# 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: | + | 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 cycloid is the path traced by a point on a rolling circle: | ||

+ | [[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). | ||

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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 to .
It is described by parametric equations which are simple to derive:

A cycloid is the path traced by a point on a rolling circle:

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

Then

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

the x-coordinate of the point relative to the center is:

so the x-coordinate of the point is:

Similarly,

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

the y-coordinate of the point relative to the center is:

so the y-coordinate of the point is:

Since a brachistochrone is an upside-down cycloid, we reverse the sign of y:
This is a brachistochrone starting at .

If you want it to start at you just shift it:

If you want it to go through you need to solve for :

I recommend first solving for in terms of using , then substituting into .