Publications

This page is a work in progress: I'm currently adding features (and less often, papers) to it. You can also visit my Google Scholar page for a list of my publications.

AnimationPak: packing elements with scripted animations
Reza Adhitya Saputra, Craig S. Kaplan, and Paul Asente
Conference Paper 
Proceedings of Graphics Interface 2020 (2020). Pages 393–403
Publisher Page Project Page
We present AnimationPak, a technique to create animated packings by arranging animated two-dimensional elements inside a static container. We represent animated elements in a three-dimensional spacetime domain, and view the animated packing problem as a three-dimensional packing in that domain. Every element is represented as a discretized spacetime mesh. In a physical simulation, meshes grow and repel each other, consuming the negative space in the container. The final animation frames are cross sections of the three-dimensional packing at a sequence of time values. The simulation trades off between the evenness of the negative space in the container, the temporal coherence of the animation, and the deformations of the elements. Elements can be guided around the container and the entire animation can be closed into a loop.
@inproceedings{saputra_kaplan_asente_2020, author = {Saputra, Reza Adhitya and Kaplan, Craig S. and Asente, Paul}, title = {Animation{P}ak: packing elements with scripted animations}, booktitle = {Proceedings of Graphics Interface 2020}, series = {GI 2020}, year = {2020}, pages = {393--403}, publisher = {Canadian Human-Computer Communications Society / Soci\'{e}t\'{e} canadienne du dialogue humain-machine}, location = {University of Toronto}, isbn = {978-0-9947868-5-2}, doi = {10.20380/GI2020.39} }
A frameless 2-coloring of the plane lattice
Craig S. Kaplan and Jeffrey Shallit
Journal Article 
Mathematics Magazine (2020). To appear
A picture frame in two dimensions is a rectangular array of symbols, with at least two rows and columns, where the first and last rows are identical, and the first and last columns are identical. If a coloring of the plane lattice has no picture frames, we call it frameless. In this note we show how to create a simple 2-coloring of the plane lattice that is frameless.
@article{kaplan_shallit_2020, author = {Kaplan, Craig S. and Shallit, Jeffrey}, title = {A frameless 2-coloring of the plane lattice}, journal = {Mathematics Magazine}, year = {2020}, pages = {To appear}, publisher = {Taylor \& Francis} }
Quasiperiodic bobbin lace patterns
Veronika Irvine, Therese Biedl, and Craig S. Kaplan
Journal Article 
Journal of Mathematics and the Arts, Volume 14 (2020). Pages 177–198
Publisher Page
Bobbin lace is a fibre art form in which threads are braided together to form a fabric, often with a very detailed and complex design. In traditional practice, each region of the fabric is filled with a periodic texture. We establish the groundwork for non-periodic lace patterns and present three new quasiperiodic families based on Sturmian words, the Penrose tiling by thick and thin rhombs and the Ammann-bar decoration of the Penrose tiling.
@article{irvine_biedl_kaplan_2020, author = {Irvine, Veronika and Biedl, Therese and Kaplan, Craig S.}, title = {Quasiperiodic bobbin lace patterns}, journal = {Journal of Mathematics and the Arts}, volume = {14}, number = {3}, year = {2020}, pages = {177--198}, publisher = {Taylor \& Francis}, doi = {10.1080/17513472.2020.1752999} }
ACM/EG Expressive Symposium
Stephen DiVerdi, Craig S. Kaplan, Angus G. Forbes, and Chiara Eva Catalano
Editor 
The Eurographics Association (2019).
