Generation And Properties For Curves On Smooth Surfaces

The interpolation, approximation and other things for curves on smooth surfaces with computer are studied. The triangular rational Bezier representation of cylindrical helix and the generation of the NURBS curves on sphere are discussed in detail.Firstly, the thesis devotes to induct and develop the methods and properties for the rational triangular Bezier and B-splines in polar coordinates system. The de Boor algorithm and a knot insertion algorithm are also given for B-spline curves.Secondly, the various conic arcs and conies are generated by the rational triangular Bezier and the B-spline method in polar coordinates system.Thirdly, the spatial cylindrical helix is approximated by rational triangular Bezier curve in cylindrical coordinate system. Two methods are given to get the control vertexes, by making the two approximated helix segments GC2 continuous at their connecting point and leting the theoretical helix and the approximated one coincident at five points respectively. The theory error of one approximated helix equals zero at the two end -points and the center and the other's equals zero at five points. How the parameters of above two helixes affect their shapes is discussed. Furthermore, error analysis and distribution are given. Any prescribed precision can be arrived by the approximated helixes.Finally, a method is put forward to construct the NURBS curves on sphere, which extends the de Boor recursive algorithm in R3 to one on the sphere by replacing the geodesic distances for the lines and studies their many geometric properties analogous to those in Euclidean spaces, such as the differential property, the local property, the parameter invariance under a projective transformation, and so on. It is also pointed out that these curves are devoid of split property possessed by NURBS curves in Euclidean spaces. At the same time, a knot insertion algorithm is also given for the NURBS curves on sphere, then interpolation for curves on sphere is presented by spherical quadratic and cubic uniform B-spline. However, the quadratic curve is only C?continuous, while the cubic's is C1 continuous. As an application of the algorithm and the properties, some examples are given.