Mesh final pzn_geo1004_2015_f3_2017

1Challenge the future

Boundary Representations

Polygon Mesh Models

Geo1004, Geomatics Master, Directed by Dr. Ir. Sisi Zlatannova

Dr. Ir. Pirouz NourianAssistant Professor of Design Informatics


General Idea

Representing Higher-Dimensional Objects with their Lower-Dimensional Boundary

e.g. representing a box only with a 6-face container; i.e. not representing the inner

volume.


What do we mean by representation

anyway?

Representing Higher-Dimensional Objects with their Lower-Dimensional Boundary

e.g. representing a box only with a 6-face container; i.e. not representing the inner

volume.

Representing a triangular surface by storing its border triangle; i.e. the triangle that

marks the boundary between what is in and out of this 2D space is itself a 1D space.


On terminology

• Geometry: Point (0D), Line(1D), Polygon(2D), Polyhedron (3D)

• Geometry: Point (0D), Curve(1D), Surface(2D), Solid (3D)

• Topology: Vertex(0D), Edge(1D), Face(2D), Body(3D)

• Graph Theory: Node, Link, (and n-Cliques)



• 0D: Point

• 1D: Curve (represented by boundary Points)

• 2D: Surface (represented by boundary Curves)

• 3D: Solid (represented by boundary Surfaces)


Boundary Representations & n-

Manifolds

• 0D: Point (the space therein everywhere resembles a 0-dimensional Euclidean Space)

• 1D: Curve (the space therein everywhere resembles a 1-dimensional Euclidean Space)

• 2D: Surface (the space therein everywhere resembles a 2-dimensional Euclidean Space)

• 3D: Solid (the space therein everywhere resembles a 3-dimensional Euclidean Space)



• 0D: Point




• Parametric surfaces (e.g.

NURBS)

• Triangulated Irregular Networks

(TINS)

• Grid (Networks)

• Constrained Triangulation

• Subdivision Surfaces

• Polygon Meshes


Nota Bene!

• All these objects might be ‘embedded’ geometrically in a 3D

space. Geometrically they are all 3D objects (3D point, 3D

curve, 3D surface, 3D solid); but their ‘inner space’ has

different topological dimensionality

• The concept of manifold is adopted from differential

geometry and it is essentially about continuous space; here

we are talking about approximations of continuous objects

and generalized notions of manifold-ness.

• In this course, we are focused on 2-Manifold Boundary

Representations modelled as ‘polygon meshes’ and NOT

polyhedral meshes.


• 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 RepresentationA 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}


• 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 ℝ3

• The topology of a Mesh can be represented based on its [topological] vertices, referring

to a ‘set’ of 3D points in ℝ3, 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:

MeshMesh Geometry versus Mesh Topology

T1: {0,4,3}T2:{0,1,4}Q1:{1,2,5,4}

T1

T2Q1

T1

T2Q1

0

3

1

45

2 0

3

1

4 5

2


Triangulated MeshesSome 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

AB

CD

E

ABC, ACD, and ADE


Mesh BasicsSome 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.


Mesh BasicsSome preliminary definitions: border denoted as 𝜹

• A 1D border is the set of connected edges incident to only one face of a 2-manifold, i.e. composed of free edges. We can conclude that:

• If every vertex has a closed fan (i.e. a star), or there is no edge of valence (number of neighbors) less than 2, the given manifold has no border. Example: a box!

non-manifold border manifold borderno [1D] border


• 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!)

Images courtesy of Dr. Ching-Kuang Shene from Michigan Technological University

Manifold MeshA 2-manifold [everywhere] locally resembles a flat surface

• 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 non-printable.


Mesh Topology: HomeomorphismEuler-Poincare Characteristic: key to homeomorphism

• N-Manifold: Each point of an n-dimensional manifold has a neighborhood

that is homeomorphic to the Euclidean space of dimension n

• Riddle: The above definition implies that for mapping some local geographic features 2D maps

are fine. What about the whole globe? That is not homoeomorphic to a rectangular surface!