@proceedings{diverdi_etal_2019, title = {A{CM}/{EG} {E}xpressive {S}ymposium}, editor = {DiVerdi, Stephen and Kaplan, Craig S. and Forbes, Angus G. and Catalano, Chiara Eva}, year = {2019}, publisher = {The Eurographics Association} }
Book review: Math Art: Truth, Beauty, and Equations
Craig S. Kaplan
Book Review 
Journal of Mathematics and the Arts, Volume 13 (2019). Pages 305–307
Publisher Page
@article{kaplan_2019_b, author = {Kaplan, Craig S.}, title = {Book review: {M}ath {A}rt: {T}ruth, {B}eauty, and {E}quations}, journal = {Journal of Mathematics and the Arts}, volume = {13}, number = {3}, year = {2019}, pages = {305--307}, publisher = {Taylor \& Francis}, doi = {10.1080/17513472.2019.1663715} }
Depth-aware image vectorization and editing
Shufang Lu, Wei Jiang, Xuefeng Ding, Craig S. Kaplan, Xiaogang Jin, Fei Gao, and Jiazhou Chen
Journal Article 
The Visual Computer, Volume 2019 (2019). Pages 1027–1039
Publisher Page
Image vectorization is one of the primary means of creating vector graphics. The quality of a vectorized image depends crucially on extracting accurate features from input raster images. However, correct object edges can be difficult to detect when color gradients are weak. We present an image vectorization technique that operates on a color image augmented with a depth map and uses both color and depth edges to define vectorized paths. We output a vectorized result as a diffusion curve image. The information extracted from the depth map allows us more flexibility in the manipulation of the diffusion curves, in particular permitting high-level object segmentation. Our experimental results demonstrate that this method achieves high reconstruction quality and provides greater control in the organization and editing of vectorized images than existing work based on diffusion curves.
@article{lu_etal_2019, author = {Lu, Shufang and Jiang, Wei and Ding, Xuefeng and Kaplan, Craig S. and Jin, Xiaogang and Gao, Fei and Chen, Jiazhou}, title = {Depth-aware image vectorization and editing}, journal = {The Visual Computer}, volume = {2019}, number = {6}, year = {2019}, pages = {1027--1039}, publisher = {Springer Nature}, doi = {10.1007/s00371-019-01671-0} }
Improved deformation-driven element packing with RepulsionPak
Reza Adhitya Saputra, Craig S. Kaplan, and Paul Asente
Journal Article 
IEEE Transactions on Visualization and Computer Graphics (2019).
Publisher Page Project Page
We present a method to fill a container shape with deformable instances of geometric elements selected from a library, creating a 2D artistic composition called an element packing. Each element is represented as a mass-spring system, allowing it to deform to achieve a better fit with its neighbours and the container. We start with an initial placement of small elements and gradually transform them using a physics simulation that trades off between the evenness of the packing and the deformations of the individual elements. Unlike previous work, elements can be given preferred orientations, and we can use shape matching to control the initial placement of elements in tight convex corners. We also explore the creation of tileable packings, and validate our approach using statistical measurements of the distributions of positive and negative space in packings. Our method produces compositions in which the negative space between elements is approximately uniform in width, similar to real-world examples created by artists.
@article{saputra_kaplan_asente_2019, author = {Saputra, Reza Adhitya and Kaplan, Craig S. and Asente, Paul}, title = {Improved deformation-driven element packing with {R}epulsion{P}ak}, journal = {IEEE Transactions on Visualization and Computer Graphics}, year = {2019}, doi = {10.1109/TVCG.2019.2950235} }
Defining hatching in art
Greg Philbrick and Craig S. Kaplan
Conference Paper 
ACM/EG Expressive Symposium (2019).
Publisher Page
We define hatching-a drawing technique-as rigorously as possible. A pure mathematical formulation or even a binary this-or-that definition is unreachable, but useful insights come from driving as close as we can. First we explain hatching's purposes. Then we define hatching as the use of patches: groups of roughly parallel curves that form flexible, simple patterns. After elaborating on this definition's parts, we briefly treat considerations for research in expressive rendering.
@inproceedings{philbrick_kaplan_2019, author = {Philbrick, Greg and Kaplan, Craig S.}, title = {Defining hatching in art}, booktitle = {ACM/EG Expressive Symposium}, editor = {Kaplan, Craig S. and Forbes, Angus and DiVerdi, Stephen}, year = {2019}, publisher = {The Eurographics Association}, isbn = {978-3-03868-078-9}, doi = {10.2312/exp.20191082} }
Animated isohedral tilings
Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture (2019). Pages 99–106
Preprint Publisher Page
Inspired by abstract mathematical animated loops like those created by David Whyte (@beesandbombs), I explore animations based on isohedral tilings. I present a complete interactive tool for designing tiling animations, and show a few results that I have obtained using this tool.
