Isometric Piecewise Polynomial Curves

The main preoccupations of research in computer-aided geometric design have been on shape-speciication techniques for polynomial curves and surfaces, and on the continuity between segments or patches. When modelling with such techniques, curves and surfaces can be compressed or expanded arbitrarily. There has been relatively little work on interacting with direct spatial properties of curves and surfaces, such as their arc length or surface area. As a rst step, we derive families of paramet-ric piecewise polynomial curves that satisfy various positional and tangential constraints together with arc-length constraints. We call these curves isometric curves. A space curve is deened as a sequence of polynomial curve segments, each of which is deened by the familiar Hermite or B ezier constraints for cubic polynomials; as well, each segment is constrained to have a speciied arc length. We demonstrate that this class of curves is attractive and stable. We also describe the numerical techniques used that are suucient for achieving realtime interaction with these curves on low-end workstations.