Polygon Mesh Representation

My lecture notes for the course Geo1004 3D (2015) modelling of the built environment at TU Delft, Faculty of Architecture and the Built Environment

1. 1 Challenge the future
Boundary Representations 2:
Polygon Mesh Models

Ir. Pirouz Nourian

PhD candidate & Instructor, chair of Design Informatics, since 2010
MSc in Architecture 2009
BSc in Control Engineering 2005
Geo1004, Geomatics Master, Directed by Dr. ir. Sisi Zlatannova

2. 2 Challenge the future








•Example:
•Important note:
the geometry of faces is not stored; only rendered when needed!
What you see on the screen is not necessarily what you store!


Mesh Representation
A light-weight model composed of points and a set of topological
relations among them.
Points >Vertices Lines > Edges Polygons > Faces Mesh
T1: {0,4,3}
T2:{0,1,4}
Q1:{1,2,5,4}

3. 3 Challenge the future
•The geometry of a mesh can be represented by its points (known as geometrical vertices
in Rhino), i.e. a list of 3D points in ℝ
3

•The topology of a mesh can be represented based on its [topological] vertices, referring
to a ‘set’ of 3D points in ℝ
3
, there are multiple ways to describe how these vertices are
spatially related (connected or adjacent) to one another and also to edges and faces of the
mesh
•Same topology and different geometries:


Mesh
Mesh Geometry versus Mesh Topology
T1: {0,4,3}
T2:{0,1,4}
Q1:{1,2,5,4}
T1
T2
Q1
T1
T2
Q1
0
3
1
4
5
2
0
3
1
4 5
2

4. 4 Challenge the future
•formally an ordered set of Vertices, Edges and Faces or ??????=(??????,�,�)
•The ordered set describes a set of vertices and a set of faces describing how the vertices construct convex polygons by the
set of edges connecting them when embedded in ℝ
3
. (In a set there is no duplicate element. So, the vertices, edges and
faces in the above definition are unique in their sets and have no duplicates.)
•Edges and faces should not intersect or self-intersect.
•Edges correspond to straight lines, faces are generators of convex and planar polygons. A mesh represents a polyhedron if
it has planar faces without self-intersection.
•Mesh has a topological structure, in which edges are tuples of vertex indices and faces are composed of (CW/CCW
ordered) tuples of integers referring to indices of the vertices or edges
•Ensuring flatness of faces is most convenient with triangular faces. This is one of many reasons that triangular meshes are
most common and used for rendering engines.
•Faces or edges should not be degenerate; e.g. an edge that effectively corresponds to a line of length zero, or a face
giving rise to a polygon that has zero surface area. Faces (polygons) should not be self intersecting; edges should not self-
intersect either.
Proper Mesh
A mesh that does not need to be corrected!

5. 5 Challenge the future
Mesh Basics
Some preliminary definitions: fan, star, strip
A triangle strip=:ABCDEF
 ABC, CBD, CDE, and EDF
A
B
C
D
E
F
A Closed Triangle Fan or a ‘star’=:ABCDEF
A
B
C
D
E
F
 ABC, ACD, ADE, and AEF
A Triangle Fan=: ABCDE
A
B
C
D
E
 ABC, ACD, and ADE

6. 6 Challenge the future
Mesh Basics
Some preliminary definitions: free* vertices, free* edges
If an edge has less than two faces adjacent to it then it is considered free ; if a vertex is part
of such an edge it is considered as free too.
* In Rhinocommon free vertices/edges are referred to as naked.

7. 7 Challenge the future
Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University
•if a mesh is supposed to be a 2D manifold then it should meet these criteria:
1.each edge is incident to only one or two faces; and
2.the faces incident to a vertex form a closed or an open fan.

Non manifold mesh examples: (note why!)
Proper Mesh
A mesh that does not need to be corrected!?
•if a mesh is supposed to be orientable then, it should be possible to find
‘compatible’ orientations for any two adjacent faces; in which, for each pair of
adjacent faces, the common edge of the two faces has opposite orders.
•Example: Möbius band is a 2D manifold mesh that is not orientable.

8. 8 Challenge the future
Mesh Basics
Some preliminary definitions: boundary denoted as ??????
•The boundary is the set of edges incident to only one face of a manifold.
Therefore: we can conceptualize the boundary as the union of all faces of a
manifold. We can conclude that:
•If every vertex has a closed fan (or there is no edge of valence less than 2),
the given manifold has no boundary. Example: a box!
non-manifold boundary
manifold boundary
no boundary

9. 9 Challenge the future
Image Source: http://prateekvjoshi.com/2014/11/16/homomorphism-vs
homeomorphism/
Mesh Topology: Homeomorphism
clay models that are all topologically equal!?
Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University
Two 2-manifold meshes A and B are
homeomorphic if their surfaces can be
transformed to one another by topological
transformations (bending, twisting, stretching,
scaling, etc.) without cutting and gluing.

