Designing Patterns using Triangle- Quad Hybrid...

Designing Patterns using Triangle-Quad Hybrid Meshes

CHI-HAN PENG (KAUST, Saudi Arabia)et al.SIGGRAPH 2018

Copyright of figures and other materials in the paper belongs original authors.

Presented by Qi-Meng Zhang

2018. 10. 25

Computer Graphics @ Korea University

Qi-meng Zhang| 2018. 10. 25 | # 2Computer Graphics @ Korea University

Abstract

• Generate mesh patterns that consist of a hybrid of both triangles and quads.

▪ Fit the surface boundaries and curvatures

• Two key advantages:

▪ Novel-looking

▪ A finer discretization of angle deficits than using triangles or quad along

Fig. 1Equilateral Triangle Squares

Mixing


• Motivation

▪ Large body of research studying the meshing of surfaces using triangles, quad or hex meshes

▪ Existing research focus on quad-dominant mesh, that mainly consist of quads with a few isolated triangles

• Question

▪ How to tile a finite domain with a given boundary using squares and regular triangles?

1. Introduction

2D Boundary

Optimization Algorithm(compute a single feasible tiling)

3D Surface

Object Function(select particular tiling)

Patch Graph(encode boundaries of individual 2D meshes)

Combine multiple 2D solutions


• Discovered novel polygonal patterns that cannot be derived from existing triangle/hex or quad meshes

• Exhaustively enumerate 2D triangle-quad hybrid mesh pattern for 2D boundary by an IP(integer programming) optimization-based method

• An extension to 3D that enables the computation of compatible 2D solutions that can be deformed and glued together to obtain a 3D design

• UI for designing a patch segmentation on 3D surface

• Applied to architecture designs and artistic patterns create

1. Introduction

Contribution


2. Related Work

2.1 Tiling and Tessellation

“Wang Tiles for image and texture generation”[Cohen(Microsoft research) et al./SIGGRAPH 2003]

Wikipedia image


• Quasicrystal pattern

2. Related Work

2.2 Crystallography

“A Guide to the Symmetry Structure of Quasiperiodic Tiling Classes”[ J Hermisson(University of Vienna) et al./Journal de Physique 1997]

Inflation Rule: square-triangle tiling with12-fold symmetry


2. Related Work

2.3 Surface Meshing

“Regular Meshes from Polygonal Patterns”[ Vaxman(Utrecht University) et al./TOG2017]

“Pattern-Based Quadrangulation for N-sides Patches”[ Takayama(ETH Zurich) et al./CGF2014]


2. Related Work

2.4 Architecture

“Polyhedral Patterns”[ Jiang (KAUST) et al./TOG 2015]

Yas Mall(Abu Dhabi)Islamic Art Exhibition(Louvre) Fiera Milano


• Definition of regular hybrid meshes

• Subdivision scheme

• Discrete coordinate system

▪ Encode the vertices in hybrid mesh

• Generate planar regular hybrid mesh

3. Regular Hybrid Meshes In the Plane


• Definition

▪ Configuration of Vertex

• The sequence of types of its adjacent faces in the counter-clockwise order

▪ Types of Vertex

• Interior Vertices and Boundary Vertices


Fig.2

➢ Four configurations for an interior vertex


• Boundary vertices: can be distinguished by boundary angle▪ any integer multiple of

▪ 12 possible directions for half-edge(*)

• Each half-edge’s direction can encode as an integer


➢ All configurations for boundary angles

Fig.2

(*)We assume the half-edges of a face circulate in the counter-clockwise order

∈ [0,11]


• Inserts all edge midpoints as new vertices

• Triangle

▪ Each midpoint is connected to

• Opposite vertex

• Two other edge midpoints

• Quad

▪ Each edge midpoint is connected to

• Two vertices

• Midpoint of the opposite edge

3.1 Subdivision Scheme

Fig.3

5 − 𝑒𝑑𝑔𝑒 − 𝑎

2 − 𝑒𝑑𝑔𝑒3 − 𝑒𝑑𝑔𝑒

1 − 𝑒𝑑𝑔𝑒

30°

≈ 30°

5 − 𝑒𝑑𝑔𝑒 − 𝑏


• Main path

▪ A connected sequence of half-edges with the same direction label

3.1 Subdivision Scheme

12 directionsFig.3

➢ Main paths at the 12 directions form a vertex

0

1

234

5

6

7

89

1011


• Uniquely encode every vertex in a regular hybrid mesh with a fixed-length sequence of integers

▪ Arbitrarily chosen origin vertex and orient the mesh in a Cartesian System

• E.g. aligning a half-edge to the x-axis

3.2 Discrete Coordinate System

？

？

？


• Edges of a regular hybrid mesh

▪ In Euclidean plane 𝐸2 , length 2

▪ Direction vectors of edges only have 𝑥, 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 from

▪ Each vertex have a coordinate vector 𝑣, with integers(A,B,C,D)

3.2.1 The 4D System

Fig.4

➢ Possible edge direction vector:

and multiplication with -1.

