Problem Definition Given a surface (mesh) S in R 3 and a domain find a bective F: S Typical Domains Cutting to a Disk disk = genus zero + boundary sphere = closed genus zero Creates artificial boundary Texture Mapping Texture Mapping 5 boundary 6 Page 1
Normal Mapping good bad 7 8 Normal Mapping 3,500 faces Remeshing 90,000 faces 9 10 Remeshing Compression = Stanford Bunny 11 1 Page
Desirable Properties Unfolding the World Low distortion Bective mapping Efficiently computable bective not bective 13 14 Spherical Coordinates Definitions [0, ), [ /, / ) f is isometric (length preserving), if the length of any arc on S is preserved on S*. S f is conformal (angle preserving), if the angle of intersection of every pair of intersecting arcs on S is preserved on S*. f is equiareal (area preserving) if the area of an area element on S is preserved on S*. f S* 15 16 Standard Map Projections More Maps orthographic stereographic Mercator Lambert preserves angles = conformal preserves area = equiareal 17 18 Page 3
Conformal Map Output Conformally flattened D mesh similarity Similarity = Rotation + Scale Preserves angles Input 3D mesh 19 0 Conformal Parameterization Conformal Minimal Stretch 1 Differential Geometry Revisited Parametric surface: x( u, v) ( x( u, v), y( u, v), z( u, v)) Conformal Minimal Stretch Regular if x(u,v),y(u,v),z(u,v) are smooth (differentiable) tangent vectors are linearly independent 3 4 Page 4
Distortion Analysis Jacobian 5 6 Distortion Analysis Fundamental Form Revisited Characterizes the surface locally Length element First fundamental form Area element 7 8 Mapping Surfaces Isometric Maps Theorem: An allowable mapping f from S to S* is isometric (length-preserving), iff the first fundamental forms of x and x*= f x are equal, i.e. parameterization of S is allowable if ( uv, ) * * ( u, v ) parameter domain is regular parameterization of S* Isometric surfaces have the same Gaussian curvature at corresponding pairs of points! 9 30 Page 5
Conformal Maps Theorem: An allowable mapping f from S to S* is conformal, iff the first fundamental forms of x and x*= f x are proportional, i.e., there exists a positive scalar function n, such that Developable (isometric to the plane) Isometric A conformal map is always (locally) bective. 31 3 Stereographic Projection conformal maps conformal 33 34 More Conformal Maps Equiareal Maps Theorem: An allowable mapping f from S to S* is equiareal, iff the determinants of the first fundamental forms of x and x*= f x are equal, i.e., Peirce's Quincuncial Projection August Map Area element: 35 36 Page 6
Relationships Theorem: Every isometric mapping is conformal and equiareal, and vice versa. Riemann Conformal Mapping Theorem Any two simply connected compact planar regions can be mapped conformally onto each other. isometric conformal + equiareal a( u, v) b( u, v) J( u, v) b( u, v) a( u, v) J a b 0 Isometric is ideal but rare. In practice, we use: conformal equiareal some balance between the two If (x,y)(u,v) is a conformal mapping, then u(x,y) and v(x,y) satisfy the Cauchy-Riemann equations: thus both u and v are harmonic: 37 38 Harmonic Maps is harmonic if satisfies (for each coordinate): Harmonic Maps Easier to compute than conformal, but does not preserve angles. May not be bective. isometric conformal harmonic Minimizes the Dirichlet energy given boundary conditions 39 40 Harmonic Maps Theorem [Rado-Kneser-Choquet]: If f : S R is harmonic and maps the boundary S homeomorphically onto the boundary S* of some convex region S* R, then f is bective. Discrete Harmonic Maps Piecewise linear map for triangulated, disk-like surface onto planar polygon f f=0 41 Laplace equation: w ( v v ) 0 ( i, j) E 4 i j Page 7
D Barycentric Drawings B = Boundary vertices Fix D boundary to convex polygon. Define drawing as a solution of Why it Works Theorem (Maxwell-Tutte) If G = <V,E> is a 3-connected planar graph (triangular mesh) then any barycentric drawing is a valid embedding. Weights w control triangle shapes 43 44 Example Spring System W w 1 Laplacian Matrix 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 5 1 1 1 0 0 0 1 1 1 5 1 0 0 1 1 1 0 1 1 5 1 3 4 b x 3 0 0 0 3 3 b y 1 1 0 0 0 1 y 8 6 3 7 9 5 x 4 Represent as configuration of springs on mesh edges 1 E( v) w v v i j ( i, j) E Minimum of E(v) reached when gradients = 0 Ev () w ( v v ) 0 i j v i ( i, j) E 45 46 Uniform Weights w 1 No shape information equilateral triangles Fastest to compute and solve Not D reproducible Harmonic Weights w cot( ) cot( ) v j Weights can be negative not always valid Weights depend only on angles - close to conformal D reproducible v i 47 48 Page 8
Mean-Value Weights General Method tan( / ) tan( / ) w V V i Result visually similar to harmonic j V j V i Select normalized weights so that No negative weights always valid D reproducible Re-express Laplace equation as weighted average constraints Then, if are positive, so are are. 49 50 Example Fixing the Boundary uniform Simple convex shape (triangle, square, circle) Distribute points on boundary Use chord length parameterization harmonic Fixed boundary can create high distortion mean-value 51 53 Non-Convex Boundary Convex boundary creates significant distortion. Free boundary is better. 54 Page 9