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  • $\begingroup$ I think this could be it -- "extreme" could be measured by the angle between $p_1$ and given vertex. Thank you! $\endgroup$ Commented Jul 30, 2015 at 12:50
  • $\begingroup$ @greenoldman: The edges incident to $p_i$ form a cone, but the cone could span more than $\pi$ (at concavities), which is why it is perhaps not straightforward to identify the extremes, the boundaries of the cone. $\endgroup$ Commented Jul 30, 2015 at 13:00
  • $\begingroup$ @greenoldman: Start with the previous edge $e_{i-1}$. Walk around the edges incident to $p_i$ clockwise until you come back to $e_{i-1}$. Then $e_i$ the the next-to-last edge before returning to $e_{i-1}$. $\endgroup$ Commented Jul 30, 2015 at 13:10