The investigator and his colleagues study the computer generation of higher dimensional curves that interpolate a given sequence of points and are free from unnecessary oscillations. From the sequence, simple geometric constraints are found on all interpolating curves that minimize oscillations subject to the general interpolation constraints and the associated theory allows for the removal of redundant constraints. Computer algorithms incorporating the geometric constraints are used to find entire families of piecewise rational curves from which particular curves are selected subject to additional application specific constraints. The study is motivated by applications that involve the design of paths along which objects are to move, such as robotics and animation. In these applications it is desirable to create motions that are smooth and free from unnecessary oscillations. This is important, for example, to reduce the energy used in controlling robots or to create accurate animations of images from measurements such as those taken to produce medical imaging data. For such applications, minimizing the number of unnecessary oscillations is accomplished by imposing the appropriate mathematical properties on a higher dimensional curve that governs the changes in certain important variables that dictate the motion. Algorithms for the automatic computer generation of these oscillation free higher dimensional curves can be used as a basis for many different applications.
