Difference between revisions of "Brachistochrone"

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A brachistochrone is the curve of fastest descent from <math>A</math> to <math>B</math>. It is described by parametric equations which are simple to derive:
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A brachistochrone is the curve of fastest descent from <math>A</math> to <math>B</math>.  
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It is described by parametric equations which are simple to derive:
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A cycloid is the path traced by a point on a rolling circle:
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[[File:Rollingcircle.png|How to Create a Cycloid]]
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If the radius of the circle is <math>a</math> and the center of the circle is moving at a speed of <math>a</math> units per second, then it moves <math>2\pi a</math>, or one revolution, every <math>2\pi</math> seconds (in other words, it revolves 1 radian per 1 second).
  
A cycloid is the path traced by a point on a rolling circle:
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Then
[asy]
 
draw((0,0)--(9,0));
 
draw(Circle((0,1),1));
 
dot((0,0));
 
draw((0,1)--(3,1),Arrow);
 
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If the radius of the circle is <math>a</math> and the center of the circle is moving at a speed of <math>a</math> units per second, then it moves <math>2\pi a</math>, or one revolution, every <math>2\pi</math> seconds (in other words, it revolves 1 radian per 1 second). Then
 
 
the x-coordinate of the center relative to the ground is: <math>at</math>
 
the x-coordinate of the center relative to the ground is: <math>at</math>
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the x-coordinate of the point relative to the center is: <math>a\cos(-\frac{\pi}{2}-t)=-a\sin t</math>
 
the x-coordinate of the point relative to the center is: <math>a\cos(-\frac{\pi}{2}-t)=-a\sin t</math>
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so the x-coordinate of the point is: <math>a(t-\sin t)</math>
 
so the x-coordinate of the point is: <math>a(t-\sin t)</math>
  
 
Similarly,
 
Similarly,
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the y-coordinate of the center relative to the ground is: <math>a</math>
 
the y-coordinate of the center relative to the ground is: <math>a</math>
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the y-coordinate of the point relative to the center is: <math>a\sin(-\frac{\pi}{2}-t)=-a\cos t</math>
 
the y-coordinate of the point relative to the center is: <math>a\sin(-\frac{\pi}{2}-t)=-a\cos t</math>
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so the y-coordinate of the point is: <math>a(1-\cos t)</math>
 
so the y-coordinate of the point is: <math>a(1-\cos t)</math>
 
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Since a brachistochrone is an upside-down cycloid, we reverse the sign of y:
 
Since a brachistochrone is an upside-down cycloid, we reverse the sign of y:
 
<math>x=a(t-\sin t)</math>
 
<math>x=a(t-\sin t)</math>
 
<math>y=a(\cos t-1)</math>
 
<math>y=a(\cos t-1)</math>
 
This is a brachistochrone starting at <math>(0,0)</math>.
 
This is a brachistochrone starting at <math>(0,0)</math>.
 
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If you want it to start at <math>(x_0,y_0)</math> you just shift it:
 
If you want it to start at <math>(x_0,y_0)</math> you just shift it:
 
<math>x-x_0=a(t-\sin t)</math>
 
<math>x-x_0=a(t-\sin t)</math>
 
<math>y-y_0=a(\cos t-1)</math>
 
<math>y-y_0=a(\cos t-1)</math>
 
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If you want it to go through <math>(c,d)</math> you need to solve for <math>a</math>:
 
If you want it to go through <math>(c,d)</math> you need to solve for <math>a</math>:
 
<math>c-x_0=a(t-\sin t)</math>
 
<math>c-x_0=a(t-\sin t)</math>
 
<math>d-y_0=a(\cos t-1)</math>
 
<math>d-y_0=a(\cos t-1)</math>
 
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I recommend first solving for <math>t</math> in terms of <math>a</math> using <math>d-y_0=a(\cos t-1)</math>, then substituting into <math>c-x_0=a(t-\sin t)</math>.
 
I recommend first solving for <math>t</math> in terms of <math>a</math> using <math>d-y_0=a(\cos t-1)</math>, then substituting into <math>c-x_0=a(t-\sin t)</math>.

Revision as of 16:28, 15 July 2019

A brachistochrone is the curve of fastest descent from $A$ to $B$. It is described by parametric equations which are simple to derive:



A cycloid is the path traced by a point on a rolling circle: How to Create a Cycloid



If the radius of the circle is $a$ and the center of the circle is moving at a speed of $a$ units per second, then it moves $2\pi a$, or one revolution, every $2\pi$ seconds (in other words, it revolves 1 radian per 1 second).

Then

the x-coordinate of the center relative to the ground is: $at$

the x-coordinate of the point relative to the center is: $a\cos(-\frac{\pi}{2}-t)=-a\sin t$

so the x-coordinate of the point is: $a(t-\sin t)$

Similarly,

the y-coordinate of the center relative to the ground is: $a$

the y-coordinate of the point relative to the center is: $a\sin(-\frac{\pi}{2}-t)=-a\cos t$

so the y-coordinate of the point is: $a(1-\cos t)$



Since a brachistochrone is an upside-down cycloid, we reverse the sign of y: $x=a(t-\sin t)$ $y=a(\cos t-1)$ This is a brachistochrone starting at $(0,0)$.



If you want it to start at $(x_0,y_0)$ you just shift it: $x-x_0=a(t-\sin t)$ $y-y_0=a(\cos t-1)$



If you want it to go through $(c,d)$ you need to solve for $a$: $c-x_0=a(t-\sin t)$ $d-y_0=a(\cos t-1)$



I recommend first solving for $t$ in terms of $a$ using $d-y_0=a(\cos t-1)$, then substituting into $c-x_0=a(t-\sin t)$.

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