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prefaceacronymschapter 1 elementary differentiaigeometry  1.1 parametric representation of surfaces  1.2 curvatures of surfaces  1.3 the fundamental equations and the fundamental theorem ofsurfaces  1.4 gauss-bonnet theorem  1.5 differential operators on surfaces  1.6 basic properties of differential operators  1.7 differential operators acting on surface and normal vector  1.8 some global properties of surfaces    1.8.1 green's formulas    1.8.2 integral formulas of surfaces  1.9 differential geometry of implicit surfaceschapter 2 construction of geometric partial differential equationsfor parametric surfaces  2.1 variation of functionals for parametric surfaces  2.2 the second-order euler-lagrange operator  2.3 the fourth-order euler-lagrange operator  2.4 the sixth-order euler-lagrange operator  2.5 other euler-lagrange operators    2.5.1 additivity of euler-lagrange operators    2.5.2 euler-lagrange operator for surfaces with graphrepresentation  2.6 gradient flow    2.6.1 l2-gradient flow for parametric surfaces    2.6.2 h-1 gradient flow for parametric surfaces  2.7 other geometric flows    2.7.1 area-preserving or volume-preserving second-ordergeometric flows    2.7.2 other sixth-order geometric flows    2.7.3 geometric flow for surfaces with graph representation  2.8 notes  2.9 related works    2.9.1 the choice of energy functionals    2.9.2 about geometric flowschapter 3 construction of geometric partial differential equationsfor level-set surfaces  3.1 variation of functionals on level-set surfaces  3.2 the second-order euler-lagrange operator  3.3 the fourth-order euler-lagrange operator  3.4 the sixth-order euler-lagrange operator  3.5 l2_gradient flows for level sets  3.6 h-1-gradient flow for level sets  3.7 construction of geometric flows from operator conversion  3.8 relationship among three construction methods of thegeometric flowschapter 4 discretization of differential geometric operators andcurvatures  4.1 discretization of the laplace-beltrami operator overtriangular meshes    4.1.1 discretization of the laplace-beltrami operator overtriangular meshes    4.1.2 convergence test of different discretization schemesof'the lb operator    4.1.3 convergence of the discrete lb operator over triangularmeshes    4.1.4 proof of the convergence results  4.2 discretization of the laplace-beltrami operator overquadrilateral meshes and its convergence analysis    4.2.1 discretization of lb operator over quadrilateral meshes    4.2.2 convergence property of the discrete lb operator    4.2.3 simplified integration rule    4.2.4 numerical experiments  4.3 discretization of the gaussian curvature over triangularmeshes    4.3.1 discretization of the gaussian curvature over triangularmeshes    4.3.2 numerical experiments    4.3.3 convergence properties of the discrete gaussiancurvatures    4.3.4 modified gauss-bonnet schemes and their convergence    4.3.5 a counterexample for the regular vertex with valence 4  4.4 discretization of the gaussian curvature over quadrilateralmeshes and its convergence analysis    4.4.1 discretization of the gaussian curvature overquadrilateral meshes    4.4.2 convergence property of the discrete gaussian curvature  4.5 consistent approximations of some geometric differentialoperators    4.5.1 consistent discretizations of differential geometricoperators and curvatures based on the quadratic fitting of surfaces    4.5.2 convergence property of discrete differential operators    4.5.3 consistent discretization of differential operators basedon biquadratic interpolation  4.6 related work on the discretization of the gaussian curvaturechapter 5 discrete surface design by quasi finite difference method  5.1 introduction  5.2 2k-th order geometric partial differential equations ofspecial forms    5.2.1 numerical solving methods    5.2.2 comparative results and application examples  5.3 fourth-order geometric partial differential equations ofgeneral forms    5.3.1 numerical solving of fourth-order geometric partialdifferential equation of general forms    5.3.2 comparative results and application examples  5.4 minimal mean curvature variation flow    5.4.1 numerical solving of the minimal mean curvature variationflow    5.4.2 application examples  5.5 a note about the convergence    5.5.1 fully discrete scheme of the boundary conditions    5.5.2 semi-discretization of boundary conditionschapter 6 spline surface design by quasi finite difference methodand finite element method  6.1 spline surface construction by quasi finite difference method    6.1.1 b-spline surface    6.1.2 construction of geometric partial differential equationb-spline surface    6.1.3 minimal b-spline surface    6.1.4 numerical experiments of convergence  6.2 spline surface construction by finite element methods    6.2.1 gpdes and their mixed variational forms    6.2.2 construction steps of gpde spline surfaces    6.2.3 numerical examples of convergence  6.3 regularization of spline surfaces    6.3.1 l2-gradient flows    6.3.2 numerical solutions of the l2-gradient flows    6.3.3 regularization of b-spline curves  6.4 about finite difference method and finite element method  6.5 numerical integration  6.6 related work    6.6.1 bezier and b-spline curves and surfaces    6.6.2 differential equation surfaces    6.6.3 geometric differential equation surfaceschapter 7 subdivision surface design by finite element methods  7.1 sobolev spaces on surfaces  7.2 finite element spaces    7.2.1 loop's subdivision scheme    7.2.2 the limit surface corresponding to vertices    7.2.3 evaluation of regular surface patches    7.2.4 evaluation of irregular surface patches    7.2.5 basis functions and classifications of surface patches    7.2.6 parametric representation and isoparametric elements  7.3 mean curvature flow and surface modeling    7.3.1 the background of surface modeling    7.3.2 a variant of the mean curvature flow    7.3.3 numerical solutions  7.4 fourth-order geometric partial differential equations    7.4.1 variational form of the fourth-order equation    7.4.2 discretization of fourth-order equations    7.4.3 applications and examples of fourth-order equations  7.5 sixth-order geometric partial differential equations    7.5.1 weak forms    7.5.2 disretization of the sixth-order equations  7.6 subdivision surfaces with boundaries    7.6.1 extended loop's subdivision surfaces    7.6.2 minimal surface construction    7.6.3 gl surface construction  7.7 related work on subdivision surfaceschapter 8 level-set method for surface design and itsapplications..  8.1 introduction  8.2 preliminaries    8.2.1 cubic b-spline interpolation    8.2.2 runge-kutta method with variable time step-size    8.2.3 eno interpolation    8.2.4 upwind scheme  8.3 local level-set method    8.3.1 algorithm outline    8.3.2 calculation of the global distance function    8.3.3 thin shell of a level set of a cubic spline function    8.3.4 initialization    8.3.5 evolution    8.3.6 re-initialization  8.4 applications of the level-set method in geometric design    8.4.1 3d surface reconstruction from scattered data set    8.4.2 biomolecular surface construction    8.4.3 surface metamorphosis    8.4.4 surface restorationchapter 9 quality meshing with geometric flows  9.1 introduction  9.2 single-domain triangular and tetrahedral quality meshing  9.3 single-domain quadrilateral and hexahedral quality meshing  9.4 multi-domain tetrahedral quality meshing    9.4.1 quality improvement algorithm and implementation    9.4.2 application examples  9.5 multi-domain hexahedral quality meshing    9.5.1 quality improvement algorithm and implementation    9.5.2 application examples and discussion  9.6 multi-domain triangular quality meshing with gaps    9.6.1 problem background    9.6.2 sketch of multi-domain meshing algorithm    9.6.3 algorithm details    9.6.4 resultsreferencesindex