I am trying to create a polyline to feed a laser cutter.

I have two elliptical arcs and two circular arcs which touch each other at a point in a way that forms a fat "C" shape.

I have calculated the geometry so I know the lines meet.

I choose: Modify/Trim 2 and successively choose each touching pair to be sure LC also thinks they meet.

I successively select each of the objects and they change in turn form white to dotted red.

I then choose: Tools/Polyline/Create polyline from existing objects. ("Polyline created" appears in the command window).

After this step, one of the objects goes white and the other three remain dotted red.

Has anything happened??

I try to select the objects for crosshatching and when I click the hatching tool, the command line shows a message that the selected object is invalid because it is not closed.

I have played around a bit and have not found a clear answer. Does your fat C consist of 4 elements or do you have 2 C's of two elements each? In the latter case I had no problems to hatch. In the first case I couldn't trim properly. You might provide your sample, prior to trimming, here as dxf to examine, if you like.

I thought your figure was something like a half-moon, crescent!
Anyway, if you zoom in and in very close at the junction points of the arcs you will see that they don't meet, by about 0.001 or 0.0001 units if you measure it. You said you have calculated the points, this might be the source of the inaccuracy, as the output of circle or ellipse calculation is being rounded. So avoid to use calculated values or positions, instead try to solve everything geometrically with CAD.
You can't always fix such discrepancies with "Trim", because not all arcs do meet somewhere when lengthening them.
Maybe you have to reconstruct your shape considering all this, if I am on the right trace.

Still struggling a bit. I am not sure what you mean by "solve everything geometrically with CAD". What command/tool would I use to position a circular arc to perfectly meet two elliptical arcs at a very specific desired point and then do the same thing in a mirror position at the other ends of the elliptical arcs.

I am getting this message in the command space:

Hatch failed due to a gap=9.39034e-05 between (44.2, 40.3266) and (44.2, 40.3267)
Invalid hatch area. Please check that the entities chosen form one or more closed contours. When I did a trim2 operation at this point (meeting of right hand circular arc with outer elliptical arc) I did not get any feedback that nothing happened.

How close does my meeting have to be for the trim2 tool to work - the points are within 1/10000 of a millimetre.

To solve this by CAD, you have to draw a couple of construction lines e.g. through center points or tangents or orthogonal lines to construct the intersection and center points and then use snap modes to draw the entities.

But anyway, your contour will not convert to a polyline because of the elliptic arcs.
A polyline can only contain straight lines and circular arcs.
Also CNC machines don't like ellipses, most if not all don't have commands to interpolate complex curves like ellipse or spline.

Some CAM software has approximation algorithm to convert ellipse or spline into a bunch of tangential connected circular arcs.
In LibreCAD was the option to save the drawing in DXF R12 format, then the ellipse should be converted to a polyline of short straight segments. But as I recognized just now, this doesn't work anymore.

So you have to try to create your shape with circular arcs only.
Maybe there is an online converter for ellipses approximation, but I can't promise that.