A Method for Creating

Papercraft Raised Relief Maps

from Digital Elevation Models

Jürnjakob Dugge, juernjakob@dugge.de

Johann Dugge, johann@dugge.de

http://www.PapercraftMountains.com

Motivation

One-off physical representations of elevation models

Wolfgang Knoll, Martin Hechinger

Architectural Models: Construction Techniques.

Munich: Deutsche Verlags-Anstalt, 2006

Low-poly aesthetic

Timothy J. Reynolds

Map Wars

2013

Timothy J. Reynolds

Untitled

2014

www.turnislefthome.com

From digital elevation model

to physical paper model

Chil / Zacharie Grossen, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16791896

Process Overview

Process Overview

Mesh Generation

Greedy Insertion [1]

Sliver triangles

Greedy Insertion

with triangle shape constraints [2] [3]

Flipped edges

Mesh Optimisation Method

Mesh Optimisation Method

1. Objective Function

2. Mesh Generation & Optimisation

3. Results

Objective Function

DEM Feature Weighting

DEM Approximation

Triangle Shape

Objective Function

DEM Feature Weighting

Chil / Zacharie Grossen, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16791896

• All grid points have weight 𝑤 ≥ 1

• Important geographic features / areas should be prioritised, 𝑤𝑖 ≫ 𝑤

Objective Function

DEM Approximation

• Minimise sum shortest distance to underlying triangle squared, weighted

• Normalised to sum of weights, and weighted DEM variance

𝑀𝐷 = 1 −1

𝑉𝑎𝑟(𝑤𝐷)

1

𝑖=1𝑛 𝑤𝑖

𝑖=1

𝑛

𝑤𝑖𝑑𝑖2

• 3D shape (not projected 2D) considered

• 𝑀𝑆 ="Mean Ratio Metric” target shape equilateral triangle [4]

• Maximise mean of triangles

Objective Function

Triangle Shape

Objective Function

Objective Weighting

Balance optimisation between deviation reduction and triangle shape

𝑀𝑇 = 𝑐𝐷𝑀𝐷 + 𝑐𝑆𝑀𝑆

𝑐𝐷 : 𝑐𝑆 1 : 0 1 : 1 0 : 1

𝑀𝑇 0.9717 0.9195 0.9057

𝑀𝐷 0.6348 0.9239 0.8369

𝑀𝑆 0.9717 0.9151 0.9057

Mesh Generation & Optimisation

DEM Feature Weight Map Generation

Vertex Placement

Quad Flipping

Vertex Shifting

Shape Heuristic

Objective Function Gradient

Mesh Generation & Optimisation

DEM Feature Weightmap Generation

• Identify areas of high local (planar) curvature [5]

• Gaussian blur to remove small scale features

Mesh Generation & Optimisation

Vertex Placement

1. Given target number of triangles n, determine number of vertices inside Pin / on

boundary Pb.

2. Total DEM Area / n ideal equilateral triangle edge length into DEM boundary length

Pb

Pin = round ( 1/2 * (n - Pb + 2) )

Pb = n – 2 * Pin + 2

3. Vertices into corner, interpolate surface

4. Next point at highest deviation * weight * discountAnimation:

https://youtu.be/3xSo2XK8UQY

Mesh Generation & Optimisation

Vertex Placement Examples

Nearest

Linear

v4

Animation:

https://youtu.be/3xSo2XK8UQY

Animation:

https://youtu.be/Lwu31t1GTTA

Animation:

https://youtu.be/wtMNen0WD2U

Mesh Generation & Optimisation

Quad Flipping

Delaunay Triangulation

Identify convex quads

Rank quads by contribution to objective total MT

Flip (next) lowest ranking quad finish when all quads tested

Contribution to MT improved?

yes

no

Mesh Generation & Optimisation

Vertex Shifting

yes

Pull vertices towards

deviation centroids

Gravity / Spring Heuristic

Pull vertices by edge

springs

TM

improved?

Steepest Ascent

Move points in direction of best

TM improvement

TM

improved?Increase step size

Undo move

Decrease step size

yes

Quad Flipping

Quad Flipping

Mesh Generation & Optimisation

Vertex Shifting – Gravity / Spring HeuristicStable, all triangles guaranteed to retain

orientation.

Large steps possible especially for shape

improvement.

Deviation

Centroid of weighted deviation for each

triangle.

Deviation centroids pull each point attached to

triangle.

Shape

Model triangle edges as springs.

Springs pull points to reach equilibrium.

Mesh Generation & Optimisation

Vertex Shifting – Steepest Gradient

Calculate gradient of TM for each vertex

Step size: smaller than radius of smallest triangle

in-circle.

Move.

If TM did not improve, reduce step size and try

again.

Mesh Generation & Optimisation

Quad Flipping & Vertex Shifting

Animation: https://youtu.be/K6HIjYyjVYU

Results

Matterhorn – 98 Triangles

Animation: https://youtu.be/TqQO4s_Uk2A

Results

Mount St Helens – 98 Triangles

Animation: https://youtu.be/_rBfJfUtr9w

Results

Torrener Joch – 196 Triangles

Animation: https://youtu.be/yvmny5qFfYo

Unfolding

Model Building

References

[1] Michael Garland and Paul Heckbert. Fast Polygonal Approximation of Terrains and Height Fields.

Technical Report CMU-CS-95-181

[2] Alper Üngör. "Off-centers: A new type of Steiner points for computing size-optimal quality-guaranteed

Delaunay triangulations." Computational Geometry 42.2 (2009): 109-118.

[3] Andrey N. Chernikov and Nikos P. Chrisochoides. "Generalized Delaunay mesh refinement: From

scalar to parallel." Proceedings of the 15th International Meshing Roundtable. Springer Berlin Heidelberg,

2006.

[4] Diachin et al., A Comparison Of Two Optimization Methods For Mesh Quality Improvement, Springer

London, 2006

[5] Anna M. Leonowicz and Bernhard Jenny, and Lorenz Hurni. Automated Reduction of Visual

Complexity in Small-Scale Relief Shading, Cartographica (volume 45, issue 1)