# Difference between revisions of "Brachistochrone"

(spacing) |
(sections) |
||

Line 1: | Line 1: | ||

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: | ||

+ | ==From Circle to Cycloid== | ||

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

+ | [[File:Rollingcircle.png|How to Create a Cycloid]] | ||

<br /> | <br /> | ||

<br /> | <br /> | ||

<br /> | <br /> | ||

<br /> | <br /> | ||

− | + | 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). | |

− | |||

<br /> | <br /> | ||

<br /> | <br /> | ||

<br /> | <br /> | ||

<br /> | <br /> | ||

− | |||

− | |||

Then | Then | ||

Line 20: | Line 20: | ||

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> | ||

− | + | <br /> | |

+ | <br /> | ||

+ | <br /> | ||

+ | <br /> | ||

Similarly, | Similarly, | ||

Line 28: | Line 31: | ||

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> | ||

− | + | ==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: | ||

<math>x=a(t-\sin t)</math> | <math>x=a(t-\sin t)</math> |

## Revision as of 02:39, 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 , 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 .