– p. 1/97 gu ˜ http://www.cs.sunysb.edu/ ity, gu ˜ David GU [email protected] Stony Brook University David Gu, Computer Science Department, Stony Brook Univers Computer Science Department http://www.cs.sunysb.edu/

Computational Lecture Notes Topology, , Complex Analysis – p. 2/97 gu ˜ α U . Such a n http://www.cs.sunysb.edu/ for which R } α of for which every point has ity, , φ ) V β α M β φ U U { ( β S φ β U V → U : αβ φ α φ is a family of charts U ) α . U atlas ( M α David Gu, Computer Science Department, Stony Brook Univers φ is a connected Hausdorfff space 2 n α R φ that is homeomorphic to an open subset U Figure 1: Manifold. chart, atlas. of dimension

Definition of Manifold manifold a neighbourhood is called a coordinate chart. An A constitute an open covering of hemeomorphism – p. 3/97 gu ˜ is a manifold of http://www.cs.sunysb.edu/ n ) β atlas. Taking all charts U ity, entiable structure. are two charts on a manifold ∩ } α β with a differentiable atlas if U ( , φ β . β φ ∞ U { C → ) β and on a manifold is called differentiable if all compatible U } } ∩ α α α , φ , φ U α α ( U U α { { φ : αβ David Gu, Computer Science Department, Stony Brook Univers φ : A differentiable manifold of dimension : A chart is called : An atlas : Suppose , the chart transition is together with a differentiable structure. 6 = n β U ∩

Differential Manifold α U , dimension compatible with a given differentiable atlas yieds a differ adding this chart to the atlas yields again a differentiable Transition function Differentiable Atlas Differential Structure differentiable manifold S charts transitions are differentiable of class • • • • – p. 4/97 gu ˜ M are differentiable, 1 http://www.cs.sunysb.edu/ − h and ity, h between differential manifolds wherever they are defined. ′ ∞ M is said to be differentiable if all the C } → ′ α , φ M ′ α : U h { and } α , φ David Gu, Computer Science Department, Stony Brook Univers α is a homeomorphism and if both U { h are differentiable of class : If :A continuous map 1 − α hφ ◦ with charts ′ β is called a diffeomorphism. ′ φ h

Differential Map M and Differential Map Diffeomorphism then maps • • – p. 5/97 gu ˜ . r , 6 = 0 v r http://www.cs.sunysb.edu/ × u r , ) , ity, , 3 ∂z ∂u )) . R , ) ∞ → u, v C ( ∂z ∂v D , , z : ) r ∂v ∂y , u, v ( ∂v ∂x , y ) =( u, v v ( r x , ) ∂z ∂u )=( , are the coordinates parameters of the surface ) are differentiable of class David Gu, Computer Science Department, Stony Brook Univers u, v ∂y ∂u ( ) , r u, v ( u, v ∂x ∂u , is a planar domain, a map ( 3 } , z R ) =( ) u r u, v are linearly independent, namely u, v ( ( { v r , y = )

Regular Surface Patch D and u, v ( u r x 2. 1. is a surface patch in Suppose satisfying – p. 6/97 gu ˜ http://www.cs.sunysb.edu/ u ity, ) 0 u ) r , v 0 0 u , v ( 0 r n u ( v v S r ) Π u, v ( David Gu, Computer Science Department, Stony Brook Univers r Figure 2: Surface patch.

Regular Surface Patch – p. 7/97 gu ˜ . 3 http://www.cs.sunysb.edu/ R → . ¯ ity, D , r 6 = 0 )) : ¯ v D, ) ) ¯ ¯ v v ∈ u, (¯ , ¯ ¯ v u ) u, u, 3 (¯ (¯ ∂ ∂ , v R ) ∂v ∂v u, v ¯ v ( → u, (¯ ) ) → D ¯ ¯ v v u ( ¯ ¯ ¯ v u D u, u, r ): (¯ (¯ ∂ ∂ ∈ Jacobin ∂u ∂u ) u, v )= ¯ v ( ¯ v r u, u, = (¯ : (¯ ) ) σ ¯ v σ ◦ David Gu, Computer Science Department, Stony Brook Univers r u, v u, (¯ ( ∂ ∂ )= ¯ v u, is bijective and the (¯ r D → ¯ D can have different parameterizations. Consider a surface

Different parameterizations : r σ then we have new parametric representation of the surface Surface and parametric transformation namely – p. 8/97 gu ˜ http://www.cs.sunysb.edu/ >, v dv, r · , ity, v r Gdv < = + dv · >, G v r F du , u r + 2 < . du = · S is its parametric representation, denote ) of >,F Edu u, v u ( = r r David Gu, Computer Science Department, Stony Brook Univers , 2 u = r ds r < , = 3 = I R E in S first fundamental form

First fundamental form is called the the quadratic differential form Given a surface – p. 9/97 gu ˜ . S . 2 ¯ v ¯ Gd http://www.cs.sunysb.edu/ + ¯ v ity, ¯ ud ¯ F d + 2 2 ¯ u ¯ Ed is invariant under parametric is invariant under the rigid motion of = S S 2 Gdv + F dudv David Gu, Computer Science Department, Stony Brook Univers + 2 2 Edu

Invariant property of the first fundamental form The first fundamental form of a surface The first fundamental form of a surface transformation, • • – p. 10/97 gu ˜ are coordinate v > r u , http://www.cs.sunysb.edu/ u n r , , v ) r ity, . < 2 u, v ( − r = Ndv = is > > > r >. S , u + 2 v u | n n n n v v , , , r r u , d u v r r r r × × u u < < < Mdudv r r < d | − − − − + 2 = = = = 2 = n > > > is defined as n n n II Ldu , , , S = uu uv vv r r r David Gu, Computer Science Department, Stony Brook Univers II < < < = = = has parametric representation , then the unit normal vector of S S L N M