@inproceedings{kaplan_2019, author = {Kaplan, Craig S.}, title = {Animated isohedral tilings}, booktitle = {Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture}, editor = {Goldstine, Susan and McKenna, Douglas and Fenyvesi, Krist\'{o}f}, year = {2019}, pages = {99--106}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-1-938664-30-4}, issn = {1099-6702} }
An ultra-stable gold-coordinated protein cage displaying reversible assembly
Ali D. Malay, Naoyuki Miyazaki, Artur Biela, Soumyananda Chakraborti, Karolina Majsterkiewicz, Izabela Stupka, Craig S. Kaplan, Agnieszka Kowalczyk, Bernard M. A. G. Piette, Georg K. A. Hochberg, Di Wu, Tomasz P. Wrobel, Adam Fineberg, Manish S. Kushwah, Mitja Kelemen, Primož Vavpetič, Primož Pelicon, Philipp Kukura, Justin L. P. Benesch, Kenji Iwasaki, and Jonathan G. Heddle
Journal Article 
Nature, Volume 569 (2019). Pages 438–442
Publisher Page
Symmetrical protein cages have evolved to fulfil diverse roles in nature, including compartmentalization and cargo delivery1, and have inspired synthetic biologists to create novel protein assemblies via the precise manipulation of protein–protein interfaces. Despite the impressive array of protein cages produced in the laboratory, the design of inducible assemblies remains challenging2,3. Here we demonstrate an ultra-stable artificial protein cage, the assembly and disassembly of which can be controlled by metal coordination at the protein–protein interfaces. The addition of a gold (i)-triphenylphosphine compound to a cysteine-substituted, 11-mer protein ring triggers supramolecular self-assembly, which generates monodisperse cage structures with masses greater than 2 MDa. The geometry of these structures is based on the Archimedean snub cube and is, to our knowledge, unprecedented. Cryo-electron microscopy confirms that the assemblies are held together by 120 S–Aui–S staples between the protein oligomers, and exist in two chiral forms. The cage shows extreme chemical and thermal stability, yet it readily disassembles upon exposure to reducing agents. As well as gold, mercury(ii) is also found to enable formation of the protein cage. This work establishes an approach for linking protein components into robust, higher-order structures, and expands the design space available for supramolecular assemblies to include previously unexplored geometries.
@article{malay_etal_2019, author = {Malay, Ali D. and Miyazaki, Naoyuki and Biela, Artur and Chakraborti, Soumyananda and Majsterkiewicz, Karolina and Stupka, Izabela and Kaplan, Craig S. and Kowalczyk, Agnieszka and Piette, Bernard M. A. G. and Hochberg, Georg K. A. and Wu, Di and Wrobel, Tomasz P. and Fineberg, Adam and Kushwah, Manish S. and Kelemen, Mitja and Vavpeti\v{c}, Primo\v{z} and Pelicon, Primo\v{z} and Kukura, Philipp and Benesch, Justin L. P. and Iwasaki, Kenji and Heddle, Jonathan G.}, title = {An ultra-stable gold-coordinated protein cage displaying reversible assembly}, journal = {Nature}, volume = {569}, year = {2019}, pages = {438--442}, publisher = {Springer Nature}, doi = {10.1038/s41586-019-1185-4} }
RepulsionPak: deformation-driven element packing with repulsion forces
Reza Adhitya Saputra, Craig S. Kaplan, and Paul Asente
Conference Paper 
Proceedings of the 44th Graphics Interface Conference (2018). Pages 10–17
Publisher Page Project Page
We present a method to fill a container shape with deformable instances of geometric elements selected from a library, creating a 2D artistic composition called an element packing. Each element is represented as a mass-spring system, allowing them to deform to achieve a better fit with their neighbours and the container. We start with an initial random placement of small elements and gradually transform them using repulsion forces that trade off between the evenness of the packing and the deformations of the individual elements. Our method produces compositions in which the negative space between elements is approximately uniform in width, similar to real-world examples created by artists. We validate our approach by performing a quantitative study using spatial statistics.
