arc180_old.svg http://inkscape.13.x6.nabble.com/file/n4967687/arc180_old.svg arc180_org.svg http://inkscape.13.x6.nabble.com/file/n4967687/arc180_org.svg
Thanks for the demo files, they illustrate the problem well. I'm just writing to confirm your calculations, and propose a solution. Attached are two demo files. The file arc180_org.svg contains two arcs, both 180 degrees, both with radius 200. The two arcs are identical except that, in the bottom arc, I have deleted all the sodipodi attributes manually. This makes the bottom arc render differently. The difference can be seen by clicking F2.
The second file, arc180_old.svg, contains the same two arcs after being converted to Bezier form. The top one was converted using 'Path->Object To Path', and the bottom one was converted using the 'Insert new nodes' button, as you discovered. The Bezier data confirms your conclusions:
Top arc: 2 Beziers, 90 degrees each, Bezier control arm length = 110.45694. This is consistent with the formula : (4/3)*tan(theta/4).
Bottom arc: 8 Beziers, 22.5 degrees each, Bezier control arm length = 26.17994. This is consistent with the formula : theta/3.
Proposed solution below:
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