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DSA 2

**Triangulation of a point set S in the plain is ...**

**a partition of the convex hull of it into interior-disjoint triangles, whose edges are segments spanned by the points of S.Also no point of S lies inside a segment or a triangle.**

DSA 2

**Triangulation of a Polygon P is ...**

**the maximal plane straight-line graph on S inside **

DSA 2

**Plane Straight-Line Graph **is

**a set of points in the plain and the set of edges spend by this points, such that no two edges cross in their interior**

DSA 2

**Maximal Plane Straight-Line Graph **is

**one where we can't add any edge without crossing another edge.**

**-> We can obtain a Maximal Plane Straight-Line Graph by adding edges, that are not crossing the other edges, as long as possible (the result is always a TRIANGULATION)**

DSA 2

How many edges and triangles can a **triangulation of a pointset** S with n points have?

**e = 3n−3−h** edges

**t = 2n−2−h** triangles *h -> extreme points*

DSA 2

How many edges and triangles can a **triangulation of a polygon** P with n vertices have?

** e = 2n−3** edges

** t = n−2** triangles

-> The number of edges and triangles is linear in the number of vertices

DSA 2

What can triangulations be used for?

- modelling surfaces
- triangular networks in cartography
- 3D modeling
- Computer Graphics
- simulation of fluid flows
- proving time bounds
- finding shortest path inside a polygon
- combinatorics
- ...

DSA 2

A triangulation for an **n-point set** can be constructed in

** O(n log n) **time

** Θ(n) **space

DSA 2

A directed Graph is called ... if it has no loops

**simple **

DSA 2

**Simplepath** contains ...

**every vertex in V at most once **

DSA 2

Length in weighted graphs is ...

**the sum of the edge weights **

DSA 2

An undirected graph is** a forest** if

**if it is acyclic **

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