@inproceedings{saputra_kaplan_asente_2018, author = {Saputra, Reza Adhitya and Kaplan, Craig S. and Asente, Paul}, title = {Repulsion{P}ak: deformation-driven element packing with repulsion forces}, booktitle = {Proceedings of the 44th Graphics Interface Conference}, series = {GI '18}, year = {2018}, pages = {10--17}, numpages = {8}, publisher = {Canadian Human-Computer Communications Society / Soci\'{e}t\'{e} canadienne du dialogue humain-machine}, location = {Toronto, Ontario, Canada}, isbn = {978-0-9947868-3-8}, doi = {10.20380/GI2018.03} }
Book review: Visualizing Mathematics with 3D Printing
Craig S. Kaplan
Book Review 
The College Mathematics Journal, Volume 48 (2017). Pages 69–72
Publisher Page
@article{kaplan_2017_b, author = {Kaplan, Craig S.}, title = {Book review: {V}isualizing {M}athematics with 3{D} {P}rinting}, journal = {The College Mathematics Journal}, volume = {48}, number = {1}, year = {2017}, pages = {69--72}, publisher = {Taylor \& Francis}, doi = {10.4169/college.math.j.48.1.69} }
Islamic Geometric Patterns: Their Historical Development and Traditional Methods of Construction
Jay Bonner
Book Chapter 
Springer-Verlag New York (2017). Pages 549–573
Publisher Page
@inbook{bonner_2017, author = {Bonner, Jay}, title = {Islamic {G}eometric {P}atterns: {T}heir {H}istorical {D}evelopment and {T}raditional {M}ethods of {C}onstruction}, year = {2017}, chapter = {4}, pages = {549--573}, publisher = {Springer-Verlag New York}, isbn = {978-1-4419-0216-0}, note = {Concluding chapter contributed to book by Jay Bonner}, doi = {10.1007/978-1-4419-0217-7} }
Marbling-based creative modelling
Shufang Lu, Yue Huang, Xiaogang Jin, Aubrey Jaffer, Craig S. Kaplan, and Xiaoyang Mao
Journal Article 
The Visual Computer, Volume 33 (2017). Pages 913–923
Publisher Page
Most mathematical marbling simulations generate patterns for texture mapping and surface decoration. We explore the application of three-dimensional deformations inspired by mathematical marbling as a suite of tools to enable creative shape design. Our tools are expressed as analytical functions of space and are volume-preserving vector fields, meaning that the modelling process preserves volumes and avoids self-intersections. Complicated deformations are easily combined to create complex objects from simple ones. To achieve smooth and high-quality shapes, we also present a mesh refinement and simplification algorithm adapted to our deformations. We show a number of examples of shapes created with our technique in order to demonstrate its power and expressiveness.
@article{lu_etal_2017, author = {Lu, Shufang and Huang, Yue and Jin, Xiaogang and Jaffer, Aubrey and Kaplan, Craig S. and Mao, Xiaoyang}, title = {Marbling-based creative modelling}, journal = {The Visual Computer}, volume = {33}, number = {6-8}, year = {2017}, pages = {913--923}, doi = {10.1007/s00371-017-1396-3}, url = {https://doi.org/10.1007/s00371-017-1396-3} }
Interwoven Islamic geometric patterns
Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture (2017). Pages 71–78
Preprint Publisher Page
Rinus Roelofs has exhibited numerous planar and polyhedral sculptures made up of two layers that weave over and under each other to form a single connected surface. I present a technique for creating sculptures in this style that are inspired by the geometric structure of Islamic star patterns. I first present a general approach for constructing interwoven two- layer sculptures, and then specialize it to Islamic patterns in the plane and on polyhedra. Finally, I describe a projection operation that bulges the elements of these designs into undulating dome shapes.
