Triangulation of a point set

Builds on Convex hull Triangle Collinear points

Definition

Let $P$ be a finite set of points in $\R^2$ , not all collinear, and let $\operatorname{conv}(P)$ be its convex hull. A triangulation of $P$ is a finite set $T$ of closed non-degenerate triangles such that:

the vertices of every triangle of $T$ belong to $P$ ;
any two distinct triangles of $T$ are either disjoint or meet in one common vertex or in one common edge;
the union of the triangles of $T$ is $\operatorname{conv}(P)$ ;
every point of $P$ is a vertex of some triangle of $T$ .

Remarks

Example: let $P = \set{(0,0), (1,0), (1,1), (0,1)}$ . The two triangles with vertices $(0,0), (1,0), (1,1)$ and $(0,0), (1,1), (0,1)$ form a triangulation: they meet along the diagonal, and their union is the unit square.

Non-example: add the centre $(1/2, 1/2)$ to $P$ and keep the same two triangles. Clauses 1 to 3 still hold, but the centre is not a vertex of either triangle, so clause 4 fails.

If $P$ has $n$ points, of which $h$ lie on the boundary of $\operatorname{conv}(P)$ , then every triangulation of $P$ has $2n - h - 2$ triangles and $3n - h - 3$ edges. Some authors drop clause 4 and allow points of $P$ that are not used as vertices.

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Tag	`6MFS8FEYHD`
URI	`m2://6MFS8FEYHD`
Kind	definition
Writing	W0 stub
MSC	52C99, 68U05, 52B70
Defines	Triangulation of a point set
Uses	Convex hull, Triangle, Collinear points

Triangulation of a point set

Definition

Remarks

Reading path

Further reading

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