Delaunay Triangulation

$\begingroup$

Wikipedia on Delaunay Triangulation says:

"In the plane[...], if there are b vertices on the convex hull, then any
triangulation of the points has at most 2n − 2 − b triangles, plus one
exterior face."

Here n is the total number of vertices in the plane.

I can't see how they obtained this result when I use Euler's formula 1=V-E+F. So V=n then F=1-n+E. what is face exterior?

$\endgroup$ 2

2 Answers

$\begingroup$

Euler's formular for planar graphs with $n$ vertices, $e$ edges and $f$ faces says that $2 = n - e + f$. Note that $f$ includes the outer, unbounded face to infinity, too. So we have $f-1$ triangles and a $b$-gon.

We can assign to each edge two tokens; one for each incident face. The number of tokens can be counted per face, too: $3(f-1)$ for $f-1$ triangles and $b$ for the outer face, so $3 (f-1) + b = 2e$.

Then we have $4 = 2n - 2e + 2f = 2n -f -k + 3$, and hence $f-1 = 2n -2 -b$, which is the number of triangles.

$\endgroup$ $\begingroup$

The exterior face is the green area.

$\endgroup$ 7

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

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