1 Background and definitions Cindy Grimm

2 Siggraph 2005, 8/1/ Overview What does it mean to be manifold? What is an atlas? Disks, charts, overlaps, transition functions Mesh example Traditional atlas definition Building an atlas for a circle manifold Constructive definition Building an abstract manifold (circle, no explicit geometry) Writing functions on manifolds Circular curve

3 Siggraph 2005, 8/1/ Disks and deformations A disk is a connected region Doesn’t have to be circular Boundary not included Disk of dimension R n in dimension R m, m ≥n Valid deformations Stretching, squishing Invalid deformations Folding, tearing Disk Not disk ok Not ok

4 Siggraph 2005, 8/1/ Traditional manifold definition A manifold M is an object that is locally Euclidean For every p in M Neighborhood (disk) U around p, U in M U maps (deforms) to disk in R n n is dimension of manifold No folding, tearing Self-intersections ok Boundaries ok Map boundary to half disk Singularities ok Smooth, closed manifold Manifold with self-intersections Singularity at tip Manifold with boundary Surface examples

5 Siggraph 2005, 8/1/ Traditional definition, cont. What is not ok? “Fins” Non-manifold Can deform to separate self-intersections

6 Siggraph 2005, 8/1/ Other manifold examples Joint angles Pendulum in 2D is a 1D manifold (circle) Angle Two joints in 2D: circle X circle One joint in 3D: Panorama (rotating a camera around its origin) Each picture is piece of the manifold Stitched-together images form manifold Rover, nasa.gov

7 Siggraph 2005, 8/1/ What is an atlas? History: Mathematicians 1880’s How to analyze complex shapes using simple maps Cartographers World atlas Informal definition Map world to pages Each page planar (printed on piece of paper) Every part of world on at least one page Pages overlap May not agree exactly Agree enough to navigate Overlap The world

8 Siggraph 2005, 8/1/ Local parameterization Atlas on a manifold Collection of pages Parameterization (local) Different properties Size, navigation, percentage Mercator Map Images: The atlas of Canada and About Geography

9 Siggraph 2005, 8/1/ Embedded vs abstract manifold M may be a surface of dimension n embedded in R m n is dimension of surface m >n Surface is of dimension n=2 (a piece of paper rolled up) Embedded in m=3 dimensions Joint angle forms n=1 dimension manifold (can increase/decrease angle) M may be abstract (topological) No specific geometry, e.g., joint angle

10 Siggraph 2005, 8/1/ Applying traditional manifold definition Familiar phrase: Assuming the mesh is manifold… How do you prove a mesh is manifold? Show: Every point on the mesh has a local neighborhood Flatten part of 3D mesh down into 2D Requires some constraints on the mesh At most two faces per edge Vertices can be flattened

11 Siggraph 2005, 8/1/ A mesh is manifold if… … we can construct local maps around points. Three cases: Point is in face Point is on edge Point is vertex

12 Siggraph 2005, 8/1/ Case 1: Point is in face Triangle goes to triangle P 3D 2D

13 Siggraph 2005, 8/1/ Case 2: Point is on edge Rule: There are exactly two faces adjacent to each interior edge, and one face adjacent to any boundary edge Can “hinge” around edge to flatten into plane P 3D 2D

14 Siggraph 2005, 8/1/ Case 3: Point is vertex Rule: The faces around a vertex v can be flattened into the plane without folding or tearing Triangles may change shape Vertices w i adjacent to v can be ordered w 0,…,w n-1 such that the triangles w i,v,w (i+1)mod n all exist P 2D 3D Bad vertex

15 Siggraph 2005, 8/1/ Traditional atlas definition Given: Manifold M Construct: Atlas A Chart Region U c in M (open disk) Region c in R n (open disk) Function  c taking U c to c Inverse Atlas is collection of charts Every point in M in at least one chart Overlap regions Transition functions:  01 =  1 o  0 -1 smooth M

16 Siggraph 2005, 8/1/ A circle manifold Uses: Closed curves Representing a single joint angle Challenges: Repeating points 0 is the same as 2  Tools exist for operating on lines/line segments Spline functions Optimization Implementation: How to represent points? How to specify charts? Overlaps? Transition functions?

17 Siggraph 2005, 8/1/ A circle manifold How to represent points on a circle? Problem: Want operations (e.g., addition) to return points on the circle In R 2, must project to keep points on circle Solution: Represent circle as repeating interval [0,2  ) Point  +2  k is same for all k integer Always store point in [0,2  ) range (1,0) These are embeddings This is not 0 2 

18 Siggraph 2005, 8/1/ ( ) Chart on a circle Chart specification: Left and right ends of U c Counter-clockwise order to determine overlap region Range c = (-1/2,1/2) Simplest form for  c Translate Center in [0,2  ) Scale - 1/2 1/2 LL RR Uc -1/2 1/2 ( ) cc LL RR

19 Siggraph 2005, 8/1/ Making a chart AddChart(  L,  R ) { // in 0,2  range  C = 0; s = 1; // Does not cross 0, 2  boundary if (  L <  R ) {  C = 1/2 * (  L +  R ); s = 1/2 / (  R -  C ); } else { // Add 2  to right end point  C = 1/2 * (  L +  R + 2  ); s = 1/2 / (  R + 2  -  C ); } // Make sure in 0,2  range if (  C ≥ 2  )  C -= 2  ; // Make new chart with  C and s … } LL RR   LL RR   R’ LL RR LL RR  =0

