### Optimizing length of planar curves

###### Vojtěch Kloud^{1}, Dušan Bednařík^{2}

1 Drtinova 825/54b, Hradec Králové 503 11, Czech Republic;
2 Univerzita Hradec Králové, Rokitanského 62, Hradec Králové 500 03, Czech Republic,
* Corresponding author: vojtech.kloud@gmail.com

###### Key words:

Optimization, calculus of variations, planar curves, constraints, obstacles.###### Full text download:

Kloud PDF DOI: https://doi.org/10.51337/JASB20201228004
This article focuses on the problem of finding a shortest path in plane with obstacles. Problems of such nature occur for instance in robotics or transport and are of great importance. The problem is analyzed using the methods of mathematical analysis and calculus of variations. Definitions of basic concepts of the problem are given. From these definitions, useful properties, such as convexity of the length functional, are proven. These properties are used to show the existence of a solution in one of the considered cases of the problem. Other case of the problem was considered, where it is established under which conditions does a shortest path attain its general form and what this form looks like.

The article focuses on the problem of finding a shortest path in plane with obstacles. Problems of such nature occur for instance in robotics or transport and are of great importance. Their optimal solutions can save not only time of travel but also fuel costs. Problem of finding a shortest path in plane with obstacles is usually treated as a discrete problem, meaning that the plane is thought of as a grid of points, between which one can move only horizontally, vertically or diagonally. Thus the problem loses its smooth nature and such approach may seem too restrictive.
Because of this we treat the problem purely analytically or continuously, which in turn allows one to follow any smooth path in the plane without too many restrictions. Because of the lack of literature concerning such continuous approach, basic definitions of continuous obstacles are given. Also, the set of all potential solutions to a problem is defined. This set is called the set of all admissible curves and the goal is to find the shortest one.
Basics of the continuous version of the problem are treaded using known methods of mathematical analysis and calculus of variations. After giving appropriate definitions, basic case of the problem with only one obstacle is given. There, we use proof by contradiction to show which curve is the shortest one under the assumption that there exist one such curve. We also prove the existence of a solution to this problem using some useful derived properties of the length of admissible curves. In mathematics, the assumption of existence of a solution is a strong one and therefore proof of such nature is of great significance.

Vojtěch Kloud studies at První soukromé jazykové gymnázium in Hradec Králové. His main focus is the study of mathematics, more precisely mathematical analysis. He placed second with his thesis on optimal paths in the Czech nation round of Students professional activities - SPA (Středoškolská odborná činnost - SOČ) and became an Intel ISEF Finalist which was supposed to také place in May 2020 in Anaheim (USA). He was also rewarded for his thesis by an award from the Learned Society of the Czech Republic. Besides academic interests, Vojtěch competes as an athlete on international level in pole vault and decathlon, and is a former U18 national champion in pole vault. His plan is to attend a college in the STEM field in the United States or the Czech Republic while continuing to pursue his interests in athletics and music. This presented paper is an extract from his 2019 thesis of the same title.