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    Definition of Manifold
    Differential Manifold
    Differential Map
    Regular Surface Patch
    Regular Surface Patch
    Different parameterizations
    First fundamental form
    Invariant property of the first fundamental form
    Second fundamental form
    normal curvature
    Normal curvature
    Gauss Map
    Weingarten Transform
    Principle Curvature
    Weingarten Transformation
    Mean curvature and Gaussian curvature
    Gauss Equation
    Fundamental theorem in differential geometry
    Fundamental Theorem
    Isometry
    Tangent Map
    Tangent Map
    Conformal Map
    Isothermal Coordinates
    using isothermal coordinates
    Complex Representation
    Laplace Operator
    $lambda ,H$ representation
    $lambda ,H$ representation
    $lambda ,H$ representation
    Isothermal Coordinates
    Fundamental Group
    fundamental group
    fundamental group
    Fundamental Group
    Canonical Fundamental Group Basis
    Simplical Complex
    Simplicial Homology
    Simplicial Homology Group
    Simplicial Complex (Mesh)
    Simplicial Complex (Mesh)
    Simplicial Homology
    Simplicial Cohomology
    Simplicial Cohomology Group
    Different one-forms
    Exterior Differentiation
    de Rham Cohomology Group
    Pull back metric
    pull back metric
    Conformal Map
    Harmonic map
    Equivalence between harmonic maps and conformal maps, $g=0$
    Stereo graphic projection
    M"{o}bius Transformation Group
    Conformal map of topological disk
    Mobius Transformation
    Mobius Transformation
    Analytic function
    holomorphic differentials on the plane
    Conformal Atlas
    Conformal Structure
    Riemann surface
    Harmonic Function
    Harmonic one-form
    Hodge Theorem
    Holomorphic one-forms
    Holomorphic differentials on surface
    Holomorphic 1-form , Hodge Star Operator
    Zero Points
    Holomorphic differentials
    Holomorphic quadratic differential forms
    Holomorphic Trajectories
    Trajectories
    Finite Trajectories
    Finite Curve System
    Holomorphic differentials on surface
    Decomposition Theorem
    Decomposition
    Decomposition
    Decomposition
    Global structure of finite circle system
    Global structure of finite circle system
    Global structure of finite circle system
    Global structure of finite circle system
    Global structure of finite circle system
    Global structure of finite circle system
    Medical Imaging-Conformal Brain Mapping
    Medical Imaging-Colon Flattening
    Manifold Spline
    Manifold Spline
    Manifold TSpline
    Surface Matching
    Surface Matching
    Texture Synthesis
    Texture Synthesis
    ComputationalConformalGeometryLectureNotes Topology, Differential Geometry, Complex Analysis David GU gu@cs.sunysb.edu http://www.cs.sunysb.edu/˜gu Computer Science Department Stony Brook University David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 1/97
    DefinitionofManifold A manifold of dimension n is a connected Hausdorfff space M for which every point has a neighbourhood U that is homeomorphic to an open subset V of Rn. Such a hemeomorphism is called a coordinate chart. An atlas is a family of charts {Uα, φα} for which Uα constitute an open covering of M. φ : U → V S Uα Uβ φα φβ R2 φα(Uα) φαβ φβ (Uβ ) David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 2/97 Figure 1: Manifold. chart, atlas.
    DifferentialManifold • Transition function: Suppose {Uα, φα} and {Uβ , φβ} are two charts on a manifold S, Uα ∩ Uβ 6= , the chart transition is φαβ : φα(Uα ∩ Uβ ) → φβ (Uα ∩ Uβ ) • Differentiable Atlas: An atlas {Uα, φα} on a manifold is called differentiable if all charts transitions are differentiable of class C∞. • Differential Structure: A chart is called compatible with a differentiable atlas if adding this chart to the atlas yields again a differentiable atlas. Taking all charts compatible with a given differentiable atlas yieds a differentiable structure. • differentiable manifold: A differentiable manifold of dimension n is a manifold of dimension n together with a differentiable structure. David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 3/97
    DifferentialMap • Differential Map:A continuous map h : M → M′ between differential manifolds M and M′ with charts {Uα, φα} and {U′α, φ′α} is said to be differentiable if all the maps φ′β ◦ hφ−1 α are differentiable of class C∞ wherever they are defined. • Diffeomorphism: If h is a homeomorphism and if both h and h−1 are differentiable, then h is called a diffeomorphism. David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 4/97
    RegularSurfacePatch Suppose D = {(u, v)} is a planar domain, a map r : D → R3, r(u, v) = (x(u, v), y(u, v), z(u, v)), satisfying 1. x(u, v), y(u, v), z(u, v) are differentiable of class C∞. 2. ru and rv are linearly independent, namely ru = ( ∂x ∂u , ∂y ∂u , ∂z ∂u ), rv = ( ∂x ∂v , ∂y ∂v , ∂z ∂v ), ∂z ∂u ), ru × rv 6= 0, is a surface patch in R3, (u, v) are the coordinates parameters of the surface r. David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 5/97
    RegularSurfacePatch n ru r(u0, v0) rv S v (u0, v0) u r(u, v) Π Figure 2: Surface patch. David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 6/97
    Differentparameterizations Surface r can have different parameterizations. Consider a surface r(u, v) : D → R3, and parametric transformation σ : (¯u, ¯v) ∈ ¯D → (u, v) ∈ D, namely σ : ¯D → D is bijective and the J acobin ∂u(¯u,¯v) ∂u(¯u,¯v) ∂ ¯u ∂ ¯v ∂(u, v) ∂(¯u, ¯v) = ∂v(¯u,¯v) ∂ ¯u ∂v(¯u,¯v) ∂ ¯v 6= 0. then we have new parametric representation of the surface r, r(¯u, ¯v) = r ◦ σ(¯u, ¯v) = r(u(¯u, ¯v), v(¯u, ¯v)) : ¯D → R3. David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 7/97
    Firstfundamentalform Given a surface S in R3, r = r(u, v) is its parametric representation, denote E =< ru, ru >, F =< ru, rv >, G =< rv, rv >, the quadratic differential form I = ds2 = Edu · du + 2F du · dv + Gdv · dv, is called the first fundamental form of S. David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 8/97
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