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 κ(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 κ = 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 (θ, ⇤) coordinates in the map are linked by the following equation, referred to as a parametric equation of a sphere: x(θ, ⇤) R cos(θ) cos(⇤) x(θ, ⇤)= y(θ, ⇤) = R sin(θ) cos(⇤) , z(θ, ⇤)⇥ R sin(⇤) ⇥ where θ [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 λ 1 and λ 2 of I are the square lengths of the semiaxes of the ellipse σ1 = λ1 σ2 = λ2 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 = (φ)dφ 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 ! ! 31.