Blending Of Three Quadratic Surfaces And Parametrization Of A Kind Of Quartic Implicit Algebraic Surfaces

Blending of surfaces is one of the core contents in Computer Aided Geometric Design. In this paper, we discuss the blending of three quadratic surfaces and parametrization of a kind of quartic implicit algebraic surfaces.In Chapter 2 we discuss the blending of three quadratic surfaces and obtain their GC1 blending under two conditions, construct the GC1 blending from their GC0 blending or construct the GC1 blending directly.On on hand, we give the general description of six kinds of pipe surfaces which take the coordinate x,y and z as axis. Here the six kinds of pipe surfaces are cylinder, sphere, circular cone, hyperboloid of one sheet, hyperboloid of two sheets and circular paraboloid. Our description classify into two classes: circular paraboloid and the other five. In regard ,o the four cases when blending three of the two classes, we obtain their GC1 blending S(f)B)y means of the control surface S(g), where f = ug + vh1h2h3, hi(i = 1,2,3) is clipping )lane, u and v are quadratic and linear polynomials of hi(i = 1,2,3), respectively. And theoomputation results show that u is definitely homogeneous for all these four cases.On the other hand, we obtain the GC1 blending 5(f) directly through solving the linear iquations and discuss the free parameters which can control and influence the shape and >roperty of 5(f).Since the visualization of implicit algebraic surface is still a difficult problem, to give onvenient and effective parametrization for some kind of concrete implicit surfaces is doubtlessly neaningful. In Chapter 3, we study the parametrization of a kind of GC1 blending of three [uadratic surfaces and give the parametrization algorithm.It used to be a process of intersect a certain plane pencil with the blending surface and get the parametrization of the blending by means of parametering their intersection when parameter the blending of two pipe surfaces. It seems natural to use this idea into the parametrization of the blending of three pipe surfaces, however, the blending of three pipe surfaces is not unique and the unappropriate choice of parameters can lead to singular blending surface. So it is not easy to turn this idea into reality.We creatively give our parametrization by using the ControlSphere-based expression of the blending surface(Wu[l]) and the plane pencil together. The intersection of the certain plane pencil and the blending surface are a planar cubic curve and a line each time, and it is interesting that we can reduce the line directly through the ControlSphere-based expression and rotation transformation of coordinate is no more needed (in the case of the parametrizaion of two pipe blending, rotation transformation of coordinate is needed to reduce a line from the intersect).For the quadric blending S(f), we study the part of 5(f) which join up with S(gl)(i = 1,2,3) and form a connected closed compounding surface with S(hi)(i = 1,2,3). We call it the Valid part' of the blending.Our basic idea is to use the plane pencils which contains any one of S(hj nhkk)(j,k = 1, 2,3; j < k) to intersect 5(f). Without loss of generality, we start from S(/h h2), using the plane pencil S(h(0), which sweeping from S(h2) to S(h1), to intersect the blending surface S(f), then the intersection of S(h(0) and S(f) are composed of a planar cubic curve and the line 5(- During the sweeping, the cubic curve can be closed or unclosd, we haveTheorem 1 Intersect S(h(6)) with S(f), if 6 , then the cubic intersect curve is closed: if 9 fa. B. then the r.iinip. i.i ii.nr.lo.sp.d. whp.rc.Next we are going to parameter the cubic intersect curve. It used to be a IntersectPoint-solving problem of intersect a set of radioactive rays with the quadratic curve when parameter the blending of two pipe, but the method is not suitable to parameter the cubic intersect curve here for the unconvenience of the choice of valid intersect point and the arrange of these points. We use a set of parallel lines to intersect the cubic curve and conveniently get those intersect points and arrange them in order.Fina...