Roger Guay wrote: Thanks to all who have responded to this Line tracing thing. Here is another iteration.
(snip)
I've briefly looked at your TurtleGraphicsDemo and I'm very impressed. I haven't had time to understand the "Turtle" aspect yet, but I will
have a closer look. I've been toying with the idea of simulating a
typical Science Museum Foucault Pendulum, where a beautiful pattern is
traced in the sand by the pendulum. I would like to generate those
patterns by running the equation of motion of the Foucault Pendulum. I
think this is not going to be easy because if I recall correctly, this
is a non-linear differential equation. But, I'm ignorant enough to
proceed anyway.
Roger,
There is a real catch in solving the Foucault Pendulum problem. The equations of motion have no closed form solution. (Although there are approximate solutions for small amplitude and for small angular velocity.) But if you employ the method illustrated in the Turtle Graphics Demo you don't need the solution, you effectively solve the equations of motion with a very simple algorithm:
repeat
xNew = xOld + velocityOld*t
velocityNew = velocityOld + acceleration*t
(put the new x into the old x and draw the line)
(put the new velocity into the old velocity)
end repeat
where t is some fixed small time increment, and the acceleration is a known function.
If for simplicity you take t = 1 (sec), the TG code for *any* problem in dynamics is:
Repeat forever --Or until you lose patience
IncrementXY vx, vy
add accx() to vx
add accy() to vy
end repeat
The line "IncrementXY vx,vy" increments the x and y coordinates of the path by vx and vy, and simultaneously draws the line.
It doesn't get much simpler. This becomes a template for solving *all* such problems in dynamics. You just have to write different acceleration functions depending on the physical circumstances. You can see why I am promoting Transcript/TG as a mean of teaching programing to high school science students.
This algorithm is in effect the solution to the differential equation of motion by the method of finite differences, but I wouldn't tell anybody--they will shun you.
May I suggest the two dimensional harmonic oscillator rather than the Foucault Pendulum. You get beautiful Lissajou figures (prettier than the Foucault path) which are open or closed depending on the ratio of the x and y spring constants. This is a very rich problem for exploration.
Good luck. You look like you are having fun.
Jim
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