@inproceedings{kaplan_2017, author = {Kaplan, Craig S.}, title = {Interwoven {I}slamic geometric patterns}, booktitle = {Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture}, editor = {David Swart, Carlo H. S\'{e}quin, and Krist\'{o}f Fenyvesi}, year = {2017}, pages = {71--78}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-1-938664-22-9}, issn = {1099-6702} }
FLOWPAK: flow-based ornamental element packing
Reza Adhitya Saputra, Craig S. Kaplan, Paul Asente, and Radomír Měch
Conference Paper 
Proceedings of the 43rd Graphics Interface Conference (2017). Pages 8–15
Publisher Page Project Page
We present a technique for drawing ornamental designs consisting of placed instances of simple shapes. These shapes, which we call elements, are selected from a small library of templates. The elements are deformed to flow along a direction field interpolated from user-supplied strokes, giving a sense of visual flow to the final composition, and constrained to lie within a container region. Our implementation computes a vector field based on user strokes, constructs streamlines that conform to the vector field, and places an element over each streamline. An iterative refinement process then shifts and stretches the elements to improve the composition.
@inproceedings{saputra_etal_2017, author = {Saputra, Reza Adhitya and Kaplan, Craig S. and Asente, Paul and M\v{e}ch, Radom\'{\i}r}, title = {F{LOWPAK}: flow-based ornamental element packing}, booktitle = {Proceedings of the 43rd Graphics Interface Conference}, series = {GI '17}, year = {2017}, pages = {8--15}, numpages = {8}, publisher = {Canadian Human-Computer Communications Society}, location = {Edmonton, Alberta, Canada}, isbn = {978-0-9947868-2-1}, doi = {10.20380/GI2017.02}, url = {https://doi.org/10.20380/GI2017.02} }
Combinatorial poppies
Karl Kattchee and Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (2016). Pages 111–118
Preprint Publisher Page
In this paper, we consider orthogonal closed paths that dwell on the edges of square grids. A certain subset of these paths are called “poppies.” We derive a formula for the number of asymmetrical poppies on an n by n grid, and we explore the aesthetics of the poppies.
@inproceedings{kattchee_kaplan_2016, author = {Kattchee, Karl and Kaplan, Craig S.}, title = {Combinatorial poppies}, booktitle = {Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture}, editor = {Eve Torrence and Bruce Torrence and Carlo S\'{e}quin and Douglas McKenna and Krist\'{o}f Fenyvesi and Reza Sarhangi}, year = {2016}, pages = {111--118}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-1-938664-19-9}, issn = {1099-6702} }
Hypocycloid juggling patterns
Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (2016). Pages 71–78
Preprint Publisher Page
When points are distributed evenly on a hypocycloid path and animated along that path, they can be seen as clustering together into “wheels”, groups of points that lie at the vertices of regular polygons and that rotate in synchrony. In some cases the points can group into two different sets of wheels, rotating in opposite directions. I derive formulas that predict the number of such wheels and the number of points on each one. The resulting animations are visually compelling and reminiscent of the motion of balls or clubs in multi-person juggling patterns
@inproceedings{kaplan_2016, author = {Kaplan, Craig S.}, title = {Hypocycloid juggling patterns}, booktitle = {Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture}, editor = {Eve Torrence and Bruce Torrence and Carlo S\'{e}quin and Douglas McKenna and Krist\'{o}f Fenyvesi and Reza Sarhangi}, year = {2016}, pages = {71--78}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-1-938664-19-9}, issn = {1099-6702} }
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture
Kelly Delp, Craig S. Kaplan, Douglas McKenna, and Reza Sarhangi
Editor 
Tessellations Publishing (2015).
@proceedings{delp_etal_2015, title = {Proceedings of {B}ridges 2015: {M}athematics, {M}usic, {A}rt, {A}rchitecture, {C}ulture}, editor = {Delp, Kelly and Kaplan, Craig S. and McKenna, Douglas and Sarhangi, Reza}, year = {2015}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-1-938664-15-1}, issn = {1099-6702} }
Modular origami halftoning: theme and variations
Zhifu Xiao, Robert Bosch, Craig S. Kaplan, and Robert J. Lang
Conference Paper 
Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015). Pages 61–68
Preprint Publisher Page
We introduce Modular Origami Halftoning, a set of techniques for representing continuous-tone greyscale images using a fixed set of origami modules folded from two-coloured paper. By arranging the modules to reveal different amounts of the paper's coloured side, each module can approximate the brightness of a single pixel from an image. We present two variations based on affixing a grid of disconnected modules to a backing, and one variation in which the modules interlock into a carpet, requiring no tape or glue.
