Triangle piercing: Difference between revisions

Revision as of 12:01, 9 February 2025

Mathematical problem involving triangles

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 pages

References

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

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-'''Triangle Piercing''' is a type of [[female genital piercing]] that passes beneath the base of the [[clitoral hood]] tissue, where it meets the [[inner labia]], and goes underneath the [[clitoris]]. It is known for the potential to increase sexual pleasure due to its placement. However, not all anatomies are suited for this piercing, and it requires a consultation with an experienced piercer to assess suitability.
+{{Short description|Mathematical problem involving triangles}}
+{{Use dmy dates|date=October 2023}}
-==Anatomy and Suitability==
+[[File:Nap08 694.jpg|thumb|right|A visual representation of the triangle piercing problem.]]
-The triangle piercing is unique in that it requires specific anatomical features to be safely performed. The piercing passes horizontally beneath the clitoral shaft, which means there must be enough space between the clitoris and the pubic bone. This space is necessary to avoid nerve damage and to ensure the jewelry can sit comfortably without causing pressure on the clitoris itself.
+'''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.
-==Procedure==
+==Problem statement==
-The procedure for a triangle piercing should always be performed by a professional piercer with experience in genital piercings. It typically involves the following steps:
+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.
-# Consultation to assess anatomy and discuss risks.
-# Selection of appropriate jewelry, usually a curved barbell or a ring.
-# Cleaning and marking the piercing site.
-# Piercing with a sterile needle, followed by jewelry insertion.
-Pain levels, healing times, and aftercare requirements can vary significantly between individuals. A general healing time is between 2 to 3 months, but complete internal healing may take longer.
+==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]].
-==Risks and Considerations==
+==Mathematical formulation==
-As with any piercing, there are risks involved with a triangle piercing. These include infection, prolonged healing, rejection or migration of the jewelry, and damage to the clitoral nerve. Proper aftercare is crucial to minimize these risks. This includes regular cleaning with saline solution and avoiding any irritation to the area during the healing process.
+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.
-Sexual activity should be approached with caution during the healing period to prevent complications. Additionally, the choice of jewelry is important; it should be made of a biocompatible material and of appropriate size to avoid embedding or other issues.
+==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.
-==Cultural and Personal Significance==
+==Related problems==
-The decision to get a triangle piercing is often personal, with motivations ranging from aesthetic preferences to the desire for enhanced sexual stimulation. It is also seen as a form of body autonomy and self-expression. However, cultural perceptions of genital piercings vary widely, and they can be subject to stigma or misunderstanding.
+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.
-==See Also==
+==Related pages==
-* [[Clitoral hood piercing]]
+* [[Combinatorial geometry]]
-* [[Labia piercing]]
+* [[Hitting set problem]]
-* [[Body modification]]
+* [[Set cover problem]]
+* [[Art gallery problem]]
-[[Category:Body piercings]]
+==References==
-[[Category:Female genital piercings]]
+* Matou_ek, J. (2002). ''Lectures on Discrete Geometry''. Springer-Verlag.
+* Pach, J., & Agarwal, P. K. (1995). ''Combinatorial Geometry''. Wiley-Interscience.
-{{Medicine-stub}}
+[[Category:Combinatorial geometry]]
+[[Category:Computational geometry]]