  :<math>{dF \over dt} = {\partial F \over \partial x}{dx \over dt} + {\partial F \over \partial y}{dy \over dt}</math>  
+  <br>  
+  Same for function of three or more variables<ref name = ref4>[http://en.wikipedia.org/wiki/Chain_rule The Chain Rule], from Wikipedia. This is a more thorough introduction to the chain rule in multivariable calculus.</ref> .  
  
  
  
  +  If we apply the chain rule to the level set <math>F(x,y,t) = C</math> with variable x fixed, we will get:  
  +  
  +  
  +  
  +  
  +  
  +  
  +  
<br><br>  <br><br>  
  : <math>F(x,y,t  +  :<math>{dF(x,y,t) \over dt} = {\partial F(x,y,t) \over \partial x}{dx \over dt} + {\partial F(x,y,t) \over \partial y}{dy \over dt} + {\partial F(x,y,t) \over \partial t}{dt \over dt}</math> 
<br>  <br>  
  : <math>{\partial F(x,y,t) \over \partial  +  Since <math>{dx \over dt} = 0</math> (x is fixed), and <math>{dt \over dt} = 1</math>, this expression can be reduced to: 
+  <br><br>  
+  :<math>{dF(x,y,t) \over dt} = {\partial F(x,y,t) \over \partial y}{dy \over dt} + {\partial F(x,y,t) \over \partial t}</math>  
<br>  <br>  
  : <math>{\partial F(x,y,t) \over \partial t} = 0 </math>  +  On the other hand, since <math>F(x,y,t) = C</math> is a constant function, we have: 
  +  <br><br>  
  +  :<math>{dF(x,y,t) \over dt} = 0</math>  
  +  <br>  
  +  So we can get:  
  +  <br><br>  
+  :<math>{\partial F(x,y,t) \over \partial y}{dy \over dt} + {\partial F(x,y,t) \over \partial t} = 0</math>  
+  <br>  
+  in which <math>{dy \over dt}</math> is the derivative of the implicit function y(t).  
+  <br><br>  
+  For points on the envelope, y is at its maximum or minimum as discussed before, so <math>{dy \over dt} = 0</math>. And the previous equation is reduced to:  
+  <br><br>  
+  :*<math>{\partial F(x,y,t) \over \partial t} = 0 </math>  
+  <br>  
+  which is the [[#Boundaryboundary condition]] we are trying to prove.  
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{{EquationRef2Eq. 1}}<math>F(x,y,t) = {x \over t} + {y \over \sqrt {a^2  t^2}} = 1</math>  {{EquationRef2Eq. 1}}<math>F(x,y,t) = {x \over t} + {y \over \sqrt {a^2  t^2}} = 1</math>  
<br>  <br>  
  Differentiate Eq.1 to get the boundary condition:  +  Differentiate Eq.1 with regard to <math>t</math> to get the boundary condition: 
<br><br>  <br><br>  
:<math>{\partial F(x,y,t) \over \partial t} = {x \over t^2} + {yt \over (\sqrt {a^2  t^2})^3} = 0</math>  :<math>{\partial F(x,y,t) \over \partial t} = {x \over t^2} + {yt \over (\sqrt {a^2  t^2})^3} = 0</math>  
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{{EquationRef2Eq. 2}}<math>{x \over t^3} = {y \over (\sqrt {a^2  t^2})^3}</math>  {{EquationRef2Eq. 2}}<math>{x \over t^3} = {y \over (\sqrt {a^2  t^2})^3}</math>  
<br>  <br>  
  +  Divide Eq.1 by Eq.2, use appropriate sides of Eq.2, we can get:  
+  <br><br>  
+  :<math>{x \over t} / {x \over t^3} + {y \over \sqrt {a^2  t^2}}/ {y \over (\sqrt {a^2  t^2})^3} = 1/{x \over t^3} = 1/{y \over (\sqrt {a^2  t^2})^3}</math>  
+  <br>  
+  Which gives us:  
+  <br><br>  
+  :<math>t^2 + (a^2  t^2) = {t^3 \over x} = {(\sqrt {a^2  t^2})^3 \over y}</math>  
+  <br>  
+  Keep reducing:  
+  <br><br>  
+  :<math>a^2 = {t^3 \over x} = {(\sqrt {a^2  t^2})^3 \over y}</math>  
+  <br>  
+  And keep reducing:  
<br><br>  <br><br>  
  :<math>  +  :<math>t = x^{1/3}a^{2/3}</math> <math>;</math> <math>\sqrt {a^2  t^2} = y^{1/3}a^{2/3}</math> 
<br>  <br>  
  +  Substituting back into Eq.1:  
<br><br>  <br><br>  
