VISUAL AND INTERACTIVE PARALLEL TRANSPORT ON SURFACES WITH MATHEMATICA

In some areas of computer graphics, computer vision, and data compression, familiarity with classical differential geometry is required to develop efficient algorithms. Learning differential geometry involves considerable amount of calculations with pencil and paper, and yet gaining intuitive understanding is often a long way off. In engineering department education, we naturally seek an alternative way to attain the goal, which is the main theme of this paper.

Curvedness of surfaces appears in various ways. We can easily recall that the sum of internal angles of a geodesic triangle on a sphere is greater than 180 degree, which shows the sphere has a positive Gaussian curvature. Curvedness can also be measured by parallel translation of tangent vectors. Parallel transport or the Levi-Civita connection is one of the basic concepts in classical differential geometry. In a curved space, a vector transported parallel from the initial point to the destination along two different paths may result in different vectors at the destination point. The amount of this difference depends on the curvedness of the space and the choice of paths. We present an interactive visualizer toolkit for developing teaching and self-learning materials in differential geometry of surfaces focusing on parallel transport targeting science and engineering department students. We give also present sample visualizers for the curvedness of surfaces.

Understanding parallel transport in a curved space is not straightforward for undergraduate learners since they have to solve differential equations of second order with initial conditions or boundary conditions according to the situations they are to tackle. Visualization of the effect of curvedness is much needed. The result of parallel transport varies according to the path. So, visualization should incorporate this variation of the result caused by the change of a path. Preferably further, the effect should appear promptly and smoothly when the user changes the configuration of a path continuously.

Our toolkit is designed to support teachers and course material developers to attain the goals above. It is built on top of Mathematica, a widely used computer algebra system, consisting of a set of library functions and accompanied by a variety of samples for developing interactive visualizers.

The toolkit makes full use of 3D graphics and depends heavily on the dynamic object technology of Mathematica, not to mention its symbolic computation power. The toolkit also depends on the advanced numerical solution technology that provides numerical solutions of differential equations in the form of interpolation functions, which enables us to treat the numerical solutions not as an array of real numbers but as a normal function; we can differentiate them, plot them, and further compose them. We make full use of this feature.

The toolkit is accompanied by a set of design patterns with which we can not only effectively develop teaching materials but also let students write their own code in hands-on laboratory sessions. Through the laboratory sessions, students gain deep insight both in mathematics and computing science.