Construction Of Mesh And Smooth Surfaces In Architectural Geometry

Architectural geometry is an emerging and interdisciplinary field starting from the problem solving freeform surface modeling in architecture.It has attracted much attention these days.Geometry lies at the core of design,analysis,digital modeling,and fabrication.With the development of modern science,on the one hand,geometry computation brings revolution to freeform surface modeling,posing challenges to scale and construction technologies of engineering and design.On the other hand,the progress in materials and technologies offers bigger and more flexible space for the exploration of geometry modeling.The needs originated from actual architecture yields new questions and goals for industry geometry,graphics and images,and geometry processing.Architectural geometry involves computational geometry,Computer-Aided Geometric Design(CAGD),Computer-Aided Design(CAD),Computer-Aided Manufacturing(CAM)and so on.The core of it is differential geometry,which focuses on the local analysis of geometry properties,such as the local curvatures of general spacial curves and surfaces.Some special but also common curves and surfaces,such as geodesics,asymptotes,curvature lines,developable surfaces,surfaces of constant mean curvature,surfaces of revolution et al.,own important research meanings in architectural geometry thanks to their differential properties.The methodology to explain architectural geometry modelings is Discrete Differential Geometry(DDG),which is the discrete counterpart of classic differential geometry,loyal to smooth theories and has a more direct and simpler representation.The research objects of DDG are polygons,polyhedron faces,and nonpolyhedron meshes.The representations of discrete curves and surfaces don't need global accurate algebraic expression.Usually,local properties of vertices,edges,and faces contribute to global geometry meanings.Furthermore,their differential expressions only depend on local characters.This enriches the possibilities of curves and surfaces constructions,paves a way for direct and interactive geometry modelings,and is convenient for designers to explore freeform surfaces with actual architectural requirements.Architectural geometry is significant for the future development of both theoretical research and practical architectures.This paper focuses on both the theory and the application of the construction of mesh and smooth surfaces in architectural geometry.Aiming at architectural applications,we first construct smooth surfaces interpolating special boundaries,which provides a theoretical basis for modeling waterproof architectural skin with boundary constraints.Then we research some special discrete parametric curve nets corresponding to the smooth cases in classical differential geometry,such as discrete constant mean curvature surface,discrete surface parameterized by discrete geodesic parallel coordinates,discrete geodesic curve net,discrete curvature lines net,discrete asymptotic curve net.These results not only enrich the theory of DDG but also present theoretical support for actual architectural application.Finally,thanks to the good properties of mesh and surface geometry,these conclusion helps to realize fabricationaware geometry designs.The main contents of this thesis are as follows:1.Smooth surface interpolating asymptotic quadrilateral.Firstly,this paper presents the necessary and sufficient conditions for a quadrilateral.With given corner data including corner coordinates,unit tangent vectors and curvatures,we design Bezier,rational Bezier,and Bspline asymptotic quadrilaterals.Secondly,compatible interpolation leads to corresponding tensorproduct Bezier,rational Bezier and B-spline surfaces through these closed quadrilaterals as boundaries.The prerequisite of the construction of free and continuous surfaces lies on the existence of free parameters left,then optimization of energy functions keeps the smoothness of curves and surfaces.Theoretically,this model generates the research on surfaces interpolating special boundaries such as geodesics and curvature lines.Practically,it provides ways to produce waterproof surfaces with the satisfaction of specific conditions on boundaries.2.Construction of meshes with spherical vertex star.Such mesh is symmetric about principal curvature lines,satisfying that each vertex and its four edgeshared neighbor vertices are in the same sphere at each vertex star.This geometry setting is compatible with discrete constant mean curvature surface and discrete minimal surface.When all the radii are the same and the network is orthogonal,the mesh surface becomes a discrete constant mean curvature surface.Especially,it becomes discrete minimal surfaces if the radii are infinity.In architectural applications,strained gridshell could be built from circular or straight developable steel lamellas.These lamellas,served as supporting beams structure,aligned normal to this(imaginary)discrete surface and intersect orthogonally with each other at torsalfree knots.The differential geometric advantages of actual curved and straight support structures enrich interactive designs and present a large number of repetitive parameters at knots,panels,and frame,which saves cost in fabrication and assembly.3.Construction of surfaces parametrized by discrete parallel coordinates.Geodesic parallel coordinates are orthogonal nets on surfaces where one of the two families of parameter lines are geodesic curves.The discrete version of these special surface parameterizations shows very useful for specific applications,most of which are related to the design and fabrication of surfaces in architecture.With the new discrete surface model,it is easy to control strip widths between neighboring geodesics.This facilitates tasks such as cladding a surface with strips of originally straight flat material or designing geodesic gridshells and timber rib shells.It is also possible to model nearly developable surfaces.They are characterized by geodesic strips with almost constant strip widths and are used for generating shapes that can be manufactured from materials which allow for some stretching or shrinking like felt,leather,or thin wooden boards.Most importantly,we show how to constrain the strip width parameters to model a class of intrinsically symmetric surfaces,which are isometric to surfaces of revolution and can be covered with doublycurved panels that are produced with only a few molds when working with flexible materials like metal sheets.This helps to solve the problem that freeform surfaces constructed from flat materials,which theoretically saves fabrication costs.4.Construction of special discrete surfaces parametrization.We study the local vertex of discrete quad meshes and model discrete surfaces parametrized by geodesics,curvature lines and asymptotes.The Guided Projection algorithm helps us to realize interactive designs of different discrete meshes in a very effective and speedy way,which makes it possible for us to visualize freeform surfaces,developable surfaces,surfaces of revolution,minimal surfaces,Weingarten surfaces,and their isometric deformation.