A Mathematical Investigation On A Strict Upper Bound For The Maximum Number Of Intersections Of Varying Number Of Segments From Each Vertex Equally Dividing The Angles Of A Triangle

Quezon City Science High School

Quezon City Science High School

STEM A Technical Committee

Edited by: Erika Martin & Keith Tecson

Abstract

Abstract

Intersections of lines within a polygon have significant applications in the fields of computer graphics, architecture, and systems engineering.

Precision and accuracy are crucial in these fields to ensure the stability and effectiveness of the projects. Understanding the properties of intersections made by lines can help optimize and improve the accuracy of calculations in quantitatively inclined

fields. This study aims to find a closed-form expression that acts as a

strict upper bound for the maximum number of intersections made by

varying numbers of segments that equally divide the vertex angles of a triangle. The study uses observations of cevians and concurrent points to derive the desired expressions given below:

S = ab + bc + ac - 1, if a, b, and c are all odd

S = ab + bc + ac + 1, if a, b, and c are not all odd

wherein a, b, and c represent the number of angle dividers from vertices A, B, and C, respectively, while S is a strict upper bound for the maximum number of intersections of varying numbers of segments in each vertex equally dividing the angles of a triangle.

Discrete geometry is a branch of discrete mathematics focused on the properties of limited-generated geometric objects, such as the configuration of lines and hyperplanes in Euclidean and other spaces. Many problems in the branch of discrete geometry emerge from questions in computational geometry related to algorithms for analyzing geometric structures. Studying and investigating these problems will help further advance known methods in combinatorial analysis, mathematical modeling, and computational geometry.

Computational geometry is a constitutive field of mathematics and computer science that deals with algorithmic solutions to geometry problems. One of the fundamental problems in this field is determining the number of intersections of straight lines inside a triangle. The calculation for such intersections plays an important role in the field of realistic rendering, physics simulation, modeling, and collision detection, and can also produce innovations in processor architecture.

This study focuses on finding a strict upper bound for the maximum number of intersections of varying numbers of segments equally dividing each vertex angle of a triangle and validating it using known methods in discrete mathematics. Its proof revolves around cevians and the fundamental counting principle.

Although purely theoretical, the researchers expect that this study will be useful in the continuation of future research on angle-dividing segments within polygons and in understanding the properties of polygons, specifically triangles. The findings of this study will also be beneficial in the fields of computational geometry, computer graphics, road map topology, and structural integrity, as the properties of the maximum number of intersections can be used to develop algorithms for 3D rendering programs and simulations, monitor and manage congestion in transportation networks, and aid in assessing the stability of infrastructure.

The study was conducted at the Library and Mathematics Laboratory of Quezon City Science High School from November 13 to November 27, 2023. The researchers utilized the Geogebra website to generate triangles with varying angle dividers from each vertex. The researchers produced figures involving three types of triangles, which they classified into the following cases: case 1 (equilateral triangles), case 2 (isosceles triangles), and case 3 (scalene triangles). Case 1 served as the primary basis for the study.

The researchers collected data by counting the maximum number of intersections between angle dividers of the triangles in the figures. The recorded data was organized in Google Docs for further analysis.

Based on the collected data on the maximum number of intersections between the angle dividers, the researchers identified patterns that were then used to formulate a closed-form expression or conjecture.

The expression was also separated into cases to account for differing patterns in the data. The researchers verified the validity of the closed-form expression using mathematical induction. Established postulates and theorems regarding cevians and the fundamental counting principle were also utilized to ensure that the closed-form expression is accurate for any number of angle dividers per vertex of a triangle.

For verifying the conjecture, smaller cases that may serve as a basis were considered. From the generated figures, it was observed that if one of the variables a, b, and c are not all odd before and after one of the variables is increased by 1, then the maximum number of intersections is increased by the sum of the other two unchanged variables. If the three variables are all odd after one of the variables is increased by one, then the maximum number of intersections is increased by the sum of the other two unchanged variables plus 2. If the three variables are all odd after one of the variables is increased by 1, then the maximum number of intersections is increased by

the sum of the other two unchanged variables minus 2.

For verifying the conjecture, smaller cases that may serve as a basis were considered. From the generated figures, it was observed that if one of the variables a, b, and c are not all odd before and after one of the variables is increased by 1, then the maximum number of intersections is increased by the sum of the other two unchanged variables. If the three variables are all odd after one of the variables is increased by one, then the maximum number of intersections is increased by the sum of the other two unchanged variables plus 2. If the three variables are all odd after one of the variables is increased by 1, then the maximum number of intersections is increased by

In this mathematical investigation, the researchers were able to derive a closed-form expression that computes a strict upper bound for the maximum number of intersections of varying numbers of segments in each vertex equally dividing the angles of a triangle. The formula below was derived by utilizing various techniques, including combinations and the fundamental principle of counting, based on the researchers’ initial observations.

S = ab + bc + ac - 1, if a, b, and c are all odd

S = ab + bc + ac + 1, if a, b, and c are not all odd

Wherein a, b, and c are the number of angle dividers from vertices A, B, and C, respectively and S is a strict upper bound for the maximum number of intersections of varying numbers of segments in each vertex equally dividing the angles of a triangle.

Conclusion

Conclusion

[1] Cornell University. (n.d.). Combinatorics and Discrete Geometry. In Department of

Mathematics. Retrieved from

https://math.cornell.edu/research/combinatorics-and-discrete-geometry

[2] Selimi, A., & Saracevic, M. (2018). Computational Geometry Applications. Southeast

Europe Journal of Soft Computing, 7(2). https://doi.org/10.21533/scjournal.v7i2.159

[3] Segura, R., & Feito, F. (2001). Algorithms to test ray-triangle intersection. Comparative

study. Retrieved from https://www.researchgate.net/publication/221546368

[4] Havel, J., & Herout, A. (2010). Yet Faster Ray-Triangle Intersection (Using SSE4).

Retrieved from https://www.researchgate.net/publication/41910471

[5] Sabharwal, C., Leopold, J., & McGeehan, D. (2013). Triangle-Triangle Intersection

Determination and Classification to Support Qualitative Spatial Reasoning. Retrieved

from https://www.scielo.org.mx/pdf/poli/n48/n48a3.pdf