7
$\begingroup$

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons.

Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not intercept any other vertices of $P$. Call the ray inward-pointing when it intersects $P$ an odd number of times. Call it outward-pointing if it intersects $P$ an even number of times.

Call a vertex $v$ convex when the set of all inward-pointing rays lies interior to the angle at $v$ and call the vertex concave when the set of all inward-pointing pointing rays lies exterior to the angle at $v$. It can be shown that this definition for vertex convexity is well-defined, i.e. each vertex is either convex or concave.

The book asserts the familiar fact that

if all vertices of a polygon are convex, then the polygon is convex

but does not provide a proof. Does anyone have a proof of this fact using this particular definition of vertex convexity?

To provide a definition, a polygon is convex if and only if any of the following holds:

  1. it is entirely contained in a closed half-plane defined by each of its edges.

  2. for each edge, the vertices not contained in the edge are on the same side of the line the edge defines.

  3. the angle at each vertex contains all other vertices in its interior (except the three vertices defining the angle).

$\endgroup$
  • $\begingroup$ If such a polygon is concave, two vertices are folding points. Then their angles are smaller than right angle by folding feature.// $\endgroup$ – Takahiro Waki Feb 8 '18 at 22:54

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.