Fairing Minimal Surface By 4-Point Interpolation Subdivision Scheme

The minimal surfaces have been extensively employed in many areas such as architecture, aviation, ship manufacture, and so on. However, the complexity of the minimal surface equation prevents people from modeling the minimal surface in CAD/CAGD. Modeling of subdivision surface has been a research focus in the world over the past 20 years in computer aided geometric design and computer graphics. To promote applications of minimal surface in the CAD/CAGD systems, Modeling of the minimal surface by 4-point interpolation subdivision scheme is studied in the paper. Our work will produce positive effects on importing the minimal surface into the CAD/CAGD systems.After reviewing the general situation of minimal surface and subdivision, the author introduces 4-point interpolation subdivision scheme, Doo-Sabin scheme, Catmull-Clark skill, Loop method etc. as examples of subdivision templates. Based on the above, a fairing minimal surface generation algorithm is proposed by 4-point interpolation subdivision algorithm. An energy optimization criterion from [1] is introduced to ensure the fairing of the limit surface. Based on the convergence and continuity theories of the non-stationary 4-point interpolation subdivision algorithm, the solution of the object function associated with surface area is calculated as the free parameter in the subdivision algorithm, and then we obtain the needed fairing minimal surface. Some numerical examples are given to illustrate the simplification and efficiency of the algorithm.Our results not only widen applications of minimal surface in practice, but also help to studies of minimal surface in the future.