A, C,B ,D

（2,0,0,0）

（1,0,0,1）

（0,1,1,0）

（0,0,2,0）（-1,0,0,1）

（0,-1,1,0）

➢ A+D,B+C 𝑡𝑎𝑘𝑒 𝑣𝑎𝑙𝑢𝑒 𝑖𝑛{0,2, −2}


• (A,B,C,D)—>(x1,y1)

▪ Main path in : 0-,3-,6-,9-th direction

become straight

▪ Length 2 and length 3 are mapped to same

vector

• 4D-coordinates(2,0,0,0),(0,1,0,0)

▪ Linear map P1

▪ 5-edge can project into (±5,0)or(0, ±5)

3.2.2 Projection 1

0

1

234

5

6

7

89

1011

Fig.4


• (A,B,C,D)—>(x2,y2)


become straight

▪ Length 2 and length 3 are mapped to same vector

• 4D-coordinates(0,1,1,0),(3/2,0,0,1/2)

• (A,B,C,D)—>(x3,y3)


become straight

3.2.3 Projection 2 and 3

Fig.4


• Three-2D coordinate cannot be independent

• Three projections are related by the equations

• Inversion formula

▪ Retrieve 6-coordinates—>4D coordinates

3.2 Combination Three Projection


3.2 Six-integer Coordinates

Fig.4

Fast checking if two vertices lie on same main path:

(1),(2);(3),(4);(5),(6);

If have same (2): lie on 0-,6- direction







• Main steps:

3.3 Hybrid Mesh Generation in a Given Boundary

(1)Initialization (2)Edge enumeration

(3)Mesh generation (4)Complete enumeration

(5)Solution optimization

Input: *boundary descriptionCompute: midpoints,6-integer coordinates

A superset S of potential edgeplacements within the boundary(based integer programming IP)

Compute a regular hybrid mesh by selecting edges in S

Setup with different objective function

4 55

4

7

4

563

5

5

7

Fig.4

*Boundary description as(4,5,5,4,7,4,5,6,3,5,7,5)


• Enumeration

▪ All pairs of boundary points (vertices and

midpoints)could lie on a main path inside the

boundary

▪ All possible edges between all pairs

• Test lie on a main path

▪ Coordinate test

▪ Inside test

▪ Decomposition test

• Decomposed into 1-edges, 3-edges, 5-edges-a/b

▪ IP problem

3.3.1 Edge Enumeration

Fig. 5.

Fig. 5.


• Integer Programming

▪ Additional constraint that some or all Variables are restricted to be integers

• Decomposition test

▪ Solving following IP problem

• V denote the 6-integer vector between the two points

• (t0,t1,t2,t3) denote the numbers of the four edge types(1-edge, 3-

edges, 5-edges-a/b)

• Ti is the six-integer vector encoding edge type i in a direction

• (x0,x1,x2,x3) denote all possible edge combinations

3.3.1 Edge Enumeration


• Enumerate unique potential placements of vertices

▪ Based on 4D coordinates

• Enumerate all possible edge configuration about vertices.

3.3.2 Mesh Generation

j

➢ 𝐸𝑖 and 𝐶𝑗, 𝑚 are Boolean various

➢ Nj is the number of edge placements adjacent to vj

The configuration active if and only if all corresponding edge are selectedand all other edges at vj are not selected


• Complete IP problem formulate as

3.3.2 Mesh Generation

Constraint 4

Constraint 3

Constraint 2

Boolean various

• Complete Enumeration


• Adding objective function to IP problem

3.3.4 Configuration Optimization

Fig.6 Fig.7

Penalizing:40 “butterfly”

Prioritizing:84“butterfly”

“butterfly” configuration

Penalizing:as-regular-as-possible

Penalizing:as-fractured-as-possible

Weight for Cj,m


• Adding following function to objective function

3.3.5 Symmetry Optimization

Fig.6

Fig.7

Scalar field


• Enumerate a superset of potential placements of boundary 2-edges close to the given boundary

• Find best approximates subset of half-edges

• Formulate IP problem as:

3.4 Approximating 2D Boundary Curves

Fig.6


• Angle defect 𝛿(𝑣)

▪ 𝑠(𝑣) is the sum of angles at the corners of the faces at 𝑣

▪ Interior vertex :𝛿 𝑣 = 360° − 𝑠(𝑣)

▪ Boundary vertex: 𝛿 𝑣 = 180° − 𝑠(𝑣)

4. Hybrid Meshes on Surfaces

Fig.8 angle defect (3D)


• In 3D

▪ Angle defect= discrete Gaussian curvature(if angles are measured in radians)

• Nearly regular 3D hybrid meshes （hybrid mesh）

▪ Combinatorial angles(c-angle)

• At the vertices of a triangle or quad to be 60° or 90°

▪ Combinatorial edge lengths (c-length)

4 Hybrid Meshes on Surfaces

*Appendix 1


• Each patch is combinatorially equivalent to a patch in a 2D regular hybrid mesh

• C-index

▪ Sum of c-angles divided by 30°

4.1 Patch Graph

Patch graph vertices: c-index≠ 6Patch graph edge=main path

Fig.9

Fig.3


• Overview

4.2 Generating Hybrid Meshes on Surfaces

Fig.10


• Re-mesh S by a hybrid mesh M(close to a regular one )

▪ Total Gaussian curvature inside patch closed to zero

• Discrete Gauss-Bonnet theorem

▪ Statement about surface which connects their geometry (in the sense of curvature) to their topology (in the sense of the Euler Characteristic).

4.2.1 Patch Graph Construction

total angle defect = 360°

=F-E+V

(Gaussian Curvature)

*Appendix 2


• Decompose S into patches for which equals 0


A patch of a regular 3D hybrid mesh has =1,TID=0,Since 𝛿 𝑣 = 0 for all interior vertices v

B

*TID: total interior angle defect


• Compute TID for a disk-like region

▪ 1) Compute B(boundary angle defect)

• Using 3D surface geometric angle

▪ 2) Calculate c-indices

• Round angle to closest integer multiple of 30

▪ 3) Find boundary points

▪ 4) Assign c-indices Ij to corners vj

• As close as to their inner angle

4.2.1 Patch graph construction

TID=

➢ Compute angles by LSCM (least squares conformal maps)➢ Mesh ->UV parametrization➢ Reference Library: Libigl


• Subdivision process



4.2.1 Patch graph construction

• Score of partitions


• To obtain a patch graph

4.2.1Turning the Final Partition into a Patch Graph

➢ Li ：actual lengths of graph edges in 3D

➢ Ea：the 6-integer vector of the 2-edge in the a-th direction

➢ Ia,fj：indicates the lists of indices of 2-edges circulating fj in the a-th direction


• First constraints

▪ Ensure that the sequential order of different edge types

• Second constraints

▪ Ensure that partial faces are consistently completed on both side

4.2.3 Cross-patch Consistency Constraints

Fig.10. differ in the sequential

orders of 2-edges, 3-edgesalong the graph edges


• Rotational symmetry optimization

▪ Euclidean Regularity

• A perfectly Euclidean-regular mesh are regular polygons, all edges must have the same length

• Global regularity optimization

• Planarity quads optimization

▪ Minimizing the difference between discrete curvatures of original and optimized mesh

4.2.4 Post-processing

“Regular Meshes from Polygonal Patterns”[ Vaxman(Utrecht University) et al./TOG2017]


• Altering the c-indices of an existing patch graph

• E.x. zigzags

4.3 Patch Graph Editing


5. Results and applications

5.1 3D Hybrid Mesh Design

Fig. 14.

Fig. 15.

ER-results


5.1 3D Hybrid Mesh Design

Fig. 17.


5.2 Architectural Geometry

Fig. 18.

planarity optimization score

fractured patterns


5.2 Architectural Geometry

Fig. 19.

Fig. 22.


• Limitation

▪ Do not have a theoretical study of necessary and sufficient conditions that a given 2D boundary admits at least one regular hybrid mesh solution.

▪ Considerable effort is needed to ensure the feasibility of 2D solutions when manually designing patch graphs.

• Future work

▪ Volumetric extensions

6. Limitation and Future work

Fig.23


Result Video


Appendix 1: Curvature


Appendix 2: Gauss-Bonnet Theorem

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