@inproceedings{xiao_etal_2015, author = {Xiao, Zhifu and Bosch, Robert and Kaplan, Craig S. and Lang, Robert J.}, title = {Modular origami halftoning: theme and variations}, booktitle = {Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture}, editor = {Kelly Delp, Kaplan, Craig S., Douglas McKenna and Reza Sarhangi}, year = {2015}, pages = {61--68}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-1-938664-15-1}, issn = {1099-6702} }
Gear-meshed tiling of surfaces with molecular pentagonal stars
Quirin Stöckl, Davide Bandera, Craig S. Kaplan, Karl-Heinz Ernst, and Jay S. Siegel
Journal Article 
Journal of the Americal Chemical Society, Volume 136 (2014). Pages 606–609
Publisher Page
The assembly of the chiral pentagonal-star-shaped 1,3,5,7,9-pentaphenylcorannulene on a Cu(111) surface has been studied with scanning tunneling microscopy. Two different long-range ordered phases coexist at 60 K, most likely racemic and homochiral phases. The principal motifs emulate a network of meshed gears. One of the observed structures resembles the densest packing of five-fold symmetric stars.
@article{stockl_etal_2014, author = {St\"{o}ckl, Quirin and Bandera, Davide and Kaplan, Craig S. and Ernst, Karl-Heinz and Siegel, Jay S.}, title = {Gear-meshed tiling of surfaces with molecular pentagonal stars}, journal = {Journal of the Americal Chemical Society}, volume = {136}, number = {2}, year = {2014}, pages = {606--609}, doi = {10.1021/ja411279r} }
The design of a reconfigurable maze
Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014). Pages 167–174
Preprint Publisher Page
A reconfigurable maze is a small set of modular components, called tiles, that can be assembled into a variety of distinct mazes. Although the idea of a reconfigurable maze is simple enough to articulate, there are some difficult mathematical challenges to overcome in order to make them a reality. I introduce the concept of a reconfigurable maze, and work through the design of a first prototype based on a 2 x 2 grid of square tiles. The prototype is a partial solution to the original problem, leaving open many opportunities for future innovation.
@inproceedings{kaplan_2014_b, author = {Kaplan, Craig S.}, title = {The design of a reconfigurable maze}, booktitle = {Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture}, editor = {Gary Greenfield, George Hart and Reza Sarhangi}, year = {2014}, pages = {167--174}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-1-938664-11-3}, issn = {1099-6702} }
The monumental geometry of E Pluribus Unum
Craig S. Kaplan and Chris Jordan
Conference Paper 
Proceedings of the Workshop on Computational Aesthetics (2014). Pages 97–102
Preprint Publisher Page
We describe the conception and construction of E Pluribus Unum, a large-scale artwork that combines text and geometry to depict over a million names of global organizations dedicated to social and environmental causes. The names are rendered in small type along lines arranged symmetrically in a disc. We examine the aesthetic choices involved in designing the artwork, and the technical challenges we faced in laying out and rendering a monumental amount of text.
@inproceedings{kaplan_jordan_2014, author = {Kaplan, Craig S. and Jordan, Chris}, title = {The monumental geometry of {E} {P}luribus {U}num}, booktitle = {Proceedings of the Workshop on Computational Aesthetics}, series = {CAe '14}, year = {2014}, pages = {97--102}, numpages = {6}, publisher = {ACM}, address = {New York, NY, USA}, location = {Vancouver, British Columbia, Canada}, isbn = {978-1-4503-3019-0}, doi = {10.1145/2630099.2630106} }
Stereoscopic 3D image stylization
Lesley Northam, Paul Asente, and Craig S. Kaplan
Journal Article 
Computers & Graphics, Volume 37 (August 2013). Pages 389–402
Publisher Page
We present a method for stylizing stereoscopic 3D images that guarantees consistency between the left and right views. Our method decomposes the left and right views of an input image into discretized disparity layers and merges the corresponding layers from the left and right views into a single layer where stylization takes place. We then construct new stylized left and right views by compositing portions of the stylized layers. Because the new left and right views come from the same stylized source layers, our method eliminates common stylization artifacts that cause viewer discomfort. We also present a stereoscopic 3D painterly rendering algorithm tailored to our layer-based approach. This method uses disparity information to assist in stroke creation so that strokes follow surface geometry without ignoring painted surface patterns. Finally, we conduct a user study that demonstrates that our approach to stereoscopic 3D image stylization leads to images that are more comfortable to view than those created using other techniques.
