Parameterized Generation And Design Of Minimal Surfaces

A minimal surface is a surface where the average curvature of any point on the surface is zero,the study of minimal surfaces begins with the study of how to span the smallest surface in a given closed curve,the soap bubbles in the famous Plateau experiment are minimal surfaces in this definition.The minimal surface problem has always been an important and active branch of differential geometry,it involves functional,algebra,geometry,topology,physics and so on,and has already achieved a lot of research results;due to their unique mathematical properties,excellent structural performance and hallucinogenic and interesting form,minimal surfaces have bringing to a steady stream of practical application in all disciplines.As these applications grow,the research of the minimal surface in computer graphics has gained great development in the last 20 years.However,although architecture design is an important part of minimal surfaces' application,the research achievements in differential geometry and computer graphics are still difficult to be used directly by architects.So the summarizes and seek of the method suitable for architects which used for minimal surface design,and then reduced it to a tool that can be used directly,has a great significance for designers to explore the minimal surface applications in architecture and improve the efficiency of the exploration.This paper is to explore tools which are suitable for designers to parameterized generate minimal surfaces and deform them,full-text based on Grasshopper which is a parametric platform of Rhino.The main research contents can be divided into three parts:The first part contains chapter 1 and 2,we sort out the research results of minimal surface in differential geometry and computer graphics from now,and find out two kinds of elementary function expressions,then summarized the application in various disciplines and mainstream modeling methods in architectural design;The second part contains chapter 3 and 4 which is the most important part in this paper,in chapter 3,based on 7 explicit function expressions and 12 implicit function expressions,we come up with algorithms of parametrized generation of minimal surfaces and make them into tools respectively,so designers can use tools to generate the minimal surface prototypes they need directly,and for implicit function expressions,there are two common algorithms to be generated from,so we also made two kinds of tools for designers'choice,chapter 4 is the deformation study of minimal surfaces which based on the prototype we made in chapter 3,and come up with the methods and tool's innovation or summary of interference,stretching,mapping,splicing and shearing,array and so on;The third part contains chapter 5,we use the existing tools to do some design in different scales with minimal surface as the prototype to verify the tool's operability and find the insufficient,and find out the research direction in the future.