[PPT] - Designing Pop-Up Cards Carleton Algorithms Seminar Zachary Abel, PowerPoint Presentation

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Designing Pop-Up Cards

Carleton Algorithms Seminar Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lebiw, Andr´ e Schulz, Diane L. Souvaine, Giovanni Viglietta, Andrew Winslow Ottawa – November 29, 2013

(Figures and animations by Sarah Eisenstat and Andr´ e Schulz) Designing Pop-Up Cards

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Pop-up cards

Pop-up cards (or books) are 3D paper models that fold flat with

ne degree of freedom.

(The Jungle Book: A Pop-Up Adventure, by Matthew Reinhart) Designing Pop-Up Cards

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Pop-up cards

Pop-up cards (or books) are 3D paper models that fold flat with

ne degree of freedom.

Can every possible shape be modeled as a pop-up card, and how efficiently?

(The Jungle Book: A Pop-Up Adventure, by Matthew Reinhart) Designing Pop-Up Cards

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Pop-up cards

Pop-up cards (or books) are 3D paper models that fold flat with

ne degree of freedom.

Can every possible shape be modeled as a pop-up card, and how efficiently? We are not concerned with practical realizability (e.g., paper thickness, feature size).

(The Jungle Book: A Pop-Up Adventure, by Matthew Reinhart) Designing Pop-Up Cards

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Outline

2D orthogonal polygon pop-ups, O(n) links. 2D general polygon pop-ups, O(n2) links. 3D orthogonal polyhedron pop-ups, O(n3) links.

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2D model for pop-ups

Desired card Cross section 2D model

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Linkages

Linkages are formed by rigid bars and flexible joints. If bars intersect only at joints, the linkage configuration is called non-crossing. Three non-crossing configurations of a 7-bar linkage.

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More general joints

Common joint: • Flap: ◦ Sliceform: ×

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Everything is a joint

ε

Flap with joints Sliceform with flaps + joints

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Problem formulation

Input: 2D polygon P (unfolded shape),

ne distinguished vertex, n edges.

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Problem formulation

Input: 2D polygon P (unfolded shape),

ne distinguished vertex, n edges.

Output: linkage L with boundary P.

L folds to a line, L is non-crossing throughout, L has one degree of freedom.

(Pop-up designed using algorithm by Hara and Sugihara, 2009) Designing Pop-Up Cards

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Problem formulation

Input: 2D polygon P (unfolded shape),

ne distinguished vertex, n edges.

Output: linkage L with boundary P.

L folds to a line, L is non-crossing throughout, L has one degree of freedom.

(Pop-up designed using algorithm by Hara and Sugihara, 2009) Designing Pop-Up Cards

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Problem formulation

Input: 2D polygon P (unfolded shape),

ne distinguished vertex, n edges.

Output: linkage L with boundary P.

L folds to a line, L is non-crossing throughout, L has one degree of freedom.

(Pop-up designed using algorithm by Hara and Sugihara, 2009) Designing Pop-Up Cards

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Orthogonal polygons

P orthogonal: every edge is either vertical or horizontal. Opening angle is 90◦ (angles 180◦, 270◦, and 360◦ are discussed later). Strategy: preserve parallelism throughout the motion (i.e., shearing motion).

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3-step construction

Subdivide P into horizontal stripes.

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3-step construction

Subdivide P into horizontal stripes. Model all degree-3 vertices as flaps.

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3-step construction

Subdivide P into horizontal stripes. Model all degree-3 vertices as flaps. Enforce a 1-dof motion by adding vertical bars connected by sliceforms.

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Larger opening angle

Strategy: combine the 90◦ shearing motions. Need to “reflect” the shear.

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Reflector gadget

The top part keeps vertical lines parallel. The two kites are similar and force the left and right halves to move symmetrically.

4 1 2 2 2 4 1

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Reflector gadget

The top part keeps vertical lines parallel. The two kites are similar and force the left and right halves to move symmetrically.

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Synchronizing shears

Cut P along the x and y axes. Reconnect the 90◦ solutions via reflector gadgets.

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Result

The resulting structure has complexity O(n).

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Result

The resulting structure has complexity O(n).

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General polygons: V-folds

Outward V-fold: sum of adjecent edges is the same.

4 3 2 1

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General polygons: V-folds

Outward V-fold: sum of adjecent edges is the same.

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General polygons: V-folds

Outward V-fold: sum of adjecent edges is the same. Inward V-fold: sum of

pposing edges is the same.

4 3 2 1

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General polygons: V-folds

Outward V-fold: sum of adjecent edges is the same. Inward V-fold: sum of

pposing edges is the same.

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Nested V-folds

Lemma The closing motion of nested outward (resp. inward) V-folds intersects only in the end configuration.

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Nested V-folds

Lemma The closing motion of nested outward (resp. inward) V-folds intersects only in the end configuration.

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Nested V-folds

Lemma The closing motion of nested outward (resp. inward) V-folds intersects only in the end configuration.

