Re: [CAD_CAM_EDM_DRO] Math for constant speed ellipse

> Todd,
>
> I used to have a nice symbolic math program between my
> ears. Let's see if it still runs at all.
>
> It looks like you want to calculate x axis and y axis speeds vs
> time for an ellipse of semimajor axes a in x and b in y centered around
> the origin.
>
> Let's define P as the time to travel all the way around the ellipse
> at constant speed.
>
> Then x(t)= a*cos(2*PI*t/P) and y(t)=b*sin(2*PI*t/P)
>
> parameterized to time t.
>
> To find the axis speeds we differentiate
>
> dx/dt= -a*2*PI*sin(2*PI*t/P)/P = x axis velocity
> dy/dt= b*2*PI*cos(2*PI*t/P)/P = y axis velocity
>
> The resulting constant speed around the ellipse is just
> the perimeter divided by the period which is also
>
> sqrt( dx/dt^2 + dy/dt^2)
>
> This is confirmed a constant by the trig identity cos()^2 +sin()^2 =1
>
> Hope I got that right. Still on my first cup of coffee goofing off here
> on a saturday morning.
>
> Les

This is the beginning of what lead up to my D.E. Unfortunately, sqrt( (dx/dt)
^2 + (dy/dt)^2 ) isn't constant. cos()^2 + sin()^2 = 1; but a^2 cos()^2 +
b^2 sin()^2 isn't a constant, unless a = b. If a = b then we have a circle.

This is my goal: find a function f(t), such that plotting the parametric
equations x = a cos(f(t)) and y = b sin(f(t)) with a linearly-increasing t
produces a constant speed.

To simplify things, I kept everything in radians; this keeps us from having
to multiply by PI during differentiation. The less constants to keep track
of, the better.

Set sqrt( (dx/dt)^2 + (dy/dt)^2 ) = constant C, square both sides of the
equation, then evaluate the dirivatives. You end up with the DE at
http://www.flemingcnc.com/Papers/help_de.png .

Thanks,
Todd