Abstract:

In some area of engineering, familiarity with classical differential geometry of curves and surfaces is preferable and related courses are included in the curriculum. This paper presents usage patterns of computer algebra systems to design an interactive mathematics class, taking up topics from differential geometry.

Learning differential geometry of curves involves getting familiar with calculating tangents, normals, and binormals with pencil and paper. Learners should carry out mathematical experiments with concrete examples as many times as they need to gain intuitive understanding of geometrical concepts such as curvature, torsion, and moving frames.

The traditional approach is still important in educational programs in science and any branches of engineering disciplines. However, there is a common and typical drawback. Tractable exercise problems with pencil and paper are very difficult to find except for in the classical textbooks, and they are very limited in number. We need more exercise problems to allow the learners to iterate mathematical experiments until they become satisfied with their intuitive understandings. Further, mathematical experiments should be a kind of activity that keeps the learners motivated.

To fulfill these requirements, we propose a new approach to these practices in classrooms and online courses. We present a special toolkit that assists the learners to gain geometrical insights through designing their own examples of curves and creating impressive animations. The toolkit also provides the teachers with facilities to prepare presentation materials for use in class.

Our method of introducing geometrical concepts depends heavily on computer calculus and animation with Mathematica. Students are asked to design a parametric curve with analytic expressions or possibly piecewise analytic ones. Then, they select an aircraft, which is a simple 3D-graphics object, from the set of predesigned 3D objects. Next, with the aid of Mathematica, they calculate the tangent, normal, and binormal vector field along the curve and make the aircraft fly along the curve with its nose pointing in the tangent direction, its wings kept parallel to the normal, the pilot’s head pointing to the binormal direction. This way, they can feel in touch with tangents, normals, binormals, curvatures, and torsions.

Our toolkit consists of a collection of library functions and a set of instruction patterns in class. The toolkit gives firstly a set of ready-made materials for learners and teachers. It gives also a set of facilities for course designers to extend the collection of impressive teaching materials.

Keywords:

Visual, interactive, differential geometry, mathematical experiments, mathematics laboratory, engineering education.