How do we do it then?


Image Source: http://prateekvjoshi.com/2014/11/16/homomorphism-vs homeomorphism/

Mesh Topology: Homeomorphismclay models that are all topologically equal!


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.


Image Source: https://www.shapeways.com/blog/archives/21752-a-3d-printed-topology-joke.html

Mesh Topology: Homeomorphismclay models that are all topologically equal!


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.

This is how!



Only for 2D boundary representations*


𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 𝑔 − 𝛿

𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 2 − 0 = −2

* The Euler characteristic equation will take the form of V-E+F-B

• 𝛿 is the number of [connected] borders

• 𝑔 is the number of “genera” (pl. of genus) or handles

• Irrespective of tessellation!




𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = −2 𝑓𝑜𝑟 𝑑𝑜𝑢𝑏𝑙𝑒 𝑡𝑜𝑟𝑖(ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ 2 ℎ𝑜𝑙𝑒𝑠)

𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 0 𝑓𝑜𝑟 𝑡𝑜𝑟𝑖 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ 1 ℎ𝑜𝑙𝑒)

𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 1 𝑓𝑜𝑟 𝑑𝑖𝑠𝑘𝑠 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑢𝑟𝑓𝑎𝑐𝑒𝑠)

𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 𝑓𝑜𝑟 𝑏𝑎𝑙𝑙𝑠 (ℎ𝑜𝑚𝑒𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑐 𝑡𝑜 𝑠𝑜𝑙𝑖𝑑𝑠 𝑤𝑖𝑡ℎ𝑜𝑢𝑡 ℎ𝑜𝑙𝑒𝑠)




𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 𝑔 − 𝛿

• 𝛿 is the number of borders

• 𝑔 is the number of “genera” (pl. of genus) or holes


𝜒 𝑀 = 𝑉 − 𝐸 + 𝐹 = 2 1 − 2 − 0 = −2


Poincare Dualitya pairing between k-dimensional features and dual

features of dimension n-k in ℝ𝑛

PRIMAL DUAL

0D vertex (e.g. as a point) 1D edge

1D edge (e.g. as a line segment) 0D vertex

PRIMAL DUAL

0D vertex (e.g. a point) 2D face

1D edge (e.g. a line segment) 1D edge

2D face (e.g. a triangle or a pixel) 0D vertex

PRIMAL DUAL

0D vertex (e.g. a point) 3D body

1D edge (e.g. a line segment) 2D face

2D face (e.g. a triangle or a pixel) 1D edge

3D body (e.g. a tetrahedron or a voxel) 0D vertex

ℝ

ℝ3

ℝ2




PRIMAL DUAL

0D vertex (e.g. as a point) 1D edge

1D edge (e.g. as a line segment) 0D vertexℝ

A hypothetical street network: a Junction-to-Junction adjacency graph (left) versus a Street-to-Street adjacency graph (right), both ‘undirected’, after Batty (Batty, 2004): red dots

represent graph nodes, and blue arcs represent graph links.

Batty, M., 2004. A New Theory of Space Syntax. CASA Working Paper Series, March.




PRIMAL DUAL

0D vertex (e.g. a point) 2D face

1D edge (e.g. a line segment) 1D edge

2D face (e.g. a triangle or a pixel) 0D vertex

ℝ2

A Voronoi tessellation of 2D space and its dual that is a Delaunay triangulation




PRIMAL DUAL

0D vertex (e.g. a point) 3D body

1D edge (e.g. a line segment) 2D face

2D face (e.g. a triangle or a pixel) 1D edge

3D body (e.g. a tetrahedron or a voxel) 0D vertex

ℝ3

representing adjacencies between 3D cells or bodies via their dual vertices (Lee, 2001)

Lee, J., 2001. 3D Data Model for Representing Topological Relations of Urban Features. San Diego, CA, s.n.


• formally an ordered set of Vertices, Edges and Faces or 𝑀 = (𝑉, 𝐸, 𝐹)

• 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 ℝ3. (In a set there is no duplicate element. Therefore, 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.

Valid Mesh (often expected to be a 2–Manifold Surface)


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

𝜒 𝑀 = 2𝜒 𝑀 = 66!?

http://4.rhino3d.com/5/rhinocommon/


Mesh Topological Structure

• Mesh Topological Data Models:

Face-Vertex: e.g. 𝐹0 = {𝑉0, 𝑉5, 𝑉4} & 𝑉0 ∈ 𝐹0, 𝐹1, 𝐹12, 𝐹15, 𝐹7

Images courtesy of David Dorfman, from Wikipedia

Face-Vertex (as implemented in Rhinoceros)


Vertex-Vertex: e.g. 𝑉0~{𝑉1, 𝑉4, 𝑉3}

Face-Vertex: e.g. 𝐹0 = 𝑉0, 𝑉5, 𝑉4 , 𝑉0 ∈ 𝐹0, 𝐹1, 𝐹12, 𝐹15, 𝐹7

Winged-Edge: e.g. 𝐹0 = 𝐸4, 𝐸8, 𝐸9 , 𝐸0 = 𝑉0, 𝑉1 , 𝐸0~ 𝐹1, 𝐹12 , 𝐸0~ 𝐸9, 𝐸23, 𝐸10, 𝐸20

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.


http://doc.cgal.org/latest/HalfedgeDS/index.html

http://doc.cgal.org/latest/HalfedgeDS/index.html


Mesh Topological Structure

• Mesh Topological Data Models:

Face-Vertex Example:

http://4.rhino3d.com/5/rhinocommon/html/AllMembers_T_Rhino_Geometry_Collections_MeshTopologyVertexList.htm


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

http://4.rhino3d.com/5/rhinocommon/html/AllMembers_T_Rhino_Geometry_Collections_MeshTopologyVertexList.htm

http://4.rhino3d.com/5/rhinocommon/html/M_Rhino_Geometry_Collections_MeshTopologyVertexList_ConnectedFaces.htm

http://4.rhino3d.com/5/rhinocommon/html/M_Rhino_Geometry_Collections_MeshTopologyVertexList_ConnectedTopologyVertices.htm

http://4.rhino3d.com/5/rhinocommon/html/M_Rhino_Geometry_Collections_MeshTopologyVertexList_ConnectedTopologyVertices_1.htm

http://4.rhino3d.com/5/rhinocommon/html/T_Rhino_Geometry_Collections_MeshTopologyVertexList.htm

http://www.grasshopper3d.com/group/plankton


Mesh Geometry

• Polygon vs Face

• Triangulate

• Quadrangulate


Mesh Intersections

• Boolean operation on Meshes:

1. 𝐴 ∪ 𝐵: Boolean Union

2. 𝐴 − 𝐵: Boolean Difference

3. 𝐴 ∩ 𝐵: Boolean Intersection


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?


How to Compute Mesh Normals?

•

00,

00 ),(vvuuv

p

u

pvun

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?


Repairing/Reconstructing Geometry

• Example: Coons’ Patch

• Code it and get bonus points!

Image courtesy of CVG Lab


Mesh Smoothing

•

for (int k = 0; k <= L - 1; k++) {List<Point3d> SmoothV = new List<Point3d>();for (int i = 0; i <= M.Vertices.Count - 1; i++) {

List<Point3d> Neighbours = Neighbors(M, i);Point3d NVertex = new Point3d(0, 0, 0);foreach (point3d neighbor in Neighbours ) {

NVertex = NVertex + neighbor;}NVertex = (1 / Ngh.Count) * NVertex;SmoothV.Add(NVertex);

}M.Vertices.Clear();M.Vertices.AddVertices(SmoothV);A = M;

}


Questions? [email protected]

mailto:[email protected]

Mesh final pzn_geo1004_2015_f3_2017

Engineering

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