Modeling with Differential Geometry

COURSE DESCRIPTION

Differential geometry uses the tools of multivariate calculus (and linear algebra) to study the “geometry” of non-linear spaces. Roughly speaking, the aim is to study and understand the possible shapes of curves and surfaces in space. Soon enough, a new concept (absent from Euclidean geometry) emerges: curvature; a good part of the course will be devoted to understanding this through examples and classical results.

The first part of the course studies curves, both in the plane and in 3-space, while the second part focuses on the geometry of surfaces in R3. Throughout the course, we will utilize the computer software package SageMath, both to make long or tedious computations easier and to gain visual intuition.

The course is complemented with examples that show how the mathematics being learned can be used to solve applied problems from different sources. In addition, the last topic of the course will be an introduction to quaternion arithmetic and how to use this formalism to understand rigid transformations of the space and applications to mechanics and robotics.

The goal of this course is for students to achieve a working intuition of the geometry of curves and surfaces (through theorems, examples, the use of visualization tools and applications), that have wide applicability.

 

Prerequisites

Students should have taken a multivariate calculus course and a linear algebra (in 2 and 3 dimensions) course.

A first course in ordinary differential equations is recommended.

COURSE GOALS

On completion of the course, students will:

  • understand the basic features of curves (in the plane, in space, inside surfaces): The invariants that determine them, many examples that are ubiquitous, how to construct and draw them, how to use certain types of curves to interpolate data, how they have been used in applications;
  • have a working intuition of surfaces in 3-space: How to parametrize them, what is the curvature and why it is an obstruction to draw accurate maps. Students will have learned the answers to beautiful classical problems that still serve as guides for contemporary science and technology. They will also have had first-hand experience with examples of surfaces that are used in modeling and applications.
    COURSE CONTENT

    1. Curves (5 weeks)

    Curves in the plane and space; local theory; global results; applications: involute gears, ribbons and supercoiling of DNA.

    2. Surfaces in 3-space (7 weeks)

    Parametrized surfaces; first fundamental form (the metric); examples; Gauss map, curvature; ruled and developable surfaces; intrinsic geometry, parallel transport and geodesics; global results, Gauss-Bonnet theorem; applications: hyperboloid gears, an industrial packing problem, rigid body motions, maps (or why is it difficult to paper wrap a ball).

    3. Quaternions (2 weeks)

    Quaternions, rotations, mechanics and applications to robotics.

     

    Bibliography

    1. Shifrin, Theodore (2016). Differential Geometry: A First Course in Curves and Surfaces.
    2. Oprea, John (2007). Differential Geometry and its Applications, Mathematical Association of America. 
    3. Ghomi, Mohammad. Lecture Notes on Differential Geometry.
    4. Murray, Richard M.; Zexiang, Li; Shankar Sastry, S. (1994). A Mathematical Introduction to Robotic Manipulation, CRC Press.
    5. Hilbert, D.; Cohn-Vossen, S. (1999). Geometry and the Imagination, AMS Chelsea Publishing.

    Support Sessions

    2 hours per week with a teaching assistant

    Grading

    There will be weekly homework counting for 20% of the final grade. The other 80% will be distributed between two midterm exams (20% each) and a final exam (40%)