– 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 Conformal Geometry Lecture Notes Topology, Differential Geometry, 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 Gaussian curvature 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 conformal map 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,
< 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 { }