Tool Path Equation
Tool Path Equation
(OP)
This is really a math question but I don't know where else to ask it.
I need the equation (if there is one)of the curve for the path of the center of a circle which is rolling around an ellipse. Any suggestions for math forums would also be appreciated.
I need the equation (if there is one)of the curve for the path of the center of a circle which is rolling around an ellipse. Any suggestions for math forums would also be appreciated.





RE: Tool Path Equation
http://mathworld.wolfram.com/Ellipse.html
Using equation 10 from that site, simply add the r of your circle to both a and b. (I think)
RE: Tool Path Equation
RE: Tool Path Equation
Isn't it the equation for an ellipse, larger-or smaller-by the radius of the circle? Am I missing something?
Regards,
Lcubed
RE: Tool Path Equation
RE: Tool Path Equation
If you know where the major and minor widths are, subtracting the diameter of the circle leaves you with the two axes of the ellipse itself.
TTFN
RE: Tool Path Equation
x^2/a^2+y^2/b^2=1 then
dy/dx=-x/y*b^2/a^2 and the normal to the ellipse which passes thru the centre of the circle is -dx/dy=y/x*a^2/b^2.
Therefore you can write the slope eq of the ellipse in terms of the new coordinates as
dy'/dx'=(-x'-rho*cos(thet))/(y'-rho*sin(thet))*b^2/a^2
where thet is the arctan of dx'/dy' and rho the radius of circle and therefore
sin(thet=dx'/dy'/sqrt(1+dx'/dy')^2)
cos(thet)=1/sqrt(1+dx'/dy')^2)
Now you have a difficult diff eq that probably doesn't have a closed solution, but may be more useful than mapping a solution.
RE: Tool Path Equation
If zekeman's method does not work, search on the Web for the mathematical solution.
RE: Tool Path Equation
RE: Tool Path Equation
"parallel" was used for straight lines only.
But it was on another continent, long time ago and not in English language. Mathematically it is solved as envelope, in CAD we call it "offset" - all of it the same.
RE: Tool Path Equation
A circle is just a special ellipse where both axis are equal.
Why wouldn't this also work for the general case?
sreid, why wouldn't it be an ellipse?
Going back to mathworld, you are just changing the constant a to a different constant, a+r. Seems like it should get you a larger ellipse.
RE: Tool Path Equation
Mathematically stated ?? The ellipse rate of curvature must be greater than the circles rate of curvature at all points.
RE: Tool Path Equation
RE: Tool Path Equation
Although it sometimes works and can provide insight, I am afraid that if you make a habit of generalizing from the special to the general case you will make a lot of mistakes!
RE: Tool Path Equation
First thing is to find the angle of the normal to the ellipse at a given point (Xe,Ye). This is done first on a circle where Ac = Bc = Be,
with points (Xc,Yc) and the tangent of the normal is (Yc/Xc).
By considering the ellipse as a X scaled circle the tangent scales as well, so that at a point (Xe,Ye) we now have the normal angle ;
Theta = ArcTan ( Yc * Ae / Xc * Be )
Then positioning the roller centre (Xr,Yr) is adding the X and Y components of a triangle whose hypotenuse is the ellipse normal of length Rc.
So the coordinates of the rolling circle on the ellipse are :
Xr = Xe + ( Rc * Cos ( ArcTan ( Yc * Ae / Xc * Be ) ) )
Yr = Ye + ( Rc * Sin ( ArcTan ( Yc * Ae / Xc * Be ) ) )
If the circle were to roll inside the ellipse then the components would be subtracted from the ellipse points.
Plug some numbers into this if it seems too good to be true.
No need to plunge into differential geometry or the Newton-Raphson method, more an understanding of conics and coordinate geometry.
A constant offset from any ellipse does not give another ellipse, the new curve is an oval.
RE: Tool Path Equation
RE: Tool Path Equation
A small point. Your equations(in parametric form) are almost ok except that your terms in Ae/Be should be (Ae/Be)^2.
If you have a graphing calculator, the curve can be shown nicely, using the parametric feature.
RE: Tool Path Equation
There is nothing inherently wrong with the idea of presenting a solution in parametric form, and nothing in the question precludes such a solution.
However as a practical matter, if sreid intends to use the solution for machining an elliptical part with a milling cutter then he would be well served to use the parametric form which is slightly incorrect as presented. They should read:
Xr=x + Rc*cos(arctan(y/x*(a/b)^2)
Yr=y + Rc*sin(arctan(y/x*(a/b)^2
The ellipse being:
x^2/a^2+y^2/b^2=1
Again for machining, the solution is not valid for the undercutting condition,
Rho<Rc
where Rho= radius of curvature at x,y coordinate of ellipse.
RE: Tool Path Equation
If so, why not just use a G41 or G42 command and the original path that you want the cutter to follow. If the control uses FANUC programming language all of this is a moot point.
RE: Tool Path Equation
My comment about undercutting applies only to cutting an internal ellipse where it is obvious you can't, if the radius of curvature at any point of the ellipse is less than the cutter radius.
No such limitation exists for the convex ellipse.
RE: Tool Path Equation
RE: Tool Path Equation
Point well taken.
I do think some of these problems are real engineering situations and others are purely academic and the author should indicate as such so that people like you can address the problem properly.
RE: Tool Path Equation
Whether or not there is any NC machining involved here I don't know and there is no shortage of CAD and NC programming packages to define geometry, but it is surely better to have some understanding of what's happening. It's very easy to just be an operator of such systems and not much more.
If a cutter is inside the ellipse, then an immediate and exact value of it's largest radius can be found ;
Minimum radius of Ellipse on major axis,
Rmin = B^2 / A
and if needed, the other case,
Rmax = A^2 / B
RE: Tool Path Equation
If you have a computer, you can solve a problem many ways
even by trial and error.
Else you can blow the result( for fix numbers ) into
a PROM or you can add a small computer module just to
calculate it. See jkmicro or bagotronics ( <www....com>
of course ) both make lowcost PC compatible singleboard computer cards.
<nbucska@pcperipherals DOT com> subj: eng-tips
read FAQ240-1032
RE: Tool Path Equation
http://mathworld.wolfram.com/ParallelCurves.html
The reason for the question is that we machine a lot of ellipse sections and we create our own tool path code. Clearly as others have suggested it's not hard to calculate the required tool center but they are kind of klunky. It's usually nice to be able to plug a number into an equation and get and answer.