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Nested V-folds

Lemma The closing motion of nested outward (resp. inward) V-folds intersects only in the end configuration.

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Nested V-folds

Lemma The closing motion of nested outward (resp. inward) V-folds intersects only in the end configuration.

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Cell decomposition

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Cell decomposition

Draw a ray from the fold to every vertex in P.

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Cell decomposition

Draw a ray from the fold to every vertex in P. Make outward V-folds for all edges between rays.

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Cell decomposition

Draw a ray from the fold to every vertex in P. Make outward V-folds for all edges between rays.

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Cell decomposition

Draw a ray from the fold to every vertex in P. Make outward V-folds for all edges between rays. Every wedge can be folded flat, but there are too many dof!

Want: wall segments rotate around fold. Want: wedge motions be synchronized.

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Restricting to rotations

For each pair of wall segment in an internal cell:

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Restricting to rotations

For each pair of wall segment in an internal cell:

Add two parallel segments.

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Restricting to rotations

For each pair of wall segment in an internal cell:

Add two parallel segments. Add two parallelograms to get two outward V-folds.

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Restricting to rotations

For each pair of wall segment in an internal cell:

Add two parallel segments. Add two parallelograms to get two outward V-folds.

Result: wall segments rotate around the apex.

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Restricting to rotations

For each pair of wall segment in an internal cell:

Add two parallel segments. Add two parallelograms to get two outward V-folds.

Result: wall segments rotate around the apex.

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Restricting to rotations

For each pair of wall segment in an internal cell:

Add two parallel segments. Add two parallelograms to get two outward V-folds.

Result: wall segments rotate around the apex. (Leaf cells are handled separately.)

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Synchronizing wedges

Strategy: link neighboring cells with with a gadget that synchronizes the independent motions of the wedges.

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Synchronizing wedges

Strategy: link neighboring cells with with a gadget that synchronizes the independent motions of the wedges. Basic sync gadget: inward V-fold + outward V-fold.

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Synchronizing wedges

Strategy: link neighboring cells with with a gadget that synchronizes the independent motions of the wedges. Basic sync gadget: inward V-fold + outward V-fold. The basic sync gadget has a 1-dof motion that makes all the cells in the same wedge fold at the same speed.

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Fitting the sync gadget

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Folding leaf cells

For cells with only one wall, use two sync gadgets and no rotation gadget.

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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Result

The resulting structure has complexity O(n2).

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3D (orthogonal) model for pop-ups

Input: orthogonal polyhedron P, one distinguished edge. Output: set of hinged rigid sheets of paper that folds from P to a flat state with a 1-dof motion.

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3D (orthogonal) model for pop-ups

Input: orthogonal polyhedron P, one distinguished edge. Output: set of hinged rigid sheets of paper that folds from P to a flat state with a 1-dof motion. Bellows theorem: every flexible polyhedron has the same volume in all configurations.

We must cut the boundary.

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Cutting into slices

Use the 3D grid induced by the vertices of P. Create slices perpendicular to the crease.

Each slice is a 2D linkage problem.

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Pinwheel construction

For each cross section, construct a pinwheel-pattern linkage, enforcing a 1-dof shearing motion.

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Pinwheel construction

For each cross section, construct a pinwheel-pattern linkage, enforcing a 1-dof shearing motion. Extrude each cross section to get a 3D model for a slice of P.

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Putting slices together

Fuse paper in adjacent slices.

But we still have holes on the sides...

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Closing holes

Add two hinged sheets of paper to close each hole. Just the left and bottom sides are hinged to the rest of the structure.

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Closing holes

Add two hinged sheets of paper to close each hole. Just the left and bottom sides are hinged to the rest of the structure.

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Closing holes

Add two hinged sheets of paper to close each hole. Just the left and bottom sides are hinged to the rest of the structure.

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Closing holes

Add two hinged sheets of paper to close each hole. Just the left and bottom sides are hinged to the rest of the structure.

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Closing holes

Add two hinged sheets of paper to close each hole. Just the left and bottom sides are hinged to the rest of the structure.

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Closing holes

Add two hinged sheets of paper to close each hole. Just the left and bottom sides are hinged to the rest of the structure.

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

Designing Pop-Up Cards

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Result

The resulting structure has complexity O(n3).

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Summary

O(n) solution for orthogonal polygons. O(n2) solution for general polygons. O(n3) solution for orthogonal polyhedra. Open problems: Can every polyhedron be a pop-up? Are our solutions optimal? Practical concerns:

Paper thickness? Paper flexibility? Feature size?

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References

Z. Abel, E.D. Demaine, M.L. Demaine, S. Eisenstat, A. Lubiw,
A. Schulz, D.L. Souvaine, G. Viglietta, and A. Winslow

Algorithms for designing pop-up cards STACS 2013

T. Hara and K. Sugihara

Computer-aided design of pop-up books with two-dimensional v-fold structures JCCGG 2009

R. Uehara and S. Teramoto

The complexity of a pop-up book CCCG 2006

Designing Pop-Up Cards