Second fundamental form Define functions tangent vectors of Suppose a surface the second fundamental form of (1) (2) (3) then the second fundamental form is represented as – p. 11/97 gu ˜ , which is ) through u, v Π ( r http://www.cs.sunysb.edu/ , a plane at point ) n ity, k . u, v ( w r = S w ) has curvature u, v ( Π r n ∩ S Γ Γ= Π along the tangent vector S S David Gu, Computer Science Department, Stony Brook Univers is a tangent vector at point v r η Figure 3: normal curvature. + u , the planar curve r ǫ w = normal curvature and w n Suppose normal called the normal curvature of – p. 12/97 gu ˜ is w 0 < 1 k 0 http://www.cs.sunysb.edu/ 2 > 2 2 n ctions are positive. On k Nη Gη ity, and negative, or zero. + + , the normal curvature along F ǫη Mǫη v r η + 2 + 2 2 + 2 u Eǫ Lǫ r ǫ 0 = = > ) ) 1 w w w k , , w w ( ( I II 0 David Gu, Computer Science Department, Stony Brook Univers )= > n w 2 ( k n k , a tangent vector S

Normal curvature saddle surface patch, the normal curvatures may be positive Suppose a surface On convex surface patch, the normal curvature along any dire Figure 4: Convex surfaceface patch patch. and saddle sur- – p. 13/97 gu ˜ http://www.cs.sunysb.edu/ n 2 S , the normal vector at point ity, ) , ) u, v ( r u, v ( n → ) u, v g ( r , 2 S n → S S : . g David Gu, Computer Science Department, Stony Brook Univers S of Figure 5: Gauss map. , the mapping ) Gauss map is a surface with parametric representation u, v

Gauss Map ( S n is ) u, v is called the Suppose ( – p. 14/97 gu ˜ is a W . http://www.cs.sunysb.edu/ ameters. , . ) 2 e tangent plane to itself. v S n ity, µ >. ) + w u >. ( n v λ Weingarten transform , W ( 2 ) , S − v v p ( T < W )= = , the normal curvature v → ( < > S S p W w is called the )= T , to the tangent space of ) v g : → ( v S ( n v k W r W µ < + u David Gu, Computer Science Department, Stony Brook Univers r λ = of Gauss map v W is a unit tangent vector of v

Weingarten Transform Weingarten transform is independent of the choiceSuppose of the par Weingarten transform is a self-conjugate transform from th • • • linear map from the tangent space of The differential map The properties of Weingarten transform – p. 15/97 gu ˜ it is . The http://www.cs.sunysb.edu/ , 2 2 1 k k , then the normal θk 2 m at the principle ity, 2 e , θ 2 principle curvatures n e onal. 2 + sin k 1 + sin 1 θk )= e 2 2 θ e ( W = cos , = cos 1 e v 1 ,v > k ) v ( 1 )= k 1 W e ( < David Gu, Computer Science Department, Stony Brook Univers W 2 )= k v ( n n k is are unit vectors. Because Weingarten map is self conjugate, v 2 e

Principle Curvature and 1 e curvature along eigen directions are called principle directions, namely The eigen values of Weingarten transformation are called where curvatures. symmetric. Therefore, the principle directionsSuppose are an orthog arbitrary unit tangent vector therefore, normal curvature reaches its maximum and minimu – p. 16/97 gu ˜ LF MF − − http://www.cs.sunysb.edu/ . = 0 NF NE ity, MF ME e 2 2 − − F M − − . ) LG MG 2 v EG LN 2 k 2 + + F k 2 − 1 u 1 NE k ( 2 EG + 1 2 u, v, F = = = = − 1 MF x y z − 2 EG − David Gu, Computer Science Department, Stony Brook Univers LG − F G EF 2 k

Weingarten Transformation LM M N Locally, a surface can be approximated by a quadratic surfac Weingarten transformation coefficients matrix is Principle curvatures satisfy the quadratic equation – p. 17/97 gu ˜ , ) D ( g under Gauss map D http://www.cs.sunysb.edu/ and , D n ity, NE 2 + . F )) ) − D MF is the image of D ( 2 ( ) 2 g 2 EG ( − D F M ( g − Area − , LG Area P 1 2 p EG LN → )= D = 2 k K )= lim + p ( 1 k K ( 2 1 David Gu, Computer Science Department, Stony Brook Univers including point S = is defined as H is defined as is a region on

Mean curvature and D mean curvature Gaussian curvature . The Gaussian curvature is the limit of the area ratio betwee The the Mean curvature is related theSuppose area variation of the surface. g – p. 18/97 gu ˜ http://www.cs.sunysb.edu/ = 0 = 0 u v ) ity, ) } G EG u E EG ) √ √ √ √ ( u ( ) E M G M √ √ − − ( v u ) ) +( E G G E √ v √ ( ) ( v N L ) G E − − √ √ v u ( ) ) ( G E and the second fundamental form are not { M M √ √ ( ( 1 EG − − √ David Gu, Computer Science Department, Stony Brook Univers v u ) ) E,F,G − E G L N √ √ ( (

