Research, Geometric Modeling Method Based On Curves And Surfaces

In this paper,we research the geometric modeling on the parametric curve and surface.With the development of the computer industry and the geometric modeling technology,Some domains(aerospace, meteorology, medicine, surrounding observation) need research the surface construction restricted to the curve and the super-surface construction restricted to surface,which include four kinds of questions: 1) the smooth surface interpolation of three-dimension datas two varables function restricted to the curve.2) the smooth hyper-surface interpolation restricted to the surface .3) the construction of a kind of closed parametric surface with geometric continuity on the sphere .4) the parametric curve and surface interpolation deformation and free deformation.Firstly,we review the history of geometric modeling and analyse the advantage and disadvantage of the existed methods about the the geometric modeling on the parametric curve and surface. Then we research the new geometric modeling methodsWe present a modified radial function to settle the smooth surface interpolation of three-dimension datas two varables function restricted to the unit circle,We can obtain good approximation by adjusting the form parameter of the radial basis function;Correspoinding to the smooth surface interpolation of three-dimension datas two varables function restricted to common parametric curve,we put forword a method of mapping which from the parametric curve to the parametric axis by curve parametrizing. We translate the smooth surface interpolation of three-dimension datas two varables function restricted to parametric curve to the two-dimension datas one varable function curve interpolation on the parametric axis,we establish a C~1 continuous interepolation function by using the minimum norm method ,At last, mapping the function defined on the parametric axis to the parametric curve.Numetrical experimations and figure examples indicate that the two methods can obtain good approximation .We present a local approximation method which ensures that the function interpolates all given datas to settle the subject of smooth hyper-surface interpolation of four-dimension datas three varables function restricted to the sphere,At each given point on the sphere,we construct a correspoinding weigh function which possesseslocal support characteristic and a local approximation function which interpolates the function values of the point, we get a function defined on the sphere by adding up the local functions in the form of weight functions sum,Numerical experiment shows that the method have a good approximation.About the construction of a kind of closed parametric surface with geometric continuity on the sphere ,we present the construction method of parametric surface by using the trigonometric B splines tensor product o fit the datas on the sphere,and get the G~1 and G~2 continuous restriction conditions .About the parametric curve and surface interpolation deformation,we put forword a new deformation method: giving the extension vector function based on interpolation B-spline, translating the interpolation datas into the deformation direction vector, the method not only can make the deformed curve and surface interpolate the given points but also can control deformation region axactly and guarantee C~2 continuity between deformed and the undeformed regions.On the parametric curve and surface free deformation, we present a new free deformation method: giving the extension vector function based on B-spline in the form of the weights, we choose some auxiliary characteristic points on the curve and surface,we can translating the points into the main deformation direction vector. The method not only can can control deformation region axactly but also guarantee C2 continuity between deformed and the undeformed regions. Figure examples show that we can get plenty of free deformation results by adjusting the values of the weight and the characteristic points .Finally,we summarize the whole paper and point out the new research domains based on the content of...