[cairo] Derivation of the mitre limit

Chris Wilson chris at chris-wilson.co.uk
Wed Sep 26 03:33:00 PDT 2007

I was reviewing https://bugs.freedesktop.org/show_bug.cgi?id=7245,
cairo_stroke_extents() gives wrong result for arcs in some cases, when I
left scratching my head over the detailed comment that derives the
condition for the mitre limit (cairo-path-stroke.c, line 276).

Namely the first identity where it is stated sec(theta) = 1/sin(theta).
Eeek, surely you meant sec(theta) = 1/cos(theta) - and I double checked
against my trigonometric identities booklet and

After that the correction is fairly straight-forward as we can use the
identity 2·cos²(psi/2) = 1+cos(psi) in place of the original
2·sin²(psi/2) = 1-cos(psi) with the result 2 <= ml² (1 + in · out).

Fixing the minus sign fixes Bug 7245 but generates a new failure with
random-intersections, so can someone (looks at Carl ;-) double check
my working and update the reference image if necessary.
Chris Wilson
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