The planar arrangement solved a line and a circle, or two circles, for their two
crossings even where the curves only touch. At a tangent contact the two roots are
rounding noise spread along the curves by its square root: a 1e-12 mm error in the
centre's distance from the line puts them 2e-6 mm either side of the contact on a
2 mm arc, well inside the tolerance of both curves; when the noise falls the other
way there is no root at all. Cutting a rounded corner (an arc tangent to the two
edges it ends on) out of a plate with planar_regions_boolean then ran that far along
the plate's edge past the arc's end before turning onto the arc: a 200 x 37 plate
turned by 0.25 degrees came back with six edges, one of them 3.45e-6 mm long.
Curves that touch within the arrangement's tolerance now meet at the one point their
centres give: the foot of the circle's centre on the line, or the point on the line
of centres of two circles tangent inside or outside. Circles of nearly one radius
about nearly one centre keep the general formula.
Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>