:<math>{x^{2/3} \over a^{2/3}} + {y^{2/3} \over a^{2/3}} = 1</math>  :<math>{x^{2/3} \over a^{2/3}} + {y^{2/3} \over a^{2/3}} = 1</math>  
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Which leads to:  Which leads to:  
<br><br>  <br><br>  
  :<math>x^{2/3} + y^{2/3} = a^{2/3}</math> , the equation of an Astroid.  +  :<math>x^{2/3} + y^{2/3} = a^{2/3}</math> , finally, the equation of an Astroid. 
<br>  <br>  
  Other proofs are similar.<ref name =  +  Other proofs are similar.<ref name = ref5>[http://en.wikipedia.org/wiki/Envelope_(mathematics) Envelope], from Wikipedia. This page was particularly helpful for me in the More Mathematical Explanation section. It also has proof for some more envelopes.</ref> 
other=Calculus  other=Calculus  
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  For more about the Envelope Theorem, please go here<ref name =  +  For more about the Envelope Theorem, please go here<ref name = ref6>[http://www.economics.utoronto.ca/osborne/MathTutorial/MEEF.HTM Martin J. Osborne, ''Mathematical methods for economic theory: a tutorial by Martin J. Osborne'', 2011]. This is a brief introduction to Envelope Theorem in Microeconomics.</ref>. 
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  A similar mechanism is used in AM (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 [[#Figure73 Figure 73]]). For more about broadcasting, please go here <ref name =  +  A similar mechanism is used in AM (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 [[#Figure73 Figure 73]]). For more about broadcasting, please go here <ref name = ref7>[http://en.wikipedia.org/wiki/Amplitude_modulation Amplitude Modulation Broadcasting], from Wikipedia. This page is a more extensive introduction to AM broadcasting.</ref>. 
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:<math>y = x{\tan \theta}  x^2{{g \over 2v^2}(1 + \tan^2 \theta)}</math>,  :<math>y = x{\tan \theta}  x^2{{g \over 2v^2}(1 + \tan^2 \theta)}</math>,  
<br>  <br>  
  in which <math>\theta</math> is the angle of projection. This is a result from simple mechanics. For more about projectile trajectory, please go here<ref name =  +  in which <math>\theta</math> is the angle of projection. This is a result from simple mechanics. For more about projectile trajectory, please go here<ref name = ref8>[http://en.wikipedia.org/wiki/Trajectory Trajectory], from Wikipedia. This is the physics behind projectile motions.</ref>. 
<br><br>  <br><br>  
For now, let's leave Physics behind and focus on the curves themselves. In this trajectory, <math>\theta</math> is the variable parameter. If we denote <math>\tan \theta</math> by <math>t</math>, we can write out a family of curves (in level set form):  For now, let's leave Physics behind and focus on the curves themselves. In this trajectory, <math>\theta</math> is the variable parameter. If we denote <math>\tan \theta</math> by <math>t</math>, we can write out a family of curves (in level set form):  
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  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<ref name =  +  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<ref name = ref9>[http://arxiv.org/pdf/physics/0410034v1.pdf JeanMarc Richard, ''Safe domain and elementary geometry'', 2008]. This is a study about safe domains in projectile motion.</ref>. 
Revision as of 17:02, 7 June 2012
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 these curves' "sweeping area". In most cases, the envelope is tangent to each member of the family at some point.


