A PRIMER on DIFFERENTIAL GEOMETRY Lecture 2 DIFFERENTIAL GEOMETRY

A PRIMER ON DIFFERENTIAL GEOMETRY Lecture 2 DIFFERENTIAL GEOMETRY

• Use of differential calculus to study problems in geometry

• For surfaces:

• Parametric representation, surface normal, tangent plane, curvature, Laplacian, etc.

2 PARAMETERIC REPRESENTATION OF CURVES

d • curve l R d =2, 3 ⇢ a u b • parameter domain [a, b] R ⇢ d • mapping f :[a, b] R p = f(u) !

f is a univariate vector function f 0(u) mapping points of the real line to points in dD space l • tangent vector at f ( u ) is f 0 ( u ) PARAMETERIC REPRESENTATION OF CURVES

• different parametrizations may describe the same curve • arc length: for [c,d] ⊂ [a,b], the length of f([c,d]) is 0 u L d ! f 0(t) dt || || • arc length parametrization: Zc d ! l :[0,L] R ! where L is the total length of the curve, and such that for all u ∈ [0,L] l l0(u) =1 || || PARAMETERIC REPRESENTATION OF CURVES

• curvature: if l is parametrized by arc length, then its curvature at p = l(u) is given by (u)= l00(u) 0 u L || || • curvature is the inverse of the radius of the osculating circle, which is the limit of circles passing through p and other two points on the curve, as such points approach p r 1 l = r PARAMETRIC REPRESENTATION OF SURFACES

• surface

• parameter domain

• mapping

f is a bi-variate vector function mapping points of the 2D plane to points in 3D space PARAMETRIC REPRESENTATION OF SURFACES i i i i • We can deﬁne an injective parametrization iff surface S is

homeomorphic34 to a (punctured) disk 3. Differential Geometry

Figure 3.3. Transforming a vector w¯ from7 parametric space into a tangent vector w of a surface described by a parameterization x. (Image taken from [Hormann S et al. 07]. c 2007 ACM, Inc. Included here by permission.)

derivatives ⌅x ⌅x x (u ,v ):= (u ,v ) and x (u ,v ):= (u ,v ) u 0 0 ⌅u 0 0 v 0 0 ⌅v 0 0 are, respectively, the tangent vectors of the two iso-parameter curves

Cu(t)=x(u0 + t, v0) and Cv(t)=x(u0,v0 + t)

at the point x(u0,v0) . In the following we drop the parameters (u0,v0) or (u, v) for notational⇥ S brevity. It is important to remember, however, that all quantities are defined point-wise and will typically vary across the surface. Assuming a regular parameterization,i.e.,x x = 0, the tangent u v ⇤ plane to is spanned by the two tangent vectors xu and xv.Thesur- face normalS vector is orthogonal to both tangent vectors and can thus be computed as x x n = u v . x x ⇧ u v⇧ In addition, we can define arbitrary directional derivatives of x.Givena T direction vector w¯ =(uw,vw) defined in parameter space, we consider the straight line parameterized by t passing through (u0,v0) and oriented by w¯ given by (u, v)=(u0,v0)+tw¯ . The image of this straight line through x is the curve Cw(t)=x(u0 + tuw,v0 + tvw).

The directional derivative w of x at (u0,v0) relative to the direction w¯ is deﬁned to be the tangent to Cw at t = 0, given by w = ⌅Cw(t)/⌅t.By

i i

i i PARAMETRIC REPRESENTATION OF SURFACES

i i

• Fori a surface S that is not homeomorphic to a punctured i

disk, parametrization32 needs to cut S at 3.the Differential image Geometry of borders of Ω

8 Figure 3.1. Cut me a meridian, and I will unfold the world! (Image taken from [Hormann et al. 07]. c 2007 ACM, Inc. Included here by permission.)

(x, y, z) coordinates in 3D space and the (, ⇤) coordinates in the map are linked by the following equation, referred to as a parametric equation of a sphere: x(, ⇤) R cos() cos(⇤) x(, ⇤)= y(, ⇤) = R sin() cos(⇤) , z(, ⇤)⇥ R sin(⇤) ⇥ where [0, 2⇥], ⇤ ⇤[ ⇥/2, ⇥⌅/2], and⇤ R denotes the⌅ radius of the ⇥ ⇥

Figure 3.2. Spherical coordinates. (Image taken from [Hormann et al. 07]. c 2007 ACM, Inc. Included here by permission.)

i i

i i TANGENT PLANE • Jacobian of

• tangent plane at

• Taylor expansion of

• ﬁrst order approximaon of

9 SURFACE NORMAL

• Surface normal at a point p is the unary normal vector n of tangent plane Tp pointing outwards from S f f f • If p = f(x) then n = u v f f || u v|| n

10 DIRECTIONAL DERIVATIVES

T • Given a direction vector ¯w =(uw,vw) ¯w f • Straight line in parameter space

! (u, v)=(ux,vx)+t ¯w Cw(t) n • Corresponds to a curve on S

Cw(t)=f(ux + tuw,vx + tvw)

11 DIRECTIONAL DERIVATIVES

¯w • Directional derivative w of f at (ux,vx) with

respect to ¯w is the tangent to C w ( t ) computed f at t = 0 w C (t) • By the chain rule w = Jf ¯w w n

12 FIRST FUNDAMENTAL FORM

• The Jacobian matrix encodes the metrics on the surface:

we can measure distances, angles and areas

• Scalar product of two tangent vectors:

T T T T ! w1 w2 =(Jf ¯w 1) (Jf ¯w 2)= ¯w 1 (Jf Jf ) ¯w 2

• First fundamental form deﬁnes the inner product on the tangent space

! EF f T f f T f I = J T J = = u u u v f f FG f T f f T f   u v v v

13 MEASURING LENGTHS

• The ﬁrst fundamental form can be used to measure lengths on the surface:

• Length of a tangent vector w : w 2 = ¯w T I¯w || || • Length of a curve f(u(t)) where u(t) is a curve in Ω:

b b T 2 2 l(a, b)= (ut,vt)I(ut,vt) dt = Eut +2Futvt + Gvt dt a a Z q Z q

14 MEASURING AREAS

• The ﬁrst fundamental form can be used to measure areas on the surface:

• Area of a region f(U) where U is a region in Ω:

A = det(I)dudv = EG F 2dudv U U ZZ p ZZ p

15 ANISOTROPY ELLIPSE

• small disk around

• image of under ﬁrst order approximation of f

• shape of on the tangent plane

• ellipse

• semiaxes and

• behavior in the limit

16 ANISOTROPY ELLIPSE • Singular Value Decomposition (SVD) of

with rotations and

and scale factors (singular values)

17 ANISOTROPY ELLIPSE

• From the ﬁrst fundamental form I:

• The eigenvectors ¯e 1 and ¯e 2 of I map to the axes e 1 and e 2 of the ellipse

• The eigenvalues 1 and 2 of I are the square lengths of the semiaxes of the ellipse 1 = 1 2 = 2 p p 18 NORMAL CURVATURE

Differential Geometry

• A direction t = J f ¯t on the tangent plane at point p, • Normal Curvature together with the normal n at p xu × xv n n = deﬁne a plane that cuts S along ∥xu × xv∥ a curve c xu xv t c p t • The curvature of c curve at p is the normal curvature w.r.t. t xu xv t = cos φ + sin φ ∥xu∥ ∥xv∥

19 Mark Pauly 54 SECOND FUNDAMENTAL FORM

• The projections of second order derivatives of the parametric function onto the normal line help studying second order differential properties of the surface (curvature)

• Second fundamental form:

LM f T n f T n ! II = = uu uv MN f T n f T n   uv vv

20 NORMAL CURVATURE

• Normal curvature can be computed by means of the ﬁrst and second fundamental forms

¯tT II¯t Lu2 +2Mu v + Nv2  (¯t)= = t t t t n ¯T ¯ 2 2 t It Eut +2Futvt + Gvt

21 PRINCIPAL CURVATURES

• Consider all possible directions t on the tangent plane at p

• Unless the normal curvature is equal at all t ‘s, there exist exactly two directions:

• t1 such that κ1 = κ(t1) is maximum

• t2 such that κ2 = κ(t2) is minimum

• κ1 and κ2 are called the principal curvatures

• t1 and t2 are called the principal directions of curvature

22 DARBOUX FRAME

• Principal directions have arbitrary orientations

• Orientations can be selected such that (t1,t2,n) form a right-handed 3D frame with t1 origin at p n t2 • The surface in a sufﬁciently small neighborhood of p can be expressed in explicit form n = f(t1,t2) with respect to the Darboux frame

23 EULER THEOREM

• the principal curvatures are mutually orthogonal

• for any direction t on the tangent plane

2 2 ! n(¯t)=1 cos + 2 sin

where is the angle between t and t1 2⇡ • Mean curvature:  +  1 H = 1 2 =  ()d 2 2⇡ n Z0 det(II) • Gaussian curvature: K =   = 1 · 2 det(I) 24 DifferentialLOCAL CLASSIFICATION Geometry OF SURFACES • A point x on the surface is called A point on a surface is called: – elliptic, if K > 0 • elliptic– parabolic if K > 0 , if K = 0 • hyperbolic– hyperbolic if K < 0 , if K < 0 κ = κ • parabolic– umbilical if K = 0, if 1 2

• An umbilical is a point with κ1 = κ2 • Developable surface K = 0 • Developable surface ⇔ K = 0 at all⇔ points

25 Mark Pauly 11 GAUSS MAP

• The Gauss map sends a point on the surface to the outward pointing unit normal vector: f (x) f (x) G(p)=G(f(x)) = u ⇥ v = n f (x) f (x) || u ⇥ v ||

26 SHAPE OPERATOR

• The shape operator Sh is the differential of the Gauss map

• It is a linear operator that, when applied to a tangent vector t tells how the surface normal varies when moving along t on S

• It is represented by a 2×2 matrix defined at each point p in the frame (fu,fv) through the coefficients of the first and second fundamental form:

! 2 1 LG MF MG NF Sh =(EG F ) ME LF NE MF ✓ ◆

27 SHAPE OPERATOR

• The principal directions of curvature are the eigenvectors of Sh

• The principal curvatures are the eigenvalues of Sh

• The shape operator is described by a diagonal matrix of principal curvatures in the frame (t1,t2)

28 CURVATURE TENSOR

• The curvature tensor C is a symmetric 3×3 matrix with

• eigenvalues κ1, κ2 and 0 1 C = PDP

• eigenvectors t1, t2 and n P =[t1, t2, n] D = diag(1,2, 0) !

• It deﬁnes a tensor ﬁeld on S that can be used to compute all curvatures

29 LINES OF CURVATURE

• A line of curvature is a smooth line on S that is tangent at each point to a principal direction of curvature

• Lines of curvature belong to two classes - maximal and minimal curvature lines - and form two orthogonal line ﬁelds on S

• Each line of curvature either is a loop, or has its endpoints at umbilical points (in rare cases it can wind in a spiral about an orbit - we ignore these cases)

30 LINES OF CURVATURE

• Lines of curvature can be arranged in parallel bundles, i.e., maximal groups of lines that either are all concentric loops, or have common endpoints

• A line of curvature that separates two distinct bundles is called a separatrix

• Umbilicals are classiﬁed depending on the arrangement of bundles and separatrices about them