Arcs in this case are defined by a start point, an endpoint and a central angle (positive angles are counterclockwise, negative angles are clockwise). In each case, we're given two arcs. The endpoint of the first arc is always the start point of the other. I'm trying to find a formula to determine whether the intersection of the two arcs at their shared start/end point is acute. Two examples are shown below.

Two examples of endpoint-sharing arcs with acute angles

This is pretty easy to do analytically on a case by case basis, but finding a formula for it has proven elusive. Any help on finding an algorithmic approach to this problem would be much appreciated

  • $\begingroup$ Are your arcs all parts of circles? If so are radii the same? $\endgroup$ – coffeemath Aug 31 '18 at 18:04
  • $\begingroup$ If the arcs represent the arcs of a circle, then you can find a nice closed-form expression. mathworld.wolfram.com/Circle-CircleIntersection.html $\endgroup$ – Maxtron Aug 31 '18 at 18:05
  • $\begingroup$ Would it be possible to look at the dot product of the two vectors that form the derivative of each curve at the point of intersection? The sign of the dot product determines whether the angle is acute/right/obtuse. $\endgroup$ – barrycarter Sep 1 '18 at 1:40
  • $\begingroup$ @coffeemath , the arcs are all circular arcs, but their radii are not the same. $\endgroup$ – Braden Kallin Sep 4 '18 at 13:41
  • $\begingroup$ @barrycarter , I didn't know this about vector dot products, but that's a start. This works when the inside angle is acute, but not when the outside angle (see the second diagram) is acute. I'm currently wondering what the conditional is that would determine which angle to check. $\endgroup$ – Braden Kallin Sep 4 '18 at 13:45

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