A unified approach to blending of constant and varying parametric surfaces with curvature continuity

In this paper, we develop a new approach to blending of constant and varying parametric surfaces with curvature continuity. We propose a new mathematical model consisting of a vector-valued sixth-order partial differential equation (PDE) and time-dependent blending boundary constraints, and develop an approximate analytical solution of the mathematical model. The good accuracy and high computational efficiency are demonstrated by comparing the new approximate analytical solution with the corresponding accurate closed form solution. We also investigate the influence of the second partial derivatives on the continuity at trimlines, and apply the new approximate analytical solution in blending of constant and varying parametric surfaces with curvature continuit

Blending two cones with Dupin cyclides

This paper presents a complete theory of blending cones with Dupin cyclides and consists of four major contributions. First, a necessary and sufficient condition for two cones to have a blending Dupin cyclide is established. Second, based on the intersection structure of the cones, finer characterization results are obtained. Third, a new construction algorithm that establishes a correspondence between points on one or two coplanar lines and all constructed blending Dupin cyclides makes the construction easy and well-organized. Fourth, the completeness of the construction algorithm is proved. Consequently, all blending Dupin cyclides are organized into one to four one-parameter families, each of which is parameterized by points on a special line. It is also shown that each family contains an infinite number of ring cyclides, ensuring the existence of singularity free blending surfaces. © 1998 Elsevier Science B.V

Differential equation-based shape interpolation for surface blending and facial blendshapes.

Differential equation-based shape interpolation has been widely applied in geometric modelling and computer animation. It has the advantages of physics-based, good realism, easy obtaining of high- order continuity, strong ability in describing complicated shapes, and small data of geometric models. Among various applications of differential equation-based shape interpolation, surface blending and facial blendshapes are two active and important topics. Differential equation-based surface blending can be time-independent and time-dependent. Existing differential equation-based surface blending only tackles time-dependen

Do blending and offsetting commute for Dupin cyclides?

A common method for constructing blending Dupin cyclides for two cones having a common inscribed sphere of radius r ? 0 involves three steps: (1) computing the (\Gammar)- offsets of the cones so that they share a common vertex, (2) constructing a blending cyclide for the offset cones, and (3) computing the r-offset of the cyclide. Unfortunately, this process does not always work properly. Worse, for some half-cones cases, none of the blending cyclides can be constructed this way. This paper studies this problem and presents two major contributions. First, it is shown that the offset construction is correct for the case of ffl 6= \Gammar, where ffl is the signed offset value; otherwise, a procedure must be followed for properly selecting a pair of principal circles of the blending cyclide. Second, based on Shene's construction in "Blending two cones with Dupin cyclides", CAGD, Vol. 15 (1998), pp. 643--673, a new algorithm is available for constructing all possible blending cyclides for ..

Do blending and offsetting commute for Dupin cyclides?

A common method for constructing blending Dupin cyclides for two cones having a common inscribed sphere of radius r\u3e 0 involves three steps: (1) computing the (-r)-offsets of the cones so that they share a common vertex, (2) constructing a blending cyclide for the offset cones, and (3) computing the r-offset of the cyclide. Unfortunately, this process does not always work properly. Worse, for some half-cones cases, none of the blending cyclides can be constructed this way. This paper studies this problem and presents two major contributions. First, it is shown that the offset construction is correct for the case of ε≠-r, where ε is the signed offset value; otherwise, a procedure must be followed for properly selecting a pair of principal circles of the blending cyclide. Second, based on Shene\u27s construction in `Blending two cones with Dupin cyclides\u27, CAGD, 15 (1998) 643-673, a new algorithm is available for constructing all possible blending cyclides for two half-cones. This paper also examines Allen and Dutta\u27s theory of pure blends, which uses the offset construction. To help overcome the difficulties of Allen and Dutta\u27s method, this paper suggests a new algorithm for constructing all possible pure blends. Thus, Shene\u27s diagonal construction is better and more reliable than the offset construction

Do Blending and Offsetting Commute for

A common method for constructing blending Dupin cyclides for two cones having a common inscribed sphere of radius r>0 involves three steps: (1) computing the (−r)-offsets of the cones so that they share a common vertex,(2) constructing a blending cyclide for the offset cones,and (3) computing the r-offset of the cyclide. Unfortunately,this process does not always work properly. Worse,for some halfcones cases,none of the blending cyclides can be constructed this way. This paper studies this problem and presents two major contributions. First,it is shown that the offset construction is correct for the case of ɛ � = −r,where ɛ is the signed offset value; otherwise,a procedure must be followed for properly selecting a pair of principal circles of the blending cyclide. Second,based on Shene’s construction in “Blending two cones with Dupin cyclides”, CAGD,Vol. 15 (1998),pp. 643– 673,a new algorithm is available for constructing all possible blending cyclides for two half-cones. This paper also examines Allen and Dutta’s theory of pure blends, which uses the offset construction. To help overcome the difficulties of Allen and Dutta’s method,this paper suggests a new algorithm for constructing all possible pure blends. Thus,Shene’s diagonal construction is better and more reliable than the offset construction