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Torsten
Posts:
1,128
Registered:
11/8/10
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Re: Brachistochrone
Posted:
Mar 13, 2013 8:26 AM
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"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.
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