2023 Faculty Courses School of Computing Department of Mathematical and Computing Science Graduate major in Mathematical and Computing Science
Topics in Geometry
- Academic unit or major
- Graduate major in Mathematical and Computing Science
- Instructor(s)
- Masaaki Umehara / Shinya Nishibata / Hideyuki Miura / Toshiaki Murofushi / Sakie Suzuki
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Mon (S3-207(S322)) / 5-6 Thu (S3-207(S322))
- Class
- -
- Course Code
- MCS.T504
- Number of credits
- 200
- Course offered
- 2023
- Offered quarter
- 2Q
- Syllabus updated
- Jul 8, 2025
- Language
- Japanese
Syllabus
Course overview and goals
When we recognize planar curves and surfaces as wave fronts, and we can consider their time evolutions. Singular points appear frequently. In this course, we review differential geometry of curves and surfaces, and also give an introduction to singularities on curves and surfaces. We introduce criteria for important singularities as well as their fundamental properties. Students attending this course will have a better familiarity with curves, surfaces and the concept of manifolds. The course itself is almost fully self-contained. So it is possible to join this course without prior knowledge of these materials.
Course description and aims
[Theme] The fundamental properties of curves and surfaces are explained from the viewpoint of differential geometry. In particular, we explain several types of curvatures on curves and surfaces. We also explain topological properties, criteria and geometric properties of singularities appearing in curves and surfaces. In each class, we try to explain the material by showing examples, sometimes using computers.
[Goal] The students are expected to understand the fundamentals of curves and surfaces for handling geometric structures appearing in mathematical and computing science, and also to be able to apply them to practical problems.
Keywords
curves, surfaces, singular points, Gaussian curvature, wave fronts
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
The course provides the fundamentals of curves, surfaces and singularities.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | planar curves (singular points, regular points, curvature) | Understand the contents covered by the lecture. |
| Class 2 | planar curves (four vertex theorem, rotation index) | Understand the contents covered by the lecture. |
| Class 3 | parallel curves, evolutes, cusps as singularities | Understand the contents covered by the lecture. |
| Class 4 | wave fronts as planar curves | Understand the contents covered by the lecture. |
| Class 5 | behaviour of curvature functions near singular points | Understand the contents covered by the lecture. |
| Class 6 | a criterion for cusps and its applications | Understand the contents covered by the lecture. |
| Class 7 | fundamentals of surface theory 1 (the first and second fundamental forms) | Understand the contents covered by the lecture. |
| Class 8 | fundamentals of surface theory 2 (Gaussian curvature, mean curvature, principal curvature) | Understand the contents covered by the lecture. |
| Class 9 | Gaussian curvature and mean curvature of parallel surfaces | Understand the contents covered by the lecture. |
| Class 10 | wave fronts as surfaces | Understand the contents covered by the lecture. |
| Class 11 | important singularities appearing in surfaces | Understand the contents covered by the lecture. |
| Class 12 | a proof of the criterion for cusps | Understand the contents covered by the lecture. |
| Class 13 | a proof of the criterion for cross caps | Understand the contents covered by the lecture. |
| Class 14 | Gauss-Bonnet type theorems for closed wave fronts | Understand the contents covered by the lecture. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
Masaaki Umehara and Kotaro Yamada, Curves and surfaces, World Scientific, 2017 (The translation of Japanese book).
Masaaki Umehara, Kentaro Saji and Kotaro Yamada, Differential Geometry of Curves and Surfaces with Singularities, World Scientific, 2022 (The translation of Japanese book).
Evaluation methods and criteria
Final report and class attendance
Related courses
- MCS.T331 ： Discrete Mathematics
Prerequisites
The student has better to have a knowledge of Topology and vector analysis.