Background and definitions

1
Background and definitions

• Cindy Grimm

2
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
Disks and deformations

• A disk is a connected region
• Doesnt have to be circular
• Boundary not included
• Disk of dimension Rn in dimension Rm, m n
• Valid deformations
• Stretching, squishing
• Invalid deformations
• Folding, tearing

Disk
Not disk
ok
Not ok
Not ok
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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 Rn
• n is dimension of manifold
• No folding, tearing
• Self-intersections ok
• Boundaries ok
• Map boundary to half disk
• Singularities ok

Surface examples
5
Traditional definition, cont.

• What is not ok?
• Fins

Non-manifold
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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
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What is an atlas?

• History
• Mathematicians 1880s
• 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

The world
Overlap
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Local parameterization
Mercator Map

• Atlas on a manifold
• Collection of pages
• Parameterization (local)
• Different properties
• Size, navigation, percentage

Images The atlas of Canada and About Geography
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Embedded vs abstract manifold

• M may be abstract (topological)
• No specific geometry, e.g., joint angle

• M may be a surface of dimension n embedded in Rm
• n is dimension of surface
• m gtn

Surface is of dimension n2 (a piece of paper
rolled up) Embedded in m3 dimensions
Joint angle forms n1 dimension manifold (can
increase/decrease angle)
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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

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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

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Case 1

• Point is in face
• Triangle goes to triangle

P
3D
2D
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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
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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 wi adjacent to v can be ordered
  w0,,wn-1 such that the triangles wi,v,w(i1)mod
  n all exist

3D
2D
Bad vertex
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Traditional atlas definition

• Given Manifold M
• Construct Atlas A
• Chart
• Region Uc in M (open disk)
• Region c in Rn (open disk)
• Function ac taking Uc to c
• Inverse
• Atlas is collection of charts
• Every point in M in at least one chart
• Overlap regions
• Transition functions
• y01 a1 o a0-1 smooth

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A circle manifold

• Uses
• Closed curves
• Representing a single joint angle
• Challenges
• Repeating points
• 0 is the same as 2p
• Tools exist for operating on lines/line segments
• Spline functions
• Optimization
• Implementation
• How to represent points?
• How to specify charts?
• Overlaps? Transition functions?

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A circle manifold

• How to represent points on a circle?
• Problem
• Want operations (e.g., addition) to return points
  on the circle
• In R2, must project to keep points on circle
• Solution
• Represent circle as repeating interval 0,2p)
• Point q2pk is same for all k integer
• Always store point in 0,2p) range

These are embeddings
0 2p
This is not
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Chart on a circle
qR
qc

• Chart specification
• Left and right ends of Uc
• Counter-clockwise order to determine overlap
  region
• Range c (-1/2,1/2)
• Simplest form for ac
• Translate
• Center in 0,2p)
• Scale

Uc
( )
qL
-1/2
1/2
qR
qL
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Making a chart
AddChart( qL, qR ) // in 0,2p range qC
0 s 1 // Does not cross 0, 2p
boundary if (qL lt qR ) qC 1/2
(qL qR ) s 1/2 / (qR - qC)
else // Add 2p to right end point
qC 1/2 (qL qR 2p ) s 1/2 /
(qR 2p - qC) // Make sure in 0,2p
range if (qC 2p) qC - 2p // Make new
chart with qC and s
qR
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Chart on a circle

• Chart a function
• Sort input point first (q?q, q2p ,q-2p )
• Euclidean distance is topological distance

a( qin ) q qin // Find the value
for q (-2p) closest to qc if ( qin - qC
p ) q qin else if ( (qin
2p) - qC p ) q qin 2p else
if ( (qin - 2p) - qC p ) q qin
- 2p return (q - qC) s
q-2p
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Chart on a circle

• Alpha inverse
• t in (-1/2, 1/2)
• Returned point in 0,2p)

a-1( t ) q t / s qC // Converts
to 0,2p range return CirclePoint(q)
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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, i1
• Transition function translate
• No overlap for charts i, i2

1/2
-1/2
Chart 1
Transition functions
1/2
-1/2
Chart 0
Chart 0
Chart 1
Overlap regions
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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

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Writing functions on manifolds

• Do it in pieces
• Write embed function per chart
• Can use any Rn technique
• Splines, Radial Basis Functions, polynomials
• Doesnt 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

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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

Map each chart
Embed
Blend
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Alternative formulation

• Define using transition functions
• Derivatives
• Pick point p in a chart
• Map p to all overlapping charts

Embed
Blend
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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

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What if we dont 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

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Constructive definition

• Construct Proto-Atlas A
• Disks
• Region c in Rn (open disk)
• Overlap regions between disks
• Ucc Ì c
• Transition functions between disks
• ycc
• To create a manifold object from A
• Glue points together that are the same, i.e.,
• ycc(p)q implies p q
• Transition functions must make sense
• Reflexive ycc(p)p
• Symmetric ycc(ycc(p))p
• Transitive yik(p) yij(yjk(p))

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Four chart proto-atlas for circle

• Four intervals, each (-1/2, 1/2)
• Overlap regions
• U1,2 (0,1/2) ? U2,1 (-1/2,0)
• U1,0 (-1/2,0) ? U0,1 (0,1/2)
• U1,1 (-1/2, 1/2)
• U1,3 empty
• Transition functions
• y1,2(t) t 1/2
• y1,0(t) t 1/2
• y1,1(t) t
• And similarly for other charts

Chart 1
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Resulting abstract manifold

• What is the form of a point on the manifold?
• List of charts and points in those charts
  lt(c,t),gt
• lt(0,0.2),(1,-0.3)gt
• 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)
• ac(lt(c,t)gt) t

0.2
Chart 0
Chart 1
-0.3
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Terminology review

• Manifold, adj An object is manifold if is
  locally Euclidean

Not manifold
Manifold with self-intersections
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Terminology review

• Manifold, noun An object M that is manifold
• Disk A connected region
• Chart A map from the disk Uc on the object M to
  a disk c in Rn. Provides a local parameterization
  ac Uc ?c

Disk
Not disk
ac
Uc
c
R2
M
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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. Uc n Uc not empty
• Transition function A map ycc from disk in Rn
  to disk in Rn

Uc
Uc
ycc
C
C
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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