Doctor of philosophy in mathematics: Shortest paths along a sequence of line segments and connected orthogonal convex hulls

In the dissertation, we consider the problem of finding the shortest path between two points along a sequence of adjacent triangles in a general setting. The sequence of triangles is replaced by a sequence of ordered line segments. The 3D space is replaced by a Euclidean space.
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Doctor of philosophy in mathematics: Shortest paths along a sequence of line segments and connected orthogonal convex hullsVIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS PHONG THI THU HUYENSHORTEST PATHS ALONG A SEQUENCE OF LINE SEGMENTS ANDCONNECTED ORTHOGONAL CONVEX HULLS DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS PHONG THI THU HUYENSHORTEST PATHS ALONG A SEQUENCE OF LINE SEGMENTS ANDCONNECTED ORTHOGONAL CONVEX HULLS Speciality: Applied Mathematics Speciality code: 9 46 01 12 DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Associate Professor PHAN THANH AN HANOI - 2021ConfirmationThis dissertation was written on the basis of my research works carried out atInstitute of Mathematics, Vietnam Academy of Science and Technology, un-der the supervision of Associate Professor Phan Thanh An. All the presentedresults have never been published by others. September 17, 2021 The author Phong Thi Thu Huyen iAcknowledgment First and foremost, I would like to thank my academic advisor, AssociateProfessor Phan Thanh An, for his guidance and constant encouragement. The wonderful research environment of the Institute of Mathematics, Viet-nam Academy of Science and Technology, and the excellence of its staff havehelped me to complete this work within the schedule. I would like to expressmy special appreciation to Professor Hoang Xuan Phu, Professor NguyenDong Yen, Associate Professor Ta Duy Phuong, and other members of theweekly seminar at Department of Numerical Analysis and Scientific Com-puting, Institute of Mathematics, as well as all the members of AssociateProfessor Phan Thanh An’s research group for their valuable comments andsuggestions on my research results. In particular, I would like to expressmy sincere thanks to Associate Professor Nguyen Ngoc Hai and PhD studentNguyen Thi Le for their significant comments and suggestions concerning theresearch related to Chapters 1, 2 and Chapter 3 of this dissertation. I would like to thank the Professor Nguyen Dong Yen, Doctor Hoang NamDung, Doctor Nguyen Duc Manh, Doctor Le Xuan Thanh, Associate Profes-sor Nguyen Nang Tam, Associate Professor Nguyen Thanh Trung, and DoctorLe Hai Yen, and the two anonymous referees, for their careful readings of thisdissertation and valuable comments. Finally, I would like to thank my family for their endless love and uncon-ditional support. iiContentsTable of Notation vList of Figures viIntroduction viiiChapter 0. Preliminaries 1 0.1 Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Graham’s Convex Hull Algorithm . . . . . . . . . . . . . . . . 3Chapter 1. Shortest Paths with respect to a Sequence of Line Segments in Euclidean Spaces 9 1.1 Shortest Paths with respect to a Sequence of Ordered Line Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Concatenation of Two Shortest Paths . . . . . . . . . . . . . . 21 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Chapter 2. Straightest Paths on a Sequence of Adjacent Poly- gons 36 2.1 Straightest Paths . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 An Initial Value Problem on a Sequence of Adjacent Polygons 38 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Chapter 3. Finding the Connected Orthogonal Convex Hull of a Finite Planar Point Set 46 3.1 Orthogonal Convex Sets and their Properties . . . . . . . . . . 46 iii 3.2 Construction of the Connected Orthogonal Convex Hull of a Finite Planar Point Set . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Algorithm, Impleme ...