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Warning, the name changecoords has been redefined
Cycloids
Here is an example to show how to use Maple to calculate and draw cycloids,
including slipping cycloids.
First we describe the motion of the wheel's center. (We assume the wheel
has radius 1.)
> p1 := a*t; p2 := 1; a := 1;
> plot( [t,1,t=0..6*Pi] , scaling=constrained);
Not very interesting so far ...
Now we describe the wheel's motion, as though the center were still:
>
> q1 := cos(b*t); q2 := -sin(b*t); b := 1;
> plot( [q1,q2,t=0..6*Pi], scaling=constrained);
That's not too interesting either. Things get better when we add the two
motions.
> r1 := p1+q1; r2 := p2+q2;
> plot( {[p1,p2,t=0..6*Pi],[q1,q2,t=0..6*Pi],[r1,r2,t=0..6*Pi]}, scaling=constrained, title = `A cycloid`);
We can introduce some slippage into the picture by playing with
the multiplicative constants a and/or b. Let's put the slippage
on the wheel
> b := 2; q1 := sin(b*t); q2 := -cos(b*t);
> plot( [ p1+q1, p2+q2, t=0..6*Pi], scaling=constrained);
Let's do something with epicycloids. The general form is
P(t) = (R+1)( cos(at), sin(at) ) + ( cos(bt), sin(bt) ), for appropriate values
of the constants R, a, and b. For example:
> R := 3; a := 1; p1 := (R+1)* cos(a*t); p2 := (R+1)*sin(a*t);
Now let's parametrize the inner circle:
> c1 := R* cos(a*t); c2 := R*sin(a*t);
> b=2; q1 := cos(b*t); q2 := sin(b*t);
Now we'll plot everything:
> plot( {[p1+q1,p2+q2,t=0..2*Pi],[c1,c2,t=0..2*Pi]}, scaling=constrained );