10. 10 Challenge the future
Mesh Topology: Homeomorphism
Euler-Poincare Characteristic: key to homeomorphism
Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University
????????????=??????−�+�=21−??????−??????
•?????? is the number of boundaries
•?????? is the number of “genera” (pl. of genus) or holes
•Irrespective of tessellation!
????????????=??????−�+�=21−2−0=−2

11. 11 Challenge the future
Mesh Topology: Homeomorphism
Euler-Poincare Characteristic: key to homeomorphism
•N-Manifold/Non-Manifold: Each point of an n-dimensional manifold has a
neighborhood that is homeomorphic to the Euclidean space of n-dimension

•Riddle: The above definition implies that for mapping some local features
2D maps are fine. What about the whole globe? That is not
homoeomorphic to a rectangular surface! How do we do it then?

12. 12 Challenge the future
Quiz:
Check Euler-Poincare Characteristic on two spherical meshes
•Question: we know that the Euler-Poincare Characteristic is independent of
tessellation; check this fact on the two spheres below; explain the discrepancy.
Hints: http://4.rhino3d.com/5/rhinocommon/ look at mesh [class] members
????????????=2 ????????????=66!?

13. 13 Challenge the future
Mesh Geometry
•Polygon vs Face
•Triangulate
•Quadrangulate

14. 14 Challenge the future
Mesh Geometry
•Boolean operation on meshes:
1.�∪�: Boolean Union
2.�−�: Boolean Difference
3.�∩�: Boolean Intersection

•why do they fail sometimes?

15. 15 Challenge the future
Mesh Topological Structure
•Mesh Topological Data Models:

Face-Vertex: e.g. �
0={??????
0,??????
5,??????
4} & ??????
0∈�
0,�
1,�
12,�
15,�
7










Images courtesy of David Dorfman, from Wikipedia
Face-Vertex (as implemented in Rhinoceros)

16. 16 Challenge the future
Vertex-Vertex: e.g. ??????
0~{??????
1,??????
4,??????
3}
Face-Vertex: e.g. �
0=??????
0,??????
5,??????
4, ??????
0∈�
0,�
1,�
12,�
15,�
7
Winged-Edge: e.g. �
0=�
4,�
8,�
9, �
0=??????
0,??????
1,�
0~�
1,�
12,�
0~�
9,�
23,�
10,�
20
Half-Edge: each half edge has a twin edge in opposite direction, a previous and a next








Mesh Topological Structures
Image courtesy of David Dorfman, from Wikipedia
What is explicitly stored as topology of a mesh: E.g.
Face-Vertex (as implemented in Rhinoceros)
http://doc.cgal.org/latest/HalfedgeDS/index.html

17. 17 Challenge the future
Mesh Topological Structure
•Mesh Topological Data Models:

Face-Vertex Example:
http://4.rhino3d.com/5/rhinocommon/html/AllMembers_T_Rhino_Geometry_Collections_MeshTopologyVertexList.htm











Face-Vertex (as implemented in Rhinoceros)
Name Description
ConnectedFaces Gets all faces that are connected to a given vertex.
ConnectedTopologyVertices(Int32) Gets all topological vertices that are connected to a given vertex.
ConnectedTopologyVertices(Int32, Boolean) Gets all topological vertices that are connected to a given vertex.
The MeshTopologyVertexList type exposes the following members.
Half-Edge: Half-Edge Mesh Data Structure, example implementation by Daniel Piker:
http://www.grasshopper3d.com/group/plankton

18. 18 Challenge the future
Repairing/Reconstructing Geometry
•Example: Coons’ Patch





•Code it and get bonus points!

Image courtesy of CVG Lab

19. 19 Challenge the future
Mesh Smoothing
•
For k As Integer=0 To L - 1
Dim SmoothV As New List(Of Point3d)
For i As Integer=0 To M.Vertices.Count - 1
Dim Ngh = Neighbors(M, i)
Dim NVertex As New Point3d(0, 0, 0)
For Each neighbor As point3d In Ngh
NVertex = NVertex + neighbor
Next
NVertex = (1 / Ngh.Count) * NVertex
SmoothV.Add(NVertex)
Next
M.Vertices.Clear
M.Vertices.AddVertices(SmoothV)
A = M
Next

20. 20 Challenge the future
Normal Vectors of a Mesh
•Topological Vertices versus Geometrical Points
•Joining mesh objects: What is a mesh box?
•Welding Meshes: how does it work?
•Face Normal versus Vertex Normal

Where do they come from and what do they represent?

21. 21 Challenge the future
How to Compute Mesh Normals?
•00,
00),(
vvuuv
p
u
p
vun












 1
( ) ( )
0
( )( )
N
x i next i i next i
i
N y y z z


   1
( ) ( )
0
( )( )
N
z i next i i next i
i
N x x y y


   1
( ) ( )
0
( )( )
N
y i next i i next i
i
N z z x x


  
Martin Newell at Utah (remember the teapot?)
Why?

22. 22 Challenge the future
(2D Curvature Analysis: Mesh)
•Discretization
Mesh

•Measurement
At vertices

•Attribution
To vertices
Curvature Estimation on Meshes?

23. 23 Challenge the future
Curvature Estimation on Meshes
•What was the definition of curvature?
•Change of tangents or change of normals?
•How to measure change of normals?


•ProVec = vec - (vec * M.Normals(i)) * M.Normals(i)

•Projected_Vector=Vector-(Vector.Normal).Normal Why?

•Maximum & Minimum change
•Estimation of Gaussian & Mean curvature on mesh vertices

24. 24 Challenge the future
Questions? [email protected]