Research On Key Technology In Shape Modification And Deformation Of Curves And Surfaces

Geometric modeling is the kernel of Computer Aided Design/Computer Aided Manufacturing (CAD/CAM) all the time. As an auxiliary means of geometric modeling, technique in shape modification and deformation of curves and surfaces is an important component of geometric modeling. It is well known that shape modeling of any complicated objects can not be accomplished at one stroke. It is a time-consuming process with repeated modification and editing. Therefore, simple and useful shape modification and deformation technique is very significant to reduce design period and improve modeling efficiency. Rearch on key technique in shape modification and deformation of curves and surfaces is investigated in depth in this paper, and the main research contents and achievements are as follows:By means of introducing some new conceptions including overlapped curve, redundant constraints and redundant curve et al, a new method for shape modification of NURBS curves based on the idea of Boolean Union is presented. First, a set of curves satisfying the geometric constrains at single parametric point are computed respectively, from which the overlapped curve is constructed. Second, redundant constraints are extracted. Finally, the modified curve is obtained which equals to redundant curve subtracted from overlapped curve. The modified curves can satisfy the geometric constraints, such as point and tangent vector constraints given by user, at more than one parametric point. Comparing with existing methods, the proposed approach has obvious geometric explanation.Through weighted-least-square optimization, a new shape modification approach of NURBS curves based on weight variation is presented, and the modified NURBS curve can satisfy the specified constraints, such as single point and multi-point constraints. By changing weighted coefficiencies, the shape of curve can also be further adjusted on the premise of satisfying the given constraits, so that the shape of modified curve seems more natural. For some special cases of shape modification for NURBS curve based on weight variation, analyses and provements are also given from the perspective of mathematics and geometry respectively.For commonly used two methods in shape modification of parametric surfaces, approach based on least-square and approach based on minimizing the bending energy of surfaces, united and explicit matrix forms are derived to compute new control vertice of the modified B-spline surfaces. The modified surfaces can satisfy the specified geometric constraints, such as single point, multi-point, normal vector, curve and surface constraints. Qualitative and quantitive comparitions and usage range of two shape modification techneques are given, and their applications in reverse engineering are also shown. When dealing with shape modification based on curve constraint, since algebra technology is used to simplify the composition and multiplication of B-spline base functions, the computation of Blossom is also avoided. So, real-time interactive modification can be carried out using the proposed methods.Applying the bar deformation in Mechanics of Materials to geometric deformation design and compute animation, a novel physics based deformation techniques is proposed. By means of interactively defining the elastic bar with circular cross section, the undeformed object is embedded in it globally or locally. After boundary constraints including the fixed and hinged support and external loads are imposed on the elastic bar, the object deforms wih the bar simultaneously. Stretch, bending, taping, twist and their combined deformation can be achieved using this method. The approach is simple and useful.Comparing with traditional physics based deformation, our method need less computation without solving lots of Langrange partical differential equations.Experimental results demonstrate the proposed methods.