Parameterization Of Implicit Blending Algebraic Surfaces Of Three Quadratic Surfaces

Blending surfaces is one of the core contents in Computer Aided Geometric Design. In this paper, we discuss parameterization of implicit blending algebraic surfaces of three quadratic surfaces. Since the visualization of implicit algebraic surface is still a difficult problem, it is doubtlessly meaningful to construct some convenient and effective parameterization method for some kind of concrete implicit surfaces. In Chapter 2, we study the parameterization of a kind of GC 1 blending surfaces of three quadratic surfaces and give the parameterization algorithm. In the case of blending two pipes, people parameterize the blending surfaces by parameterizing the intersection of a certain plane pencil with the blending surface. It seems natural to use this idea into the parameterization of the blending surfaces of three pipe surfaces, however, the blending surface of three pipe surfaces is very complicated and the inappropriate choice of parameters can lead to singular blending surface. So it is not easy to turn this idea into reality. We creatively give our parameterization by using the expression of the blending surface in affine coordinate system ([9]) and the plane pencil together. The intersection of the certain plane pencil and the blending surface consists of a planar cubic curve and a line, and it is interesting that we can reduce the line directly through the affine expression and rotation transformation of coordinate is no more needed(in the case of the parameterization of blending surface of two pipes, rotation transformation of coordinate is needed to reduce a line from the intersection). For the quartic blending surface S ( f), we study the part of S ( f) which meets S ( gi )(i=1,2,3) and forms a connected closed surface together with S ( hi )(i=1,2,3). We call it the 'valid part' of the blending surface and will only consider the parametrization of this valid part. Our basic idea is to use the plane pencils which contains any one of S ( hj ,hk)(j,k=1, 2,3;j < k) to intersect S ( f). Without loss of generality, we start from S ( h1 , h2 ), using the plane pencil S ( h(θ)), which sweeps from S ( h1) to S ( h2), to intersect the blending surface S (f), then the intersection of S ( h(θ)) and S ( f) are composed of a planar cubic curve and the line S ( h1 ,h2). Parameterization can be done under two conditions: One is that axes l1 , l2,l3 of three quadratic surfaces intersect at O and l1 , l2,l3 are on the same plane, the other is that axes l1 , l2,l3 of three quadratic surfaces intersect at O and l1 , l2,l3 are not on the same plane. Then our task is to parameterize the intersection of S ( h(θ)) and the valid part of S ( f) for some fixed θ. In the parameterization of the blending surface of two pipes, one carried out the parameterization byparameterizing the intersections of the blending surface and the plane pencil which pass one point. But this method don't suit for parameterizing planar cubic curve because of choosing and sorting the effective points. Here we use parallel planes, namely choose h (t) that is parallel to h3 , to intersect the planar cubic curve and select the effective points by a reasonable restriction on the range of the intersection points. If three axes intersect one point and are on the same plane, we can solve x, y directly because S ( h(θ)) is independent of z . Then we obtain a univariate quadratic equation after substituting x, y into S ( f). Solving this quadratic equation we obtain points ( x , y , z1 )and( x , y , z 2) on the intersection curve. After θand t having taken all values pre-assigned, we can get a coordinates sequence of efficient point by solving equations successively. If three axes intersect at one point and are not on the same plane, we must solves equations S ( f ) = 0, S ( h (θ)) = 0 and h ( t ) = 0. Generally speaking, this resulting univariate equation has three real roots(possible has multiple root) or has a real root and a pair of conjugate complex roots. From a geometric point of view, a plane S ( h ) intersecting a cubic curve may result in no intersection point (namely the plane and cuver do not meet), or one intersection point (in this case the plane are tangent to the curve at this ponit), or two intersection points (the plane cuts the curve). Therefore at each time we get the roots of this cubic equation, we can decide if we keep or reject each root according to the continuity of the curved surface, the method is as follows: If θ=0, the relative cubic plane curve is just a circle, this circle can begotten by solving 2 2 2g 1 = y sin θ1 2 ? 2 yzΘ1 + z sin θ31 ? r1= 0, h1 = x + y cos θ1 2 + z cos θ31 ? d1= 0. Cuting the circle with the parallel plane pencil h (t i ) = x cos θ31 + y cosθ23 ? d3+ ti of equidistant, we get discrete point dates s and the t ivalue of the corresponding plane h (t i ). Denote the maximum and minimum value of t iby A( 0) and B (0). We divide the interval ((0, π/2)) and get a sequence of nodes { }θj. Then we consider the computation of discrete points on blending surface. At θ= θ1 and t i ( B ( 0)??t