First Fundamental Form

Spring 2015 CSCI 599: Digital Geometry Processing

Hao Li http://cs599.hao-li.com 1 Outline

• Parametric Curves

• Parametric Surfaces

2 Surfaces

What characterizes shape?! • shape does not depend on Euclidean motions • metric and curvatures • smooth continuous notions to discrete notions

3 Metric on Surfaces

Measure Stuff! • angle, length, area • requires an inner product • we have: • Euclidean inner product in domain • we want to turn this into: • inner product on surface

4 Parametric Surfaces

Continuous surface

x(u, v) x(u, v)= y(u, v) z(u, v) ⇥ n ⇤ ⌅ Normal vector xu xv p x x n = u v x x ⇥ u v⇥

Assume regular parameterization

x x = 0 normal exists u v ⇥ 5 Angles on Surface

Curve [ u ( t ) ,v ( t )] in uv-plane deﬁnes curve on the surface x(u, v)

c(t)=x(u(t) ,v(t))

Two curves c 1 and c 2 intersecting at!p • angle of intersection? n • two tangents t and t 1 2 xu xv p ! ti = ixu + ⇥ixv c2 • compute inner product c1 tT t = cos t t 1 2 1 2 6 Angles on Surface

Curve [ u ( t ) ,v ( t )] in uv-plane deﬁnes curve on the surface x(u, v)

c(t)=x(u(t) ,v(t))

Two curves c 1 and c 2 intersecting at p

T T t1 t2 =(1xu + ⇥1xv) (2xu + ⇥2xv)

T T T = 12xu xu +(1⇥2 + 2⇥1) xu xv + ⇥1⇥2xv xv

T T xu xu xu xv 2 =(1, ⇥1) T T xu xv xv xv ⇥2 ⇥ ⇥ 7 First Fundamental Form

First fundamental form

EF xT x xT x I = := u u u v FG xT x xT x ⇥ u v v v⇥

Deﬁnes inner product on tangent space

T 1 , 2 := 1 I 2 ⇥ ⇥ ⇥ ⇥ ⇤ 1⇥ 2⇥⌅ 1⇥ 2⇥

8 First Fundamental Form

First fundamental form I allows to measure! (w.r.t. surface metric)

Angles t>t = (↵ , ), (↵ , ) 1 2 h 1 1 2 2 i

Length ds2 = (du, dv), (du, dv) squared ! ⇥ inﬁnitesimal ! = Edu2 +2F dudv + Gdv2 length

Area dA = x x du dv ⌅ u ⇤ v⌅ T T T 2 = x xu x xv (x xv) du dv inﬁnitesimal ! u · v u Area ⇥ = EG F 2du dv cross product → determinant with unit vectors → area 9 Sphere Example

Spherical parameterization cos u sin v x(u, v)= sin u sin v , (u, v) [0, 2) [0, ) cos v ⇥ ⇥ Tangent vectors⇤ ⌅ sin u sin v cos u cos v x (u, v)= cos u sin v x (u, v)= sin u cos v u v ⇥ 0 ⇥ sin v ⇤ ⌅ ⇤ ⌅ First fundamental Form

sin2 v 0 I = 01 ⇥ 10 Sphere Example

Length of equator x(t, ⇡/2)

2 2 2 2 1ds = E (ut) +2Futvt + G (vt) dt 0 0 ⇥ 2 = sin v dt 0

=2 sin v =2

11 Sphere Example

Area of a sphere

2 2 1dA = EG F 2 du dv 0 0 0 0 ⇥ 2 = sin v du dv 0 0 =4

12 Normal Curvature

Tangent vector t …

xu xv p t

xu xv t = cos φ + sin φ ∥xu∥ ∥xv∥

unit vector

13 Normal Curvature

… deﬁnes intersection plane, yielding curve c(t) normal curve

t c(t) p

xu xv t = cos φ + sin φ ∥xu∥ ∥xv∥

14 Geometry of the Normal

Gauss map!

• normal at point

• consider curve in surface again

• study its curvature at p

• normal “tilts” along curve

15 Normal Curvature

Normal curvature  n ( t ) is deﬁned as curvature of the normal curve c ( t ) at point p(t)=x(u, v)

With second fundamental form ef xT nxT n II = := uu uv fg xT nxT n ⇥ ⇤ uv vv ⌅ normal curvature can be computed as T 2 2 ¯t II¯t ea +2fab + gb t = axu + bxv (¯t)= = n ¯tT I ¯t Ea2 +2F ab + Gb2 ¯t =(a, b)

16 Surface Curvature(s)

Principal curvatures!