Gauss Equation Discrete interpretation. The first fundamental form Codazzi equations are independent, they satisfy the following Gauss equation – p. 19/97 gu ˜ or , 3 R . r → U ): and ) u, v ( u, v R http://www.cs.sunysb.edu/ ( , G ) ity, u, v ( ,F ) , and a surface u, v D ( E ⊂ U are the first and the second fundamental forms of David Gu, Computer Science Department, Stony Brook Univers satisfying the Gauss equation and Codazzi equations, then f ) L,M,N u, v ( and , N ) , there exists a neighborhood D is a planar domain, given functions u, v ( Fundamental theorem in differential geometry ∈ E,F,G D ) ,M ) u, v ( u, v ( any L Suppose such that – p. 20/97 gu ˜ are β du α du as unknowns, αβ b is positive definite. n , = 2 http://www.cs.sunysb.edu/ αβ r ψ , g ( 1 r } , ity, and r η equation group to be αβ β . ∂u ∂g 2 , uations of the natural frame of du , with , ) ) ) α , − β β γ r α n n r ∂ du ∂ = 1 α ∂u ∂u ∂ ∂u βη ( ( αβ ( αβ b are symmetric, ∂u α α g β ∂g ) ∂ ∂ ∂ + , α, β ∂u ∂u ∂u = + γ γ αβ r r φ b β ( γ β , αη )= )= )= b γ αβ α , construct Christoffel symbols ∂u α α β r α ∂g − r n r and ∂ { ∂ γα ∂ ∂u ∂u ∂u = = Γ = ) b ( ( ( γη α β β α β γ β g r n αβ βγ r ∂ ∂ ∂ ∂ g ∂ g 2 1 ∂ ∂u ∂u ∂u ( ∂u ∂u ∂u , = = David Gu, Computer Science Department, Stony Brook Univers D β α b γ αβ , is a planar region, Γ 1 } − 2 ) ,u αβ 1 g u ( { )=( =

Fundamental Theorem αβ D g ( differential forms defined on Suppose Denote Then consider the first order partial differential equation solvable (equivalent to Gauss Codazzi equations )are This partial differential equation isthe to surface. solve The the sufficient motion and eq necessary conditions for the – p. 21/97 gu ˜ . ˜ have S D ˜ . C to ∈ 2 ˜ v ) S and ˜ Gd ˜ ˜ v v u, v ∂ ∂ ∂v ∂u ( C + , ) ˜ v . If ) ˜ ˜ u u ˜ ud u, v ∂v ∂ ∂u ∂ C ( ( http://www.cs.sunysb.edu/ ˜ r F d σ = is = + 2 = r σ ˜ ity, 2 C is a bijection from σ ˜ u , J σ are ˜ S . , ˜ ) ˜ Ed S 3 ˜ v on R u, ) ) ˜ )= C (˜ and ˜ v 2 , where ˜ s S u, v u, v T σ d u, ( ( (˜ J u u ˜ I . )= = ˜ = ˜ ˜ ˜ ˜ ˜ u u and F G u, v 2 ( 2 ˜ ˜ F E ds isometry Gdv are two surfaces in + , their first fundamental forms are ˜ is mapped to curve S σ ˜ D is an J S ∈ David Gu, Computer Science Department, Stony Brook Univers σ = and ) on ˜ v F dudv S C u, (˜ + 2 , ) 2 ˜ v F G EF u, Edu (˜ ˜r = )=

Isometry r u, v ( then namely, and Suppose the parametric representation of the isometry Isometry: Suppose Suppose the parametric representations of I An arbitrary curve the same length, then • • – p. 22/97 gu ˜ . , v r b =0 http://www.cs.sunysb.edu/ is t , take a curve | + ) S ˜ v u = 0 ∂ ∂v r on ity, b a t , p ˜ S + ) ˜ v v ∂ ∂u ( and (0) = a ∗ ˜ (0) S ( S σ ) dt ˜ dv v ˜ v p dt d v ( γ = ˜ v r σ , and is independent of the choice of γ + ˜ ˜v T v =0 t (0) + | and → → ) (0) + ˜ ˜ u ˜ dt σ u du dt ∂v ∂ d , the tangent vector at u v b S ˜ ) u r p p γ + ( T = : σ ˜ u . is a tangent vector at point ∂ ∂u ∗ σ =0 σ a t S (0) = ˜ | ( p David Gu, Computer Science Department, Stony Brook Univers ˜ γ ˜ u γ (0) = T dt dt d γ ˜ γ such that ∈ , p, ˜ S S v only depends on = ˜ = r b on + (0) = ˜v )) γ t u ( r tildev a , v ) = t ( Tangent Map is a curve on v u ( ) . t r ( γ ˜ γ is called the tangent map of )= t ∗ ( This induces a map between the tangent spaces on σ then Suppose γ Tangent vector curve – p. 23/97 gu ˜ ˜ ˜ v u ˜r ˜r http://www.cs.sunysb.edu/ σ ity, J = >. w , ˜ ˜ v u ˜r ˜r v < = > ) ˜ ˜ v v ∂ ∂ ∂v ∂u w ( is an isometry if and only if for any two tangent ∗ ˜ S ˜ ˜ u u ,σ ) ∂v ∂ ∂u ∂ v ( and ∗ S = David Gu, Computer Science Department, Stony Brook Univers <σ ) ) v u r r ( ( ∗ ∗ σ σ between surfaces , σ

