Digital Papercutting

Yanxi Liu and James Hays Carnegie Mellon University

Ying-Qing Xu and Heung-Yeung ShumMicrosoft Research Asia

Motivation

Fold and Cut Example

Contribution• We present a general computational

formalization of this particular art form for the first time.

• We combine the analysis and synthesis of a papercut pattern into one automatic process, as well as the generation of a physically feasible folding plan.

Related work

• Computational geometry– Demaine, Erik., Demaine, Martin, and Lubiw, Anna.

Folding and cutting paper. In JCDCG’98.– Demaine, Erik., Demaine, Martin, and Lubiw, Anna.

Folding and One Straight Cut Suffice SODA'99.

• Papercraft– Mitani, J., and Suziki, H. Making papercraft toys from

meshes using strip-based approximate unfolding. SIGGRAPH 2004.

Formal treatment of papercutting

P S1 S2 S3

Characterization by Symmetry Groups

Frieze Group Dihedral Group

Dihedral group examples

Dihedral group examples

Dihedral group examples

Dihedral group examples

Dihedral group examples

Dihedral group examples

Fundamental Regions

Dihedral group examples

Fundamental Regions

3 folds 3 folds 4 folds 4 folds 4 folds 4 folds

Dihedral group examples

Fundamental Regions

3 folds 3 folds 4 folds 4 folds 4 folds 4 folds

Evaluating a Fold

Evaluating a Fold

Evaluating a Fold

Evaluating a Fold

Evaluating a Fold

Evaluating a Fold

Evaluating a Fold

The Space of Possible Folds

The Space of Possible Folds

Searching for good folding axes

Image space Parameter Space

X, Y R, Theta

Searching for good folding axes

Image space Parameter Space

X, Y R, Theta

Searching for good folds

Dihedral and Frieze detection• Rules for Dihedral

Groups– Folds intersect at

common point– Folds have

correct angular spacing

• Rules for Frieze Pattern– Folds have

uniform pixel spacing

– Folds are parallel

Dihedral and Frieze detection• Rules for Dihedral

Groups– Folds intersect at

common point– Folds have

correct angular spacing

• Rules for Frieze Pattern– Folds have

uniform pixel spacing

– Folds are parallel

Axes of Dihedral Group

Finding additional symmetries

First search iteration Second search iteration

Dominant peaks correspond to dihedral group

Dominant peak corresponds to inner ring

Finding additional symmetries

Folding Plan

Dihedral Group Folding Plans

• Infinitely many patterns, but only finitely many dihedral groups which share folding plans

Dihedral Group Folding Plans

Editing papercuttings

Editing papercuttings

Editing papercuttings

Results

Future/ Current Work

Future/ Current Work

Future/ Current Work

Future/ Current Work

Future/ Current Work

Future/ Current Work

Future/ Current Work

Future/ Current Work

Future/ Current Work

Future Work• A formal characterization

of the space of existing papercutting patterns.

• Explore more perceptually motivated symmetryscores.

• Applications of symmetry based analysis and categorization in other domains: quilting, Oriental rugs, architecture.