20 Siggraph 2005, 8/1/ Chart on a circle Chart  function Sort input point first (  →   ) Euclidean distance is topological distance  (  in ) {  =  in ; // Find the value for  (+-2  ) closest to  c if ( |  in -  C | ≤  )  =  in ; else if ( | (  in + 2  ) -  C | ≤  )  =  in + 2  ; else if ( | (  in - 2  ) -  C | ≤  )  =  in - 2  ; return (  -  C ) * s; } cc ( ) cc    cc ( )  

21 Siggraph 2005, 8/1/ Chart on a circle Alpha inverse t in (-1/2, 1/2) Returned point in [0,2  )  -1 ( t ) {  = t / s +  C ; // Converts to 0,2  range return CirclePoint(  ); }

22 Siggraph 2005, 8/1/ An example circle atlas Four charts, each covering ½ of circle 4 points not covered by two charts Overlaps ½ of chart for charts i, i+1 Transition function: translate No overlap for charts i, i+2 Chart 0 Chart 1 Chart 2 Chart 3 Chart 0 Chart 1 Overlap regions -1/2 1/2 -1/2 1/2 Transition functions: Chart 1 Chart 0

23 Siggraph 2005, 8/1/ Why do I care? Charts Avoids wrapping problem Get chart over area of interest Run existing code as-is Optimize pendulum position Atlas Embed circle using existing techniques E.g., splines, polynomials No special boundary cases, e.g., duplicated end points, geometric constraints

24 Siggraph 2005, 8/1/ Writing functions on manifolds Do it in pieces Write embed function per chart Can use any R n technique Splines, Radial Basis Functions, polynomials… Doesn’t have to be homogenous! Write blend function per chart k derivatives must go to zero by boundary of chart Guaranteeing continuity Normalize to get partition of unity Spline functions get this for free

25 Siggraph 2005, 8/1/ Final embedding function Embedding is weighted sum of chart embeddings Generalization of splines Given point p on manifold Map p into each chart Blend function is zero if chart does not cover p Continuity is minimum continuity of constituent parts Embed Blend Map each chart

26 Siggraph 2005, 8/1/ Alternative formulation Define using transition functions Derivatives Pick point p in a chart Map p to all overlapping charts Embed Blend

27 Siggraph 2005, 8/1/ Embedding the circle Embed each chart Quadratic polynomial Blend function for each chart Blend function B-spline basis Divide by sum to normalize Blended result -1/2 1/2 1 Normalized blend function -1/2 1/2 Proto blend function

28 Siggraph 2005, 8/1/ What if we don’t have a circle? Previous approach relied on having an existing manifold Cover manifold with charts Suppose you want to make a manifold from scratch? Create manifold object from disks and how they overlap Think of someone handing you an atlas; you can “glue” the pages together where they overlap to re- create the manifold Resulting object is an abstract manifold Requires some care to ensure glued-together object is actually manifold

29 Siggraph 2005, 8/1/ Constructive definition Construct: Proto-Atlas A Disks Region c in R n (open disk) Overlap regions between disks U cc’  c Transition functions between disks  cc’ To create a manifold object from A: “Glue” points together that are the same, i.e.,  cc’ (p)=q implies p == q Transition functions must make sense Reflexive:  cc (p)=p Symmetric:  c’c (  cc’ (p))=p Transitive:  ik (p)=  ij (  jk (p))

30 Siggraph 2005, 8/1/ Four chart proto-atlas for circle Four intervals, each (-1/2, 1/2) Overlap regions: U 1,2 = (0,1/2) → U 2,1 (-1/2,0) U 1,0 = (-1/2,0) → U 0,1 (0,1/2) U 1,1 = (-1/2, 1/2) U 1,3 empty Transition functions:  1,2 (t) = t – 1/2  1,0 (t) = t + 1/2  1,1 (t) = t … And similarly for other charts Chart 0 Chart 1 Chart 2

31 Siggraph 2005, 8/1/ Resulting abstract manifold What is the form of a point on the manifold? List of charts and points in those charts How do you map from a point on the manifold to a point in a chart? Extract the point from the list (if it exists)  c ( ) = t Chart 0 Chart

32 Siggraph 2005, 8/1/ Terminology review Manifold, adj: An object is manifold if is locally Euclidean Manifold with self-intersections Not manifold

33 Siggraph 2005, 8/1/ Terminology review Manifold, noun: An object M that is manifold Disk: A connected region Chart: A map from the disk U c on the object M to a disk c in R n. Provides a local parameterization  c : U c →c Disk Not disk UcUc M cc c R2R2

34 Siggraph 2005, 8/1/ Terminology review Manifold, noun: An object that is manifold Atlas: Collection of charts that cover the manifold Overlap region: Where two charts cover the same part of the manifold. U c ∩ U c’ not empty Transition function: A map  cc’  from disk in R n to disk in R n UcUc U c’  cc’ C C’

35 Siggraph 2005, 8/1/ Terminology review Constructive manifold: An abstract manifold built from a proto-atlas Given: Disks Overlaps between disks Transition functions between overlap regions Get: Abstract object that is manifold