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

Leslie Watts
L M Watts Furniture
Tiger, Georgia USA
http://www.alltel.net/~leswatts/wattsfurniturewp.html
engineering page:
http://www.alltel.net/~leswatts/shop.html
Surplus cnc for sale:
http://www.alltel.net/~leswatts/forsale.html

----- Original Message -----
From: "Todd Fleming" <todd@...>
To: <CAD_CAM_EDM_DRO@yahoogroups.com>
Sent: Friday, December 27, 2002 9:36 PM
Subject: [CAD_CAM_EDM_DRO] Math for constant speed ellipse


> I was trying to figure out how to machine an ellipse at constant speed
when
> I ran into a differential equation that I can't solve. Is there anyone
with
> a nice symbolic math package that's willing to try this eq? I posted an
> image of the equation here:
>
> http://www.flemingcnc.com/Papers/help_de.png
>
> Thanks!
>
> Todd Fleming
> flemingcnc.com