Re: [CAD_CAM_EDM_DRO] Math for constant speed ellipse

> I suppose you want constant speed so that you chew up your workpiece
> at a constant volume per unit time. However there are at least four
> other factors that you might want to think about as you choose a
> speed. These should only be a concern when the ellipse is very long
> and skinny.
>
> - If you are milling around the outside of the ellipse, then you
> will be chewing up a higher volume per time of material as you go
> around the end than when going along the sides, if you go at a
> constant speed along the ellipse that you are milling.

Interesting; can you explain why this is so? Better yet, do you know of an
approach that will chew the same volume of material per unit time? This
would be much better.

> - If the center of your tool is tracing the ellipse, or if you are milling
> the inside of the ellipse, and the tool width is a substantial
> fraction of the minor axis of the ellipse, then as you come away
> from the corner of the ellipse, you'll be cutting less volume per
> time because some material has already been removed. (For a proper
> inside cut, check that your tool radius is smaller than the radius
> of curvature of the ellipse at the ends!)

I have the same questions as above. I do see the problem that occurs that
you noted in parentheses.

> - Even if you move at constant speed, the acceleration is not
> constant - it peaks at the ends of the ellipse.

Yep. This I understand.

> - If your step size of theta is too large, you will have flat, polygonal
> sides on the ellipse at the ends.

I'm trying for direct elliptical interpolation; the sides of polygon will be
at most sqrt(2 * d), where d is the step size (or micro-step size) of the
machine. I'm starting to realize this may be impossible for a constant
speed. It worked out quite nicely for quadratics though; see my paper at
http://www.flemingcnc.com/ .

> At any rate, since whatever calculus and DE brain cells I may have
> had have mostly died, I would look for a less math-intensive
> approach to the problem: pick an approximate step size for theta,
> calculate the new x,y position that you are headed for, calculate
> the distance from where you are now to over there, and if the
> distance is too far (i.e you are trying to go too fast), just move
> an appropriate fraction of that distance, or if you have time (the
> processor is fast enough), interpolate another value of theta and
> head for that point. Or slow down the clock so the machine takes
> the appropriate amount of time to get there.
>
> BETTER YET!! Look up Breshenham's Algorithm on the net or in a computer
> graphics book. It's how graphics hardware draws straight lines into

I'm using Breshenham's Algorithm for quadratic interpolation. This provides
super-smooth accelleration.

> graphics memory, or controls an XY display. It's probably used in robotics
> as well. The idea is that if you are drawing a straight line whose
> slope is between -1 and 1, you go to the starting point, then take
> one-pixel steps in the x axis. At each step you decide whether to
> keep the same y value, or step up or down one pixel. This can be
> done with just an addition and a comparison at each step - no
> multiplication or division needed! On the other hand, if the line
> has a steeper slope than +-1, you step one pixel at a time in the y
> direction and decide at each step whether to move the x position.
>
> Breshenham's algorithm can be adapted to draw circles - very nice,
> round, smooth-looking ones, better than you can get by calculating
> points along the circle with sines and cosines. As I recall (from
> implementing it once, long ago) it takes some preliminary
> calculations to find where the circle has slopes of +-1. While
> plotting the points, you only need about one multiplication, one or
> two additions, and a comparison at each step. No trig or square
> roots needed during plotting. It's very fast!

Do you know off-hand how much variation in speed Breshenham's approach will
produce when applied to circles?

> Apparently the algorithm can be adapted to work with any conic section
> (ellipses, parabolas, hyperbolas), but I haven't gotten involved in the
> details.

cool!

> Enjoy!
> - Jan

Todd Fleming
flemingcnc.com