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Topic: Brachistochrone
Replies: 1   Last Post: Mar 13, 2013 8:26 AM

 Torsten Posts: 1,717 Registered: 11/8/10
Re: Brachistochrone
Posted: Mar 13, 2013 8:26 AM

"Melissa" wrote in message <khp06m\$seh\$1@newscl01ah.mathworks.com>...
> Hey! I'm trying to model a brachistochrone curve given a start and end point(randomly chosen). I'm not sure how to implement the parametric equations of the model with just two points to start off. Any suggestions?

Let A=(x1,y1) and B=(x2,y2) be the two randomly chosen points
(where the points are ordered such that y2>y1).
Then the parametric Brachistochrone curve between the two points is given by
x(t)=x2+R*(t-sin(t))
y(t)=y2+R*(cos(t)-1).
The other condition the curve has to fulfill is that it passes through A.
Thus
x1=x2+R*(t*-sin(t*))
y1=y2+R*(cos(t*)-1)
for a certain value t*.
This is a system of two equations in two unknowns (R and t*).
Once you have determined R and t*, the curve (x(t),y(t)) for t in [0:t*]
is the brachistochrone curve you want to determine.
You may solve the two-equation-system using MATLAB's fsolve or you may
eliminate R and solve the equation
(x1-x2)/(y1-y2)=(t-sin(t))/(cos(t)-1)
for t* using MATLAB's fzero.

Best wishes
Torsten.