Triangle piercing

Mathematical problem involving triangles

A visual representation of the triangle piercing problem.

Triangle piercing is a problem in combinatorial geometry that involves determining the minimum number of points required to "pierce" or "stab" all triangles in a given family of triangles. This problem is a specific case of the more general problem of finding a transversal for a family of sets.

Problem statement

The triangle piercing problem can be formally stated as follows: Given a family of triangles in the plane, find the smallest set of points such that each triangle in the family contains at least one of these points. The points are said to "pierce" the triangles.

Historical background

The problem of triangle piercing is a variant of the hitting set problem, which is a well-known problem in computer science and operations research. The triangle piercing problem is particularly interesting in the context of computational geometry, where it has applications in areas such as sensor networks and geographic information systems.

Mathematical formulation

Let \( \mathcal{T} \) be a family of triangles in the plane. A set \( P \) of points is called a piercing set for \( \mathcal{T} \) if every triangle \( T \in \mathcal{T} \) contains at least one point from \( P \). The goal is to find a piercing set of minimum cardinality.

Applications

Triangle piercing has applications in various fields, including:

• Wireless sensor networks: Ensuring coverage of a region by placing the minimum number of sensors.
• Geographic information systems: Efficiently managing spatial data by reducing redundancy.
• Computer graphics: Optimizing rendering by minimizing the number of necessary calculations.

Related problems

The triangle piercing problem is related to several other problems in combinatorial geometry, such as:

• The set cover problem, where the goal is to cover a universe of elements with the fewest number of sets.
• The art gallery problem, which involves determining the minimum number of guards required to cover an art gallery.

Related pages

• Combinatorial geometry
• Hitting set problem
• Set cover problem
• Art gallery problem

References

• Matou_ek, J. (2002). Lectures on Discrete Geometry. Springer-Verlag.
• Pach, J., & Agarwal, P. K. (1995). Combinatorial Geometry. Wiley-Interscience.