@article{northam_asente_kaplan_2013, author = {Northam, Lesley and Asente, Paul and Kaplan, Craig S.}, title = {Stereoscopic 3{D} image stylization}, journal = {Computers \& Graphics}, volume = {37}, number = {5}, month = {August}, year = {2013}, pages = {389--402}, publisher = {Elsevier}, doi = {10.1016/j.cag.2012.11.005} }
Patch-based geometric texture synthesis
Zainab AlMeraj, Craig S. Kaplan, and Paul Asente
Conference Paper 
Proceedings of the Workshop on Computational Aesthetics (2013). Pages 15–19
Publisher Page
Inspired by the results of recent studies on the perception of geometric textures, we present a patch-based geometric synthesis algorithm that mimics observed synthesis strategies. Our synthesis process first constructs an overlapping grid of copies of the exemplar, and then culls individual motifs based on overlaps and the enforcement of minimum distances.
@inproceedings{almeraj_kaplan_asente_2013_b, author = {AlMeraj, Zainab and Kaplan, Craig S. and Asente, Paul}, title = {Patch-based geometric texture synthesis}, booktitle = {Proceedings of the Workshop on Computational Aesthetics}, series = {CAe '13}, year = {2013}, pages = {15--19}, publisher = {ACM Press}, location = {Anaheim, California}, doi = {10.1145/2487276.2487278} }
Towards effective evaluation of geometric texture synthesis algorithms
Zainab AlMeraj, Craig S. Kaplan, and Paul Asente
Conference Paper 
Proceedings of the Symposium on Non-Photorealistic Animation and Rendering (2013). Pages 5–14
Publisher Page
In recent years, an increasing number of example-based Geometric Texture Synthesis (GTS) algorithms have been proposed. However, there have been few attempts to evaluate these algorithms rigorously. We are driven by this lack of validation and the simplicity of the GTS problem to look closer at perceptual similarity between geometric arrangements. Using samples from a geological database, our research first establishes a dataset of geometric arrangements gathered from multiple synthesis sources. We then employ the dataset in two evaluation studies. Collectively these empirical methods provide formal foundations for perceptual studies in GTS, insight into the robustness of GTS algorithms and a better understanding of similarity in the context of geometric texture arrangements.
@inproceedings{almeraj_kaplan_asente_2013, author = {AlMeraj, Zainab and Kaplan, Craig S. and Asente, Paul}, title = {Towards effective evaluation of geometric texture synthesis algorithms}, booktitle = {Proceedings of the Symposium on Non-Photorealistic Animation and Rendering}, series = {NPAR '13}, year = {2013}, pages = {5--14}, publisher = {ACM Press}, location = {Anaheim, California}, doi = {10.1145/2486042.2486043} }
Rasterizing and antialiasing vector line art in the pixel art style
Tiffany C. Inglis, Daniel Vogel, and Craig S. Kaplan
Conference Paper 
Proceedings of the Symposium on Non-Photorealistic Animation and Rendering (2013). Pages 25–32
Publisher Page
Pixel artists rasterize vector shapes by hand to minimize artifacts at low resolutions and emphasize the aesthetics of visible pixels. We describe Superpixelator, an algorithm that automates this process by rasterizing vector line art at a low resolution pixel art style. Our technique successfully eliminates most rasterization artifacts and draws smoother curves. To draw shapes more effectively, we use optimization techniques to preserve shape properties such as symmetry, aspect ratio, and sharp angles. Our algorithm also supports "manual antialiasing," the style of antialiasing used in pixel art. Professional pixel artists report that Superpixelator's results are as good, or better, than hand-rasterized drawings by artists.