Surprisingly, as shown in Figure 12, the area swept by a moving straight line does not necessarily have a straight boundary. In fact, its envelope is the firstquadrant portion of an astroid . One may notice the Astroid is always tangent to the ladder at some point during the sliding process, as stated in the definition of an envelope.
If we slide the ladder in other three quadrants, we will get a complete starshaped envelope, as shown in Figure 13. In fact, the name astroid comes from the Greek word for "star".
For the math behind this envelope, please go to the More Mathematical Explanation section.
A gallery of beautiful envelopes
Envelope of lines
As we have seen the ladder example, a moving straight line can have a curve as its envelope. Here are more examples:


In Figure 21, line m, our sweeping line, is perpendicular to segment OA at its midpoint M. Point O is fixed in space.
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.


Similar to what we did in Figure 21, our sweeping line is still the perpendicular bisector of segment AB. The only difference is that point A now slides on a circle, rather a straight line.
The result is an Ellipse with O and B as its foci, as shown in Figure 24.


Similar to what we did in the previous example, our sweeping line is still the perpendicular bisector of segment AB. The only difference is that point B is outside the circle.
The result is a Hyperbola with O and B as its foci, as shown in Figure 26.
So far we have got all of the three Conic Section Curves as envelopes of straight lines. However, the sweeping curve of envelopes are in no ways restricted to straight lines. Circle, Ellipse, and other curves can make fantastic envelopes as well.
Envelope of circles
This section shows some interesting envelopes generated by moving a circle around.


In Figure 31, we begin with a base circle O, which is fixed in space, then select two points A and B on the base circle. Our sweeping circle is centered at A, and passes through B.
If we fix point B and slide point A along the fixed circle, circle A will sweep out a Cardioid, as shown in Figure 32.
The name "Cardioid" comes from the Greek word for "heartshaped". For more information about the Cardioid, please go to this page.


Similar to what we did in Figure 31, we still have a fixed base circle O, and a sweeping circle that has its center A sliding on the base circle. The only difference is that our sweeping circle is now tangent to a vertical line l, rather than passes through a fixed point.
The result is a Nephroid, which is the Greek word for "kidneyshaped". For more information about Nephroid please go here.


In Figure 35, we begin with a hyperbola, with points F1 and F2 as its foci and A as its center. Our sweeping circle has its center O on the hyperbola, and passes through 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]}. For more information about Lemniscate please go here.


In Figure 35, if instead of having A as center of the hyperbola, we move it to an arbitrary position between the hyperbola's two halves, 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 as a result 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.
2.Deltoid as envelope of WallaceSimson lines
The WallaceSimson line is related to an interesting theorem in geometry proposed by William Wallace in 1796. The theorem itself is not hard to prove, and with a little manipulation we can get one of the most beautiful envelopes out of it.
This envelope was firstly discovered and proved by Swiss mathematician Jakob Steiner. In 1856 he published a paper, giving a lengthy proof of why we get a Deltoid when moving the WallaceSimson Line. A simplified version of this proof can be found here^{[3]}.
Notice that Astroid, Cardioid, Nephroid, and Deltoid all belong to the Roulette Family, which means they can also be constructed by rolling one circle around another. For more information about Roulettes, please go to this page.
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 their sweeping area.
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 original 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 cannot be expressed unless we use two functions, and . So we have to introduce new ways to describe curves.
Second, "boundary of the sweeping area" 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.
Resolving the first problem: the power of level sets
The first problem is easily resolved if we describe 2D curves using level sets, rather than graphs of single variable 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 graph 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. For example, like the circle in Figure 61, can not be rewritten as y = f(x) unless we use multiple functions. In a more extreme case, the level set in Figure 62 is not even possible to be reduced to y = f(x) form, because is a transcendental function.


