Research On Constrained Deformation Technology Of Curves And Surfaces Based On Affine Transformation

Geometric modeling is an important research content in areas of CAD/CAM, CAGD, CG and soon. As an important part of geometric modeling, deformation technique of curves and surfaces is oneof the key techniques in the rapid development and innovative design of digital products. A flexibleand handy deformation technique is extremely significant to shorten the design cycle of products andimprove the efficiency of modeling. Based on the essence of the deformation and the idea of affinetransformation, research on constrained deformation technology of curves and surfaces is deeplystudied in this paper. The major contents and achievements of the study are as follows:In order to realize the constrained deformation of parametric curves, a new constraineddeformation algorithm based on scaling matrix is proposed for parametric curves. Firstly, constructinga new polynomial scaling factor with parameters, the new factor not only brings the excellence of theexisted scaling factors, but can also get its maximum in the region. And then, multiplying the equationof the curve by the scaling matrix, the constrained deformation and periodic deformation of the curvecan be achieved. The new algorithm is simple and easy to control without any auxiliary tools. It alsocan achieve good results in controlling the region of deformation, the amplitude of deformation, thedirection of deformation, the continuity and smoothness of junction between deformed andundeformed portions. The closure property can be kept in the deformations of Bézier curves, B-splinecurves and NURBS curves.According to scaling matrix, a new constrained deformation algorithm for parametric surfaces isproposed. On this basis, a periodic deformation algorithm is presented. By applying scaling matrix tothe pending equation of the surface, the algorithm can achieve the point-constraint deformation of thesurface. Every certain control parameter in the algorithm has its own geometric property. The newalgorithm can intuitively and effectively control the region of deformation, the region of peak value,the amplitude of deformation, the smoothness of boundary, etc. In order to obtain more deformationeffects of parametric surfaces, the region of deformation can be extended from circle area to rectanglearea. With the introduction of the ridge curves, the line-constraint deformation of parametric surfacescan be achieved by the ridge curves. Besides, the periodic deformation of parametric surfaces can alsobe achieved. The example proves that the algorithm is simple, intuitive and easy to control withoutany auxiliary tools. The deformation algorithm can flexibly realize the feature modeling design ofparametric surfaces. Extending from the constrained deformation algorithm for parametric surfaces, a new localconstrained deformation algorithm driven by parametric surfaces is proposed for triangular meshsurfaces. The mapping between the mesh surface and the parametric surface can be created after thetriangular mesh surface is parameterized to a specification parameter domain. And then, a B-splinesurface could be reconstructed from the triangular mesh surface. The constrained deformation of themesh surface can be achieved through the mapping and the constrained deformation method ofparametric surfaces. Based on the above method, the periodic deformation method and localconstrained deformation method with point-constraint and line-constraint are fulfilled by introducingsome different scaling factors. Compared with traditional deformation methods of triangular meshsurfaces, the new algorithm has definitely geometric meanings, and is easy to operate. At the sametime, it can exactly control the region of deformation, the amplitude of deformation and thesmoothness of boundary, etc.Combining with the affine transformation and the generalized metaball deformation technology,a constrained deformation algorithm for mesh models based on space transformation is carried out. Byspecifying a series of constraint sources and targets, applying the constrained deformable model withpotential function to the skeleton model or vertexes of the mesh, two constrained deformationalgorithms can be achieved. One is based on skeleton-driven, and the other is based on geodesicdistance. The former constructs a skeleton model which has the same topological relation to the meshmodel, and then the deformation of the mesh model can be obtained by what has built. The latterobtains the deformation weights of the vertices in the deformation region, which can be determined bythe geodesic distances between the vertices and the constraint sources, and then the constraineddeformation of the mesh model can be achieved directly. Both of the two algorithms are simple andintuitive, and can commendably meet the need of specified geometric constraints and flexibly realizethe model deformation such as rotation, translation, scaling and other operations. The former is moresuitable for the large deformation of the mesh models with obvious skeleton features, but theconstraint source is just limited to one point. The latter is suitable for all mesh models, and theconstraint sources may consist of points, lines and faces.