ABSTRACT

The chapter addresses the fundamental problem of differential geometry, the finding of geodesic curves, that has practical implications in manufacturing. Physicists use the space-time continuum as a four-dimensional space and find geodesic paths in that space. On a general three-dimensional surface, these difficulties increase significantly and may render using the differential equation of the geodesic curve unfeasible. Differential geometry is a classical mathematical area that has become very important for engineering applications. This importance is based on the rise of computer-aided visualization and geometry generation technologies. Finding a geodesic curve on a surface is a classical problem of differential geometry. The concept of geodesic curves may be generalized to spaces of higher dimensions. The geodesic curve notation, however, while justified on a three-dimensional surface, needs to be generalized as well.