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, the the method of level sets is sufficient to describe all 2D curves that we care about.
Resolving 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:
.
 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:
This condition is easy to prove using the implicit function theorem, which states that, if we have a level set , then y can be viewed as an implicit function of x. If the value of y changes, the value of x also has to change, since the condition must always be satisfied. Sometimes we can derive an explicit function out of it, sometimes we can't, as shown in the transcendental function example. But the failure of derivation doesn't mean x and y are unrelated. They are still related through this "implicit function".
This theorem can be generalized to functions of three or more variables. For example, look at a function of three variables. If we fix the value of one variable, say x, then we will have y as an implicit function of z.
"Well", one may ask, "this is an interesting theorem, but how does it relate to envelopes?" The trick is to do the same thing to the level set that's going to sweep our envelope. In the level set , if we fix the value of x, then y will be an implicit function of t. More importantly, the maximum and minimum value of that implicit function must lie on the envelope, since the envelope should also be the boundary of this implicit function by definition.
For example, let's revisit the ladder problem and Astroid envelope. First, we can fix an x value by drawing a vertical line, as shown in Figure 63. Each phase of the ladder intersects this line at a different point, and the ycoordinate of the intersection is an implicit function of the ladder's position. What we want, however, is the highest and lowest among these intersections, because they must lie on the envelope. If the position of these two points can be determined, then we can locate two points like this for each fixed x value, and the envelope is just a collection of these highest and lowest points (see Figure 64).


Now, the only problem remains is to determine the maximum and minimum value of this implicit function. This could be solved by using the chain rule in multivariable calculus. The chain rule is a formula for computing the derivative of the composition of two or more functions. For example, if we have a function
in which and are differentiable functions of t. Then the chain rule claims that:
Same for function of three or more variables^{[4]} .
If we apply the chain rule to the level set with variable x fixed, we will get:
Since (x is fixed), and , this expression can be reduced to:
On the other hand, since is a constant function, we have:
So we can get:
in which is the derivative of the implicit function y(t).
For points on the envelope, y is at its maximum or minimum as discussed before, so . And the previous equation is reduced to:
which is the boundary condition we are trying to prove.
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 a simple case: 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 with regard to to get the boundary condition:
From which we can get:
Divide Eq.1 by Eq.2, use appropriate sides of Eq.2, we can get:
Which gives us:
Keep reducing:
And keep reducing:
Substituting back into Eq.1:
Which leads to:
 , finally, the equation of an Astroid.
Other proofs are similar.^{[5]}
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^{[6]}.
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 AM (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 ^{[7]}.
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 for some fireworks with big and heavy particles such as blueaerialshell. 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^{[8]}.
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 to eliminate t, 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^{[9]}.
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Related Links
Additional Resources
 1.http://jwilson.coe.uga.edu/Texts.Folder/Envel/envelopes.html. I used this page as a starting point.
 2.http://poncelet.math.nthu.edu.tw/disk3/summer01/work/861/02/ex2.html. Here are some animations for more cool envelopes.
 3.http://www.dynamicgeometry.com/. This is a very helpful geometric software called Geometer's Sketchpad. I used this software to create most of my pictures.
References
 ↑ Lemniscate of Bernoulli, from Wikipedia. This is an introduction to Lemniscate and how it was discovered.
 ↑ Simson Line, from Wikipedia. This is a simple proof of the existence of WallaceSimson line.
 ↑ M. de Guzman, 2001, A simple proof of the Steiner theorem on the deltoid. This is a simplified version of Jakob Steiner's proof.
 ↑ The Chain Rule, from Wikipedia. This is a more thorough introduction to the chain rule in multivariable calculus.
 ↑ Envelope, from Wikipedia. This page was particularly helpful for me in the More Mathematical Explanation section. It also has proof for some more envelopes.
 ↑ Martin J. Osborne, Mathematical methods for economic theory: a tutorial by Martin J. Osborne, 2011. This is a brief introduction to Envelope Theorem in Microeconomics.
 ↑ Amplitude Modulation Broadcasting, from Wikipedia. This page is a more extensive introduction to AM broadcasting.
 ↑ Trajectory, from Wikipedia. This is the physics behind projectile motions.
 ↑ JeanMarc Richard, Safe domain and elementary geometry, 2008. This is a study about safe domains in projectile motion.
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