@inproceedings{inglis_vogel_kaplan_2013, author = {Inglis, Tiffany C. and Vogel, Daniel and Kaplan, Craig S.}, title = {Rasterizing and antialiasing vector line art in the pixel art style}, booktitle = {Proceedings of the Symposium on Non-Photorealistic Animation and Rendering}, series = {NPAR '13}, year = {2013}, pages = {25--32}, publisher = {ACM Press}, location = {Anaheim, California}, doi = {10.1145/2486042.2486044} }
Grid-based decorative corners
Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013). Pages 317–324
Preprint Publisher Page
I explore a space of geometric, decorative corner designs based on paths through a square grid. I discuss the problems of enumerating corners of a given size efficiently, and exploring them interactively in software. I then impose a higher-level connectivity constraint on corners and discuss the effect of this constraint on the mathematical and aesthetic properties of corner designs.
@inproceedings{kaplan_2013, author = {Kaplan, Craig S.}, title = {Grid-based decorative corners}, booktitle = {Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture}, editor = {George Hart and Reza Sarhangi}, year = {2013}, pages = {317--324}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, location = {Enschede, Netherlands} }
Animating line-based Op Art
Tiffany C. Inglis and Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013). Pages 159–166
Preprint Publisher Page
We describe a tiling approach to animating line-based Op Art. The tiling must be seamless along the tile edges, and we seek to minimize the number of bidirectional tiles because they interfere with the Op Art illusion. We introduce several algorithms for creating these tilings, and demonstrate that an optimal tiling can be found in polynomial time. Examples of animated Op Art can be found on our website https://sites.google.com/site/tiffanycinglis/research/animating-line-based-op-art.
@inproceedings{inglis_kaplan_2013, author = {Inglis, Tiffany C. and Kaplan, Craig S.}, title = {Animating line-based {O}p {A}rt}, booktitle = {Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture}, editor = {George Hart and Reza Sarhangi}, year = {2013}, pages = {159--166}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, location = {Enschede, Netherlands} }
Depiction using geometric constraints
Craig S. Kaplan
Book Chapter 
Image and Video-Based Artistic Stylization (2012). Pages 167–187
Publisher Page
Hand made and computer generated drawings usually place marks with regard only for the role they play in communicating the overall content of the drawing. However, novel drawing styles emerge when extra constraints are applied to force the marks themselves to behave in an eye-catching way. I explore three drawing styles based on complex geometric relationships between paths: continuous line drawing with a closed loop, drawings from tree structures, and mazes that depict images.
@incollection{kaplan_2012, author = {Kaplan, Craig S.}, title = {Depiction using geometric constraints}, booktitle = {Image and Video-Based Artistic Stylization}, editor = {Paul Rosin and John Collomosse}, volume = {42}, year = {2012}, pages = {167--187}, publisher = {Springer London}, doi = {10.1007/978-1-4471-4519-6_9} }
Op Art rendering with lines and curves
Tiffany C. Inglis, Stephen Inglis, and Craig S. Kaplan
Journal Article 
Computers & Graphics, Volume 36 (October 2012). Pages 607–621
Publisher Page
A common technique in Op Art is the use of densely packed line segments to depict simple shapes such as circles and squares. Some artists have attempted to create more complex images using this technique but are faced with the difficulty of avoiding undesirable artifacts such as line breaks and T-junctions within their artworks. We introduce an algorithm that takes an arbitrary image and automatically generates the corresponding Op Art composition in this style. For 2-colour images, the algorithm produces artworks without any unwanted artifacts; for images with more colours, the basic algorithm cannot guarantee the removal of all artifacts, but we use a global optimisation technique to minimise their number. We also examine the use of curves in creating the illusion of 3D forms and present a corresponding algorithm based on a physical simulation of heat flow. The algorithms for generating Op Art with straight lines and with curves can be combined to create an interesting new style of art. The results have applications in graphic design, puzzle creation and non-photorealistic rendering.
@article{inglis_inglis_kaplan_2012, author = {Inglis, Tiffany C. and Inglis, Stephen and Kaplan, Craig S.}, title = {Op {A}rt rendering with lines and curves}, journal = {Computers \& Graphics}, volume = {36}, number = {6}, month = {October}, year = {2012}, pages = {607--621}, issn = {0097-8493}, note = {An earlier version of this article appeared as a conference paper in the proceedings of Computational Aesthetics 2011. A poster on the work was presented at SIGGRAPH and the Grace Hopper Celebration in 2011, and was subsequently a finