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>. | 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 | + | It is described by parametric equations which are simple to derive. |
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==From Circle to Cycloid== | ==From Circle to Cycloid== | ||
A cycloid is the path traced by a point on a rolling circle: | A cycloid is the path traced by a point on a rolling circle: | ||
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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). | + | 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> units, or one revolution, every <math>2\pi</math> seconds (in other words, it revolves 1 radian per 1 second). |
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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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==From Cycloid to Brachistochrone== | ==From Cycloid to Brachistochrone== | ||
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: | ||
Latest revision as of 02:41, 27 July 2019
A brachistochrone is the curve of fastest descent from 
to 
.
It is described by parametric equations which are simple to derive.
From Circle to Cycloid
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 
units, 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: 
From Cycloid to Brachistochrone
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 .
