ABSTRACT
Differential geometry is a classical mathematical area that has become very important for engineering applications in the recent decades. This importance is based on the rise of computer-aided visualization and geometry generation technologies. The fundamental problem of differential geometry, the finding of geodesic curves, has practical implications in manufacturing. Development of nonmathematical surfaces used in ships and airplanes has serious financial impact in reducing material waste and improving the quality of the surfaces. This chapter focuses on analytically solvable problems, the methods and concepts introduced provide a foundation applicable in various engineering areas. It demonstrates the difficulties of finding the geodesic curves even on regular surfaces like the sphere or the cylinder. On a general three-dimensional surface these difficulties increase significantly and may render using the differential equation of the geodesic curve unfeasible. Physicists use the space-time continuum as a four-dimensional (Minkowski) space and find geodesic paths.
