Algebraic Surface Blending Based On Theory Of Resulant Elimination

Nowadays, with the development of the automobile industry and mold manufacturing inindustrial design field, the product are not only required with perfect function, but also have thehigh requirment on the design and visual effect. It focus on how to find a simple and effectivemethod so as to achieve the effect of product. At present, many scholars at home and abroad domany works on algebraic blending methods. However resultant as one of the efficientelimination method in the symbolic computation was seldom used in algebraic surface blending.This paper will use the efficient and simple resultant elimination theory to algebraic surfacewhose degree is as low as possible. This method needn’t to consider the variable in algebraicsurface, it only requires to eliminate the new variable in the construction of blending equations.Therefore, it achieves simple and high efficiency. This paper’s main research is about threeaspects, the first is about two algebraic surface blending, the second is about trying to findlow-degree algebraic blending conditions, and the last is about more than two algebraic surfaceblending. Details are as follows:1. Introduce the homotopy mapping theory, construct two algebraic surface equation. Using thetheory of Sylvester resultant to eliminate the variable, so we can get the blending equation, thenintroduce the Grobner Basis method to check the factors in the equation whether or not be theblending equation. Sometimes the blending surface often appears not smooth or uncontinuous.So we introduce the parameter adjustment. In the process of adjustment, we add the mixingfactor and compensation factor, successive adjustment to find out the regular, so as to improvethe surface blending effect, then achieve it by adjusting the parameters in Maple.2. Discuss the condition that the degree of blending surface is lowest and present the relationshipbetween the coefficients of auxiliary surface hiand that of given main surface. Firstly, twocylinders, when their axes vertical and we drive the result on the base of this, we discuss theresult. On the base of this, we discuss the general cases and the conditions are given whenblending two quadeic surface and auxiliary surface.3. Use Sylvester and Dixon resultant elimination method respectively to implement threesurfaces blending with G1smooth condition, and get their respective characteristics on thecondition of different graphics concatenated. Then, apply Dixon resultant blending algorithm tothe blending of the four surfaces and we obtain a blending surface with degree12. Finally, wedemonstrate that the Dixon method is efficient than Sylvester method by using examples.