• Maximum curvature κ1 = max κn(φ) φ • Minimum curvature κ2 = min κn(φ) φ 2 2 • Euler theorem n(⇥)=1 cos ⇥ + 2 sin ⇥

• Corresponding principal directions e 1 , e 2 are orthogonal

17 Surface Curvature(s)

Principal curvatures!

• Maximum curvature κ1 = max κn(φ) φ • Minimum curvature κ2 = min κn(φ) φ 2 2 • Euler theorem n(⇥)=1 cos ⇥ + 2 sin ⇥

• Corresponding principal directions e 1 , e 2 are orthogonal

Special curvatures! 1 + 2 • Mean curvature H = extrinsic 2 • Gaussian curvature K = κ1 · κ2 intrinsic (only ﬁrst FF)

18 Invariants

Gaussian and mean curvature! • determinant and trace only

! • eigenvalues and orthovectors

19 Mean Curvature

Integral representations

20 Curvature of Surfaces

+ Mean curvature! H = 1 2 2 • H =0 everywhere minimal surface !

soap ﬁlm 21 Curvature of Surfaces

+ Mean curvature! H = 1 2 2 • H =0 everywhere minimal surface !

Green Void, Sydney Architects: Lava 22 Curvature of Surfaces

Gaussian curvature! K = κ1 · κ2

• K =0 everywhere developable surface ! surface that can be ﬂattened to a plane without distortion (stretching or compression)

Disney, Concert Hall, L.A. Timber Fabric Architects: Gehry Partners IBOIS, EPFL

23 Shape Operator

Derivative of Gauss map! • second fundamental form

! • local coordinates

24 Intrinsic Geometry

Properties of the surface that only depend on the!

ﬁrst fundamental form!

• length

• angles

• Gaussian curvature (Theorema Egregium) remarkable theorem (Gauss)

6πr − 3C(r) K = lim r→0 πr3

Gaussian curvature of a surface is invariant under local isometry

25 Classiﬁcation

Point x on the surface is called!

• elliptic, if K>0

• hyperbolic, if K<0 Gaussian curvature K • parabolic, if K =0

• umbilic, if κ1 = κ2 or isotropic

26 Classiﬁcation

Point x on the surface is called

27 Gauss-Bonnet Theorem

For any closed manifold surface with Euler characteristic =2 2g

K =2⇥ Z

K()=K()=K()=4 Z Z Z

28 Gauss-Bonnet Theorem

Sphere

1 = 2 =1/r

2 K = 12 =1/r

1 K =4r2 =4 · r2 Z

when sphere is deformed, new positive and negative curvature cancel out

29 Differential Operators

Gradient! @f @f ! f := ,..., r @x @x ✓ 1 n ◆ !

• points in the direction of the steepest ascend

30 Differential Operators

Divergence!

! @F @F divF = F := 1 + ...+ n ! r· @x1 @xn

• volume density of outward ﬂux of vector ﬁeld

• magnitude of source or sink at given point

• Example: incompressible ﬂuid • velocity ﬁeld is divergence-free

31 Differential Operators

Divergence

@F @F divF = F := 1 + ...+ n r· @x1 @xn

high divergence low divergence 32 Laplace Operator

gradient 2nd partial Laplace operator derivatives operator

∂2f ∆ = div∇ = f f ! 2 i ∂xi Cartesian coordinates function in divergence Euclidean space operator

33 Laplace-Beltrami Operator

Extension of Laplace fo functions on manifolds

Laplace- gradient Beltrami operator

…of the surface ∆S f = divS ∇S f

function on divergence manifold S operator Laplace on the surface

34 Laplace-Beltrami Operator

Laplace- gradient Beltrami operator mean curvature

∆S x = divS ∇S x = −2Hn

surface function on divergence normal manifold S operator

35 Literature

• M. Do Carmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976

• A. Pressley: Elementary Differential Geometry, Springer, 2010

• G. Farin: Curves and Surfaces for CAGD, Morgan Kaufmann, 2001

• W. Boehm, H. Prautzsch: Geometric Concepts for Geometric Design, AK Peters 1994

• H. Prautzsch, W. Boehm, M. Paluszny: Bézier and B-Spline Techniques, Springer 2002

• ddg.cs.columbia.edu

• http://graphics.stanford.edu/courses/cs468-13-spring/schedule.html

36 Next Time

Discrete Differential Geometry

37 http://cs599.hao-li.com

Thanks!