Tangent Map w , v Under natural frame, the tangent map is represented as A bijection vectors – p. 24/97 gu ˜ http://www.cs.sunysb.edu/ ity, satisfy ˜ S and S I. · 2 to be conformal is there exists a positive λ σ , if it preserves the angles between arbitrary = ˜ I David Gu, Computer Science Department, Stony Brook Univers is a ˜ S → S : σ Conformal Map , such that the first fundamental forms of λ function A bijection two intersecting curves. The sufficient and necessary condition of – p. 25/97 gu ˜ http://www.cs.sunysb.edu/ rm is represented as , 0 ity, , there exists a neighborhood ,λ> S ) 2 . dv + 2 du on a surface )( p u, v ( 2 λ = I isothermal coordinates David Gu, Computer Science Department, Stony Brook Univers is called the ) u, v ( Isothermal Coordinates , such that it can be conformally mapped to a planar region. p U (S S Chern): For an arbitrary point Under conformal parameterization, the first fundamental fo then • • – p. 26/97 gu ˜ http://www.cs.sunysb.edu/ ity, λ r ) 2 . ) ln 2 2 ∂ 2 ∂v ∂ ∂v + + 2 2 2 ∂ 2 ∂u ∂ ( ∂u 2 ( 1 λ 2 1 λ = − n = H 2 K David Gu, Computer Science Department, Stony Brook Univers

using isothermal coordinates Gaussian curvature is Mean curvature • • Isothermal coordinates is useful to simplify computations – p. 27/97 gu ˜ http://www.cs.sunysb.edu/ , let iv ) + ∂ ∂v ity, u i = + z ∂ ∂u ( λ. 2 1 ln = ¯ z 2 ¯ z ∂ ∂ ∂ ∂z∂ , ) 2 4 λ ∂ ∂v − i = − K ∂ ∂u ( 2 1 David Gu, Computer Science Department, Stony Brook Univers = ∂ ∂z

Complex Representation then For convenience, we introduce the complex coordinates – p. 28/97 gu ˜ , ∂g ∂v http://www.cs.sunysb.edu/ f . U C ity, = . ) vv g >dA f ∇ + f, uu ∇ f ( is outward normal of < 2 1 v λ U , = C f = S + ∆ ∂U , gdA David Gu, Computer Science Department, Stony Brook Univers U S ∆ f U operator on surface

Laplace Operator is the boundary of C Laplace The where The Green formula is λ, H representation

Suppose (u, v) is the isothermal coordinates, then

r r 1 r r r r < z, z > = 4 < u i v, u i v > ( − − = ru, ru > < rv,v> 2i< ru, rv >) < − − because (u, v) is isothermal,

< ru, ru >=< rv, rv >,< ru, rv >= 0 we can get

λ2 < rz, rz >= 0,< rz¯, rz¯ >= 0,< rz, rz¯ >= ,< n, n >= 1,< rz, n >=< rz¯, n >= 0. 2 therefore λ2 rzz¯ = Hn. 2

Let Q =< rzz, n >, then Q is a locally defined function on S.

David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 29/97 λ, H representation

Differentiate above equations, we get

< rz, rzz >=< rz, rzz¯ >= 0,< rzz, rz¯ >= λλz,< nz, rz >= < rzz, n >= Q, < nz, rz¯ =< −

Therefore, the motion equation for the frame rz, rz¯, n is { }

2 rzz = λzrz + Qn λ2 λ rzz¯ = Hn

2 2 nz = Hrz 2λ Qrz¯ − − −

From rzzz¯ = rzzz¯, we get the Gauss-Codazzi equation in complex form

2 2 Q λ 2 (ln λ)zz¯ = | 2| H (Gauss equation) λ2 4 λ − Qz¯ = 2 Hz (Codazzi equation)

David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 30/97 λ, H representation

Given a planar domain D R2, (u, v) are parameters, and 2 functions λ(u, v) and ⊂ H(u, v) satisfying Gauss-Codazzi equations, with appropriate boundary condition, then there exists a unique surface S, such that, (u, v) is its isothermal parameter, H(u, v) is its mean curvature function, and the surface first fundamental form is

ds2 = λ(u, v)2(du2 + dv2).

From Codazzi equation, Q can be reconstructed, then the motion equation of the natural frame rz, rz¯, n can be solved out. { } The quadratic differential form

2 2 Ψ= < rz, nz > dz = Qdz − is called the Hopf differential. It has the following speical properties • If all points on a surface S are umbilical points, then Hopf differential is zero. • Surface S has constant mean curvature if and only if Hopf differential is holomorphic quadratic differentials.

David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 31/97 – p. 32/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Isothermal Coordinates Figure 7: Isothermal coordinates on surfaces. – p. 33/97 gu ˜ are homotopic, if http://www.cs.sunysb.edu/ M ity, and S , . M 1 2 f f → 1] = = , [0 0 1 × × × between manifolds S S S | | : F F M F → S : David Gu, Computer Science Department, Stony Brook Univers 2 , f 1 f . 2 f ∼

Fundamental Group 1 f Two continuous maps we write with there exists a continuous map – p. 34/97 gu ˜ . γ http://www.cs.sunysb.edu/ . , 1 2 ity, p γ β = = S } 0 1 1] 1 , p p S, [0 , not homotopic to ×{ α → 1] }× , β 1 { [0 1] (0) = (1) = | | , 2 2 G G γ γ [0 × 1 0 p γ 1] , = = : [0 } 1] (0) = (1) = 0 , 1 1 γ G [0 γ γ ×{ 1] }× , 0 David Gu, Computer Science Department, Stony Brook Univers { [0 | | be curves with G G 2 , is homotopic to = 1 are homotopic, if there exists a continuous map α . S, i 2 2 γ γ → ∼ 1] and fundamental group 1 , γ 1 γ : [0 i γ such that Let we say we write Figure 8: – p. 35/97 gu ˜ http://www.cs.sunysb.edu/ . ] ity, 1] 1 2 , , 1 2 [0 [ ∈ ∈ 2 γ , for t for t (0) 2 1 0 γ p γ 1) 2 γ − ) t t (1) = 1 (2 (2 γ 1 2 γ γ 1 γ David Gu, Computer Science Department, Stony Brook Univers ) := t ( is defined by γ be curves with γ M := 2 → γ 1 1] γ ,

fundamental group : [0 Figure 9: product of two closed curves. 2 , γ 1 γ Let the product of – p. 36/97 gu ˜ as initial and homotopy classes. 0 http://www.cs.sunysb.edu/ p . ity, induces a homomorphism 0 f p are isomorphic. ) ) ≡ 0 1 0 γ , then N, q ) is the group of homotopy classes of ( M,p 0 ( 1 p 1 0) π ( _ π f → M, ) := , i.e. closed paths with ( and 0 0 0 1 ) p q π 0 M,p ( 1 M,p (1) = ( π γ 1 : π ∗ f David Gu, Computer Science Department, Stony Brook Univers (0) = γ with , the groups M M , the fundamental group → ∈ be a continuous map, and M 1 1] N , ∈ is a group with respect to the operation of multiplication of ,p Fundamental Group 0 0 ) → : [0 p p 0 γ M : M,p ( f 1 of fundamental groups. For any For any terminal point. π paths If The identity element is the class of the constant path Need more contents for topology

36-1 – p. 37/97 gu ˜ 1 − 1 1 a − 1 1 b b , we write the basis ) 2 1 http://www.cs.sunysb.edu/ 0 a a , through any point M,p , ( ity, 1 1 2 − 2 b π 1 b = 0 − 2 j a b , the surface can be sliced open · ) i M ,b ( j i 1 δ π 1 = a j b · 1 i b ,a = 0 j a · , such that David Gu, Computer Science Department, Stony Brook Univers } i g a ,b g ,a 2 2 a b ··· , . 2 ) ,b 0 closed surface, there exist canonical basis for 2 g Canonical Fundamental Group Basis ,a 1 represents the algebraic intersection number. Especially · , we can find a set of canonical basis for ,b 1 M,p a M ( { 1 ∈ p where For genus Figure 10: canonical basis ofπ fundamental group along them and form a canonical as – p. 38/97 gu ˜ , } 0 ≥ standard simplex i . , λ K is also a common http://www.cs.sunysb.edu/ , the 2 k = 1 σ , v i . λ ity, σ ··· and K , k of =0 1 1 i σ , i , v v 0 i . v facet λ , σ n k =0 is a R i τ = x | n 3 2 v of the simplex ∈R v x { , then all its facets also belongs to , then the intersection of K ]= 6 = k David Gu, Computer Science Department, Stony Brook Univers vertices 2 0 , v σ v is a union of simplices, such that ∩ ··· 1 , as the K 1 σ belongs to k , , v 1 σ , v 0 K v v is also a simplex, then we say is the minimal convex set including all of them, points in the general positions in ⊂ ··· ] σ , =[ k 2 1 ⊂ + 1 σ , v Simplical Complex ,σ , v τ k 1 0 σ v ··· , facet. 1 , v 2. If 1. If a simplex simplicial complex 0 v Suppose [ Suppose A we call – p. 39/97 gu ˜ k , ] n , v chains, we denote http://www.cs.sunysb.edu/ n i ··· σ , n ∂ +1 ity, i ex. i λ , v 1 i − , i . K = , v i ∈ Z σ ··· i i , λ 0 , λ v i [ i i σ i 1) n λ simplicies in − ( , ∂ i k 1 n =0 − i = n ) C σ K ]= is a linear space formed by all the ( n → n David Gu, Computer Science Department, Stony Brook Univers n , v C C : ··· , n 1 ∂ , v 0 chain space v [ n ∂ operator defined on a simplex is

Simplicial Homology dimensional n boundary chain is a linear combination of all k dimensional chain space as The The Associate a sequence of groupsA with a finite simplicial compl The boundary operator acts on a chain is a linear operator – p. 40/97 gu ˜ . = 0 http://www.cs.sunysb.edu/ T2 ∂σ y ves, which are not the osed chains are ity, α etween closed chains and β . 0 ≡ n ∂ ◦ 1 − n ∂ S2 γ David Gu, Computer Science Department, Stony Brook Univers chain, if it has no boundary, namely chain, if it is the boundary of some other chain, namely exact closed Figure 12: Idea of homology. is called a Simplicial Homology Group is called a σ σ . ∂τ = σ It can be easily shown that all exact chains are closed. Namel A chain The topology of the surface is indicated by the differences b the exact chains. For example,boundaries on (exact). a But genus on zero a surface, all torus, cl there are some closed cur boundaries of any surface patch. A chain – p. 41/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 13: Triangle mesh.

Simplicial Complex (Mesh) – p. 42/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 14: Triangle mesh.

Simplicial Complex (Mesh) – p. 43/97 gu ˜ http://www.cs.sunysb.edu/ is . ity, S K ⊂ . Σ , Σ +1 n 1 n ∂ = ker∂ 2 img∂ γ − )= 1 Z γ are homologous if and only if their difference is a of a simplical complex 2 K, ) ( , γ n 1 γ H M, Z ( David Gu, Computer Science Department, Stony Brook Univers k H dimensional patch, 2

Simplicial Homology -th homology group n The For example, two closed curves boundary of some – p. 44/97 gu ˜ ]) 0 , v 2 v http://www.cs.sunysb.edu/ ([ ω ity, . ]) + ) 2 is a linear operator, such that Z ) , v 1 Z v M, ( ([ ]) k M, 2 ω ( . C , v ∈ +1 1 k ]) + form. , v 1 → Z C , ω 0 k k v , v [ 0 +1 → C 2 v k : ) ∂ ∂ ([ ( Z ω ◦ ω ω is a 2-form, such that ω ω M, 1 ( δ = is linear space formed by all linear functionals defined k := ) C ω Z : k David Gu, Computer Science Department, Stony Brook Univers ]) = δ k 2 δ M, , v ( 1 k C , v -form, then -cochain is also called 0 1 k v ([ ω is a 1 ω δ . The )

Simplicial Cohomology Z M, cochain space ( k k cochain is a linear function C k The A The coboundary operator For example, on – p. 45/97 gu ˜ , such that τ http://www.cs.sunysb.edu/ ; the set of all exact -form k 1 − ity, kerδ k . . 1 1 − k − k k kerδ , denoted as imgδ . If there is a k imgδ δ = 0 )= Z is defined as the quotient group δω ) M, Z ( k M, , denoted as H ( 1 -form, if k − k David Gu, Computer Science Department, Stony Brook Univers k H δ -forms is the kernal of k is exact. ω is called a closed , then

Simplicial Cohomology Group ω ω = -th cohomology group τ k 1 -form − k -forms is the image set of k δ The set of all closed k The A – p. 46/97 gu ˜ , then a real ) α , v α http://www.cs.sunysb.edu/ u ( ) , β ity, with β α } ,g dv dv β α ) ) f β α , φ , v α , v =( β α U u { u ( ( g g α β β α + + ∂v ∂v ∂v ∂u β α du du ) ) β β α α β α ∂v , v ∂u ∂u ∂u , v continuity. β α u u ∞ (  ( β ) C α f α f David Gu, Computer Science Department, Stony Brook Univers = ,g = , α } ω f ω β ( , φ β has the parametric representation on local chart U ω { are functions with is a surface with a differential structure α

Different one-forms S ,g α f Suppose On different chart then where different one-form – p. 47/97 gu ˜ on divergence http://www.cs.sunysb.edu/ and 2 ity, dω du ∧ ∧ curlex 1 1 ) ω dv dv ω ) dv + ∧ ∂v ∂f f 2 2 2 ∂v ∂f ∂ ω + ω ∂u∂v + ∧ can be defined on differential forms, such − fω − du 1 d du f ∂f ∂u dω = = = 0 2 ∂f ∂u ∂ ∂v∂u ( d 2 ω ω ω ∧ is the generalization of ) = = ( = ∧ ∧ 2 1 d ω f ω ω ) = ,e.g, ∧ 0 df David Gu, Computer Science Department, Stony Brook Univers 1 ◦ u, v ≡ ( ω d ( d d df ◦ d can be defined on differential forms, such that ∧

Exterior Differentiation A special operator The so called exterior differentiation operator The exterior differential operator vector fields. It can be verified that that – p. 48/97 gu ˜ oup mology groups. http://www.cs.sunysb.edu/ ity, Ker d Img d = . , if and only if the difference : df. →R f . 0 = S : 2 ≡ ω − dω df, f exact forms 1 closed forms cohomologous ω = ω )= are 2 R ω S, ( 1 and David Gu, Computer Science Department, Stony Brook Univers H 1 ω satisfies satisfies ω ω 1-form 1-form

de Rham Cohomology Group exact closed between them is a gradient of some function A An All exact 1-forms are closed. The first de Rham cohomology group is defined as the quotient gr Two closed 1-forms de Rham cohomology groups are isomorphic to simplicial coho • • • • • • – p. 49/97 gu ˜ is a ) u, v ( , 2 2 2 N ˜ v d dv http://www.cs.sunysb.edu/ ds ) ) ∗ ˜ v dv du φ . Suppose u, u, v (˜ 2 N ( ity, G G ds + + ˜ ˜ v u ˜ v and ∂v ∂v ∂ ∂ , represented as ˜ ud 2 M d dudv , N ) ) ) ˜ ˜ v u is denoted as ˜ v ds ∂ ∂ ∂u ∂u → φ u, u, v u, v (˜ ( ( M ˜ φ F F : = φ + 2 + 2 )= ˜ 2 v 2 ˜ u u, d induced by dv du du (˜ ) ) ˜ v M . A map u, u, v ∗ N (˜ ( are φ ˜ E E of David Gu, Computer Science Department, Stony Brook Univers N = ) ˜ v = = with Riemannian metrics, u, and (˜ ˜ ˜ v u N , d d M 2 N 2 M M ds and ds M

Pull back metric then the metrics on local parameter of Two surfaces The so called pull back metric on – p. 50/97 gu ˜ . , the length dv du N ⊂ ) ∗ γ φ ( http://www.cs.sunysb.edu/ φ ) γ ( )) )) φ ity, N u, v u, v ( ( φ φ ( ( ˜ ˜ G F )) )) , this metric is the pull back metric. N u, v u, v PHI ( ( φ φ on ( ( ) ˜ ˜ F E γ ( φ is mapped to a curve segment T ) ∗ M γ φ ⊂ )( David Gu, Computer Science Department, Stony Brook Univers γ M du dv )=( u, v ( 2 N is defined as the length of pull back metric ds ∗ M φ on γ Then the parametric representation of pull back metric is Intuitively, a curve segment of – p. 51/97 gu ˜ is N → M : φ http://www.cs.sunysb.edu/ . A map 2 N ity, ds and , 2 M 2 N ds ds ∗ φ 2 . λ + satisfies = 2 N 2 M →R ds ds ∗ M φ : λ David Gu, Computer Science Department, Stony Brook Univers with Riemannian metrics, N and M

Conformal Map , if the pull back metric is a positive function λ conformal Two surfaces where – p. 52/97 gu ˜ , then 3 http://www.cs.sunysb.edu/ R ity, M > dA k f ∇ is embedded in , k N f ∇ < M , if it minimizes the following harmonic energy is a map, k N → )= David Gu, Computer Science Department, Stony Brook Univers f harmonic M ( : E f , the map is ) 3 , f Harmonic map 2 , f 1 f =( Suppose a smooth map f – p. 53/97 gu ˜ is harmonic if f http://www.cs.sunysb.edu/ ity, are genus zero closed surfaces, N and David Gu, Computer Science Department, Stony Brook Univers M , where N → = 0 is conformal. f M

Equivalence between harmonic maps and conformal maps, g : f A map and only if – p. 54/97 gu ˜ http://www.cs.sunysb.edu/ ) u, v ity, ( ) y x z z 2 2 − − 1 1 S x, y, z N ( is a conformal map = = 2 v u →R 2 S : φ David Gu, Computer Science Department, Stony Brook Univers

Stereo graphic projection The stereo graphic projection – p. 55/97 gu ˜ ion. Möbius dimensional http://www.cs.sunysb.edu/ 6 ity, , , form a 0 0 . . 2 S = 1 = 1 → bc bc 2 S − − : dimensional Möbius transformation group, φ 3 , ad , ad b b d d + + + + cz cz az az )= )= z z ( ( φ φ David Gu, Computer Science Department, Stony Brook Univers are complex numbers. is mapped to the complex plane using stereo-graphic project z 2 S and are real numbers.

Möbius Transformation Group a,b,c,d a,b,c, where where group. Suppose All the conformal map from sphere to sphere Then each map can be represented as The conformal map from disk to disk form a – p. 56/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 15: Conformal Map.

Conformal map of topological disk – p. 57/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 16: Conformal Map.

Mobius Transformation – p. 58/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 17: Mobius Transformation.

Mobius Transformation – p. 59/97 gu ˜ http://www.cs.sunysb.edu/ ity, ) ∂v ∂x u, v ∂v ∂y ( − → ) = = x, y :( ∂y ∂x ∂u ∂u f David Gu, Computer Science Department, Stony Brook Univers C→C : φ Analytic function is analytic, if it satisfies the Riemann-Cauchy equation A function – p. 60/97 gu ˜ http://www.cs.sunysb.edu/ ity, . 2 z = w David Gu, Computer Science Department, Stony Brook Univers Figure 18:

holomorphic differentials on the plane – p. 61/97 gu ˜ ) http://www.cs.sunysb.edu/ β U ∩ ity, α U ( β φ → . ) β M U ∩ α , if all chart transition functions U } ( α α φ , φ : α 1 U − α { φ = ◦ David Gu, Computer Science Department, Stony Brook Univers β A φ is a conformal atlas for = A αβ φ with an atlas M

Conformal Atlas A manifold are holomorphic, then – p. 62/97 gu ˜ is still a } α , φ α http://www.cs.sunysb.edu/ U ity, tructure. A ∪ { onformal atlas. , if the union A with an atlas David Gu, Computer Science Department, Stony Brook Univers compatible is } α , φ α Conformal Structure U { A chart conformal atlas. Two conformal atlas are compatible if their union is still a c Each conformal compatible equivalent class is a conformal s – p. 63/97 gu ˜ http://www.cs.sunysb.edu/ ctly. ity, ntial forms can be is called a . } α , φ α U { = A David Gu, Computer Science Department, Stony Brook Univers with a conformal structure

Riemann surface S A surface The definition domains of holomorphicgeneralized functions from and the differe complex plane to Riemann surfaces dire – p. 64/97 gu ˜ http://www.cs.sunysb.edu/ ity, f > dA. ∇ f, ∇ < M )= f ( E David Gu, Computer Science Department, Stony Brook Univers is harmonic, if it minimizes the harmonic energy →R S : f Harmonic Function A function – p. 65/97 gu ˜ , there is a M http://www.cs.sunysb.edu/ ∈ p ity, , f ∇ →R p = U ω : f , if and only if for each point David Gu, Computer Science Department, Stony Brook Univers harmonic is ω , there is a harmonic function p U , p

Harmonic one-form . p U such that A differential one-form on neighborhood of – p. 66/97 gu ˜ . ) R S, ( 1 H http://www.cs.sunysb.edu/ lass in ity, David Gu, Computer Science Department, Stony Brook Univers

Hodge Theorem There exists a unique harmonic one-form in each cohomology c – p. 67/97 gu ˜ with complex } α http://www.cs.sunysb.edu/ , φ with complex α } U β { ity, , φ β . U α { dz β α , dz dz α )) β dz ) α , on each chart dz z α ω ) ( z ( β β z z α ( ( f β β f f = ω = = ω David Gu, Computer Science Department, Stony Brook Univers , , α β is still a holomorphic function. z z β is a holomorphic function. On a different chart Holomorphic one-forms α dz α dz f β f then A holomorphic one-form is a differential form where coordinates coordinates – p. 68/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Holomorphic differentials on surface Figure 19: Holomorphic 1-forms on surfaces. – p. 69/97 gu ˜ http://www.cs.sunysb.edu/ is a harmonic 1-form τ ∗ , ity, dv ) dv ) u, v OMEGA ( u, v g τ, ( ∗ f + 1 . If we illustrate the operator intuitively + S du − ) √ du N ) + u, v ( τ u, v f ( = g = − ω τ = τ ∗ TAU David Gu, Computer Science Department, Stony Brook Univers is called the , τ ∗ is a holomorphic 1-form, then

Holomorphic 1-form , Hodge Star Operator ω is a real harmonic 1-form, τ the operate Suppose where as follows: conjugate to – p. 70/97 gu ˜ http://www.cs.sunysb.edu/ on a surface, if at point α e local coordinates. ity, dz ) α z ( f = ω . zero point is called a David Gu, Computer Science Department, Stony Brook Univers p , on one local coordinates ω , then point

Zero Points ) = 0 p ( f , S ∈ p A holomorphic 1-form Figure 21: Theform. zero point ofThe definition of a zero point doesn’t holomorphic depend on the choice 1- of th – p. 71/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Holomorphic differentials Figure 22: All holomorphic 1-forms on a Riemann – p. 72/97 gu ˜ with } α , φ α U { http://www.cs.sunysb.edu/ with complex } β ity, , φ . β 2 α U { dz 2 ) β , on each chart α , ω 2 α dz dz dz ) ))( β α α z dz z ( ) ( α β β f z z ( ( β β = f f ω = = David Gu, Computer Science Department, Stony Brook Univers ω , α z is still a holomorphic function. , 2 β ) z β α

Holomorphic quadratic differential forms is a holomorphic function. On a different chart dz dz α ( f β f A holomorphic quadratic form is a differential form where then coordinates complex coordinates – p. 73/97 gu ˜ . http://www.cs.sunysb.edu/ . 0 . > 0 ity, , 2 < S ω 2 , ω γ , γ critical trajectories , if along , if along David Gu, Computer Science Department, Stony Brook Univers horizontal trajectory vertical trajectory is called a is called a γ γ is a holomorphic 1-form on a Riemann surface

Holomorphic Trajectories ω A curve A curve The trajectories through zero points are called • • • Suppose – p. 74/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Trajectories Figure 23: The red curvesjectories, are the blue the curves horizontal are tra- vertical trajectories. – p. 75/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers , if its total length is finite. A finite trajectory is finite

Finite Trajectories either a closed circle. finite curve segment connecting zero points. finite curve segment intersecting boundaries. finite curve segment connecting zero point and a boundary. • • • • A trajectory is – p. 76/97 gu ˜ http://www.cs.sunysb.edu/ ity, ro points form the so called . are finite, then they are called 2 ω David Gu, Computer Science Department, Stony Brook Univers . .

Finite Curve System If the critical graph is finite, then the curve system is finite If all the horizontal offinite a curve holomorphic system quadratic form The horizontal trajectories through zerocritical points, graph and the ze – p. 77/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Holomorphic differentials on surface Figure 24: Holomorphic 1-forms on surfaces. – p. 78/97 gu ˜ , which induces a 2 http://www.cs.sunysb.edu/ φdz ity, mally mapped to a ies partition the surface to φdz. 0 p p )= p ( w with a quadratic holomorphic form S David Gu, Computer Science Department, Stony Brook Univers

Decomposition Theorem Suppose a Riemann surface finite curve system, then thetopological critical disks horizontal and trajector cylinders, eachparallelogram segment by can integrating be confor – p. 79/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Decomposition Figure 25: Criticaltem graph will partition of the a surfaceeach to finite segment topological is curve disks, conformally sys- mappedlelogram. to a paral- – p. 80/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Decomposition Figure 26: Criticaltem graph will partition of the a surfaceeach to finite segment topological is curve disks, conformally sys- mappedlelogram. to a paral- – p. 81/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Decomposition Figure 27: Criticaltem graph will partition of the a surfaceeach to finite segment topological is curve disks, conformally sys- mappedlelogram. to a paral- – p. 82/97 gu ˜ http://www.cs.sunysb.edu/ ical points. ity, ges, and different patches are David Gu, Computer Science Department, Stony Brook Univers

Global structure of finite circle system Different parallelograms are glued together along their ed met at the zero points. The edges and zero points form the crit – p. 83/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Global structure of finite circle system Figure 28: Different parallelograms are glued to- – p. 84/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Global structure of finite circle system Figure 29: The ciritical graph partition the surface – p. 85/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Global structure of finite circle system segments, each segment is conformally pa- 6 Figure 30: The ciritical graph partitionto the surface rameterized by a rectangle. – p. 86/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Global structure of finite circle system segments, each segment is a cylinder. 6 Figure 31: The ciritical graph partitionto the surface – p. 87/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Global structure of finite circle system segments, each segment is a cylinder, and 2 Figure 32: The ciritical graph partitionto the surface can be conformally mapped to a rectangle. Applications

David Gu, Computer Science Department, Stony Brook University, http://www.cs.sunysb.edu/˜gu – p. 88/97 – p. 89/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 33: Thanks Yalin wang

Medical Imaging-Conformal Brain Mapping – p. 90/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Medical Imaging-Colon Flattening Figure 34: Thanks Wei Hong, Miao Jin – p. 91/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 35: Thanks Ying

Manifold Spline – p. 92/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers Figure 36: Thanks Ying

Manifold Spline – p. 93/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Manifold TSpline – p. 94/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Surface Matching – p. 95/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Surface Matching – p. 96/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Texture Synthesis Figure 39: Thanks Lujin Wang, Miao Jin – p. 97/97 gu ˜ http://www.cs.sunysb.edu/ ity, David Gu, Computer Science Department, Stony Brook Univers

Texture Synthesis Figure 40: Thanks Lujin Wang, Miao Jin