Key Idea #2: Partition Requirements Given [P 1 , …, P i-1 , R i-1 ], partition P i-1 (i>1) into P i and R i such that 1. [P i-1 , P i , R i ] is interlocking 2. P i is disassemblable in [P i , R i ] 3. P i is connected; R i is connected P 1 P 2 R 2 R 3 [P 1 , P 2 , P 3 , R 3 ] [P 1 , P 2 , R 2 ]
Key Idea #2: Partition Requirements Given [P 1 , …, P i-1 , R i-1 ], partition P i-1 (i>1) into P i and R i such that 1. [P i-1 , P i , R i ] is interlocking 2. P i is disassemblable in [P i , R i ] 3. P i is connected; R i is connected P 1 P 1 P 2 P 2 R 2 P 3 R 3 [P 1 , P 2 , P 3 , R 3 ] [P 1 , P 2 , R 2 ]
Key Idea #3: Constructive Approach Construct the key piece P 1 (movable only along +x) Blocking & Blockee Voxel Pair y Seed Voxel Blocking & Blockee Selected Path Voxel Pair z x
Key Idea #3: Constructive Approach Construct the 2 nd piece P 2 (immobilized by the key) Blocking & Blockee Voxel Pair y x Seed Voxel Blocking & Blockee Selected Path Voxel Pair z
Our Result
Our Result
Our Result
Our Result
Summary of the project • A formal model to directly guarantee recursive interlocking based on building local interlocking groups (LIGs) • Requirements to ensure local interlocking of intermediate assemblies when extracting each puzzle piece • A constructive approach to iteratively generate geometry of each puzzle piece
Follow-up Work: Interlocking Objects for 3D Printing Song et al. Printing 3D Objects with Interlocking Parts . CAGD (Proc. of GMP), 2015
Follow-up Work: Interlocking Objects for 3D Printing Song et al. Printing 3D Objects with Interlocking Parts . CAGD (Proc. of GMP), 2015
Overview Recursive Interlocking Puzzles DESIA: A General Framework for Designing Interlocking Assemblies SIGGRAPH Asia 2012 SIGGRAPH Asia 2018
Motivation The Recursive Interlocking approach can explore only a limited design space LIG design space
Motivation The Recursive Interlocking approach can explore only a limited design space LIG design space
Motivation The Recursive Interlocking approach can explore only a limited design space LIG design space
Motivation The Recursive Interlocking approach can explore only a limited design space LIG design space
Motivation The Recursive Interlocking approach can explore only a limited design space LIG design space
Motivation The Recursive Interlocking approach can explore only a limited design space LIG design space
Motivation The Recursive Interlocking approach can explore only a limited design space LIG design space Full design space
Our Goal: Design Interlocking Assembly Can we have a general framework to design interlocking assemblies that can explore the full search space of all possible interlocking configurations? 1. Provide more design flexibility 2. Useful for designing new interlocking assemblies …
Our Key Idea: Graph-based Representation Test and design Directional blocking graphs interlocking assemblies Invented by Wilson [1992] +y G(+x, A ) G(+y, A ) A +x
Contribution #1: Test Interlocking Polynomial time complexity!!! All graphs are strongly connected The 3D assembly is interlocking (except the key part) +y G(+x, A ) G(+y, A ) A +x
Test Interlocking: Directional Blocking Graph Given an assembly A and a certain axial direction d +y +x
Test Interlocking: Directional Blocking Graph Create a directed edge from P i to P j iff P j blocks P i from translating along d +y +x
Test Interlocking: Directional Blocking Graph +y +x
Test Interlocking: Directional Blocking Graph +y +x
Test Interlocking: Directional Blocking Graph +y +x
Test Interlocking: Directional Blocking Graph +y +x
Test Interlocking: Directional Blocking Graph +y +x
Test Interlocking: Directional Blocking Graph Create a directional blocking graph for d = +x +y G(+x, A ) +x A
Test Interlocking: Directional Blocking Graph Create a directional blocking graph for d = +y +y G(+x, A ) G(+y, A ) +x A
Test Interlocking: Directional Blocking Graph The two graphs are called base directional blocking graphs of the assembly +y G(+x, A ) G(+y, A ) +x
Test Interlocking: Directional Blocking Graph An assembly is interlocking if all base directional blocking graphs are strongly connected except the key. The complexity of this testing approach is polynomial since finding all strongly connected components can be done in O(n 2 ) +y G(+x, A ) G(+y, A ) +x
Contribution #2: Design Interlocking Construct interlocking parts geometry Make all graphs strongly connected +y G(+x, A ) G(+y, A ) A +x
Design Interlocking: Iterative Construction Geometry Space Graph Space R 0 R R +y G(+x, A 0 ) G(+y, A 0 ) +x
Design Interlocking: Iterative Construction Split nodes 1 1 R 0 R R +y G(+x, A 1 ) G(+y, A 1 ) +x
Design Interlocking: Iterative Construction Graph Design (make graphs strongly connected except the key) 1 1 R 0 R R +y G(+x, A 1 ) G(+y, A 1 ) +x
Design Interlocking: Iterative Construction Geometry Realization Graph Design (realize blocking relations in the graphs) (make graphs strongly connected except the key) P 1 1 1 R 1 R R +y G(+x, A 1 ) G(+y, A 1 ) +x
Design Interlocking: Iterative Construction Geometry Realization Graph Design P 1 1 1 R 1 R R +y G(+x, A 1 ) G(+y, A 1 ) +x
Design Interlocking: Iterative Construction Geometry Realization Graph Design P 1 1 1 R 1 2 R 2 R +y G(+x, A 2 ) G(+y, A 2 ) +x
Design Interlocking: Iterative Construction Geometry Realization Graph Design P 1 1 1 R 1 2 R 2 R +y G(+x, A 2 ) G(+y, A 2 ) +x
Design Interlocking: Iterative Construction Geometry Realization Graph Design P 1 1 1 P 2 2 R 2 R R 2 +y G(+x, A 2 ) G(+y, A 2 ) +x
Design Interlocking: Iterative Construction Geometry Realization Graph Design P 1 1 1 P 2 2 3 2 3 P 3 R 3 R R +y G(+x, A 3 ) G(+y, A 3 ) +x
Design Interlocking: Iterative Construction Geometry Realization Graph Design P 1 1 1 P 2 2 3 2 3 P 4 P 3 4 5 4 5 +y P 5 G(+x, A 4 ) G(+y, A 4 ) +x
Design Interlocking: Tree-traversal Search Search space is explored in a tree traversal process with automatic backtracking
Results: Interlocking Voxelized Structures 9-part Interlocking Cube 1 1 1 2 3 2 3 2 3 4 5 4 5 4 5 6 7 6 7 6 7 8 9 8 9 8 9 G(+x, A) G(+y, A) G(+z, A) +y +z +x
Results: Interlocking Voxelized Structures 14-part Interlocking Dog G(+x, A) G(+y, A) G(+z, A) +y +z +x
Recommend
More recommend
