Differential geometry and parametric design

• Calculate the SVD decomposition of matrices – in particular regular 2×2 and 3×3 matrices.
• Apply basic kinematic concepts and methods to analyze rigid motions in plane and space.
• Find kinematically motivated parametrizations of curves and surfaces.
• Calculate lengths, areas, and volumes of parametrized objects.
• Calculate the Frenet-Serret data for curves in plane and space.
• Find and apply the first and second fundamental forms for parametrized surfaces.
• Calculate the Weingarten matrix, the principal curvatures, and the principal directions.
• Calculate the Gaussian and the mean curvature for parametrized surfaces.
• Apply the curvature concepts for curves and surfaces to analyze and solve design-related problems.
• Apply the general theory of surfaces for surfaces of revolution, ruled surfaces, and other design-motivated standard surfaces.
• Recognize and suggest applications of geometric methods for the solution of design problems.
• Apply the respective methods and concepts in an individually chosen project exercise and present the findings in a report.

Time-dependent deformation matrices and their SVD decompositions. Rotation matrices. Kinematics in plane and space. The Frenet-Serret basis for curves and the ensuing curvature and torsion. Parametrization of curves and surfaces, in particular via sweeping and rolling. Analysis of surfaces via the first and second fundamental form. The Weingarten matrix and the principal curvatures and principal directions. Ruled surfaces and developable surfaces. The Gauss curvature and mean curvature of surfaces. Special curves on surfaces: geodesics, curvature lines, asymptotic curves. Applications of these methods and concepts in architectural engineering and sculptural design. Applications of computer experiments, illustrations, and calculations to support all the learning elements of the course.

01002/01004/01005/01920

Lectures and exercises including computer experiments with Maple and Möbius. Project exercises for two weeks at the end of the semester.