The Constructions And Applications Of Subdivision Curves

In recent years, the subdivision methods have been widely used in many areasincluding computer aided geometric design, computer graphics, animation andrelated areas. This thesis designs several effective curve subdivision schemes, andthe major work are as follows:Firstly, a modified three point ternary interpolation subdivision scheme isdescribed that can deal with open control polygons unlike the original three pointinterpolation subdivision scheme. The precision set and convergence analysis arepresented.Secondly, considering that cosx, sinx and coshx, sinhx can be unified byextending the calculation to complex numbers, we design a non-stationary uniformternary three point reproducing conics subdivision scheme. Its convergence andsupport are discussed.Thirdly, we present a unified ternary three point subdivision scheme forinterpolating and approximating by introducing a tension parameter. The tensionparameter can be used to control the shape of the limit curve.Fourthly, a new binary five point approximating subdivision scheme isdescribed. The generating polynomial method has been used to investigate theuniform convergence andC~k-continuity of this subdivision scheme. Thesubdivision scheme generates a family ofC~n(n=1,2,3,4,5) limiting curves for certainrange of tension parameterμ and aC~7limiting curves for μ=9/256.Fifthly, a new relaxation of binary four subdivision scheme is presented,which keeps the second-order divided difference at the old vertices unchangedwhen the new vertices are inserted. Using the generating polynomial of subdivisionscheme, we show that the limit curve generated by our proposed scheme isC~3continuous. Shape preserving of curve is an important subject in the geometricshape design. Hence, conditions on the initial points for the monotonicitypreserving and convexity preserving of the given limit curve are discussed,respectively.