Im building a public skatepark and want to design one of the ramps with a Brachistochrone curve as its profile. How can I create one of these in MOI3d?
Based upon reading numerous papers, which include lots of calculus, the problem needs to be defined. What are the coordinates of the start point A,
and the end point B? Can we assume zero friction?
Without friction, the Brachistochrone curve is a cycloid. (There are more complex equations in the case of friction.)
The problem is determining what radius of cycloid fits the start point A, and the end point B, if they are not level. This requires a "solver" program.
This link shows a cycloid, and simple solution by Gary Brookfield, for the case that A and B are level, and separated by horizontal length L: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.509.2999&rep=rep1&type=pdf
R is the radius of a cycloid, and theta the angle of rotation of the disk.
The parametric equations, for level, or nonlevel A and B, are:
x = R * (theta - sin(theta))
y = (-1) * R * (1 - cos(theta)) {The negative 1 is applied so that the y axis increases with height.}
For level A and B, R = L / (2*PI)
Travel time on the cycloid is equal to the SquareRoot(2*PI*L / g)
g = 9.8 meters per second per second, which is earth gravity.
This particular cycloid can be plotted in MoI relatively easily, e.g. with FXGraph, or a simple script.
If your design starts with A and B at different levels, a mathematical solver will be needed in order to solve for the correct values of R and theta for the cycloid...
I do not currently know how to program a solver. There are math software programs which do solving...
I just find how have the English Page of the Mathcurves.com !
One more time the Symdrom of Edgard Allan Poe of he Purloined Letter has made a victim ! :)
And what about my silly conjecture of the Orient / Line - Line use ?
@Bemfarmer here are the rules guiding the curve(shown in red but not accurate).
Basically point A needs to be tangent to ground plane and the height of point B needs to be 5 feet or so but cannot go beyond vertical(ideally 70-80 degrees).
The existing CycloidInterp script will work, with aRadius = bRadius = 2.5 feet+, mirrored upside down, using half of one cycle.
At a radius of 2.5 feet, the height difference is 5 feet, and the upper tangent is 90 degrees.
At a radius of 3 feet, the height difference is 6 feet, and with one vertical foot removed from the top of the cycloid,
the tangent is about 65.9052 degrees from the horizontal, as measured with Michael's Angle script.
Some more options may be added to the script...
- Brian
(No calculus needed. It has already been proven that without friction, the brachistochrone is a cycloid)
(The CycloidInterp script uses the Trochoid formula. When aRadius = bRadius, the trochoid is a common cycloid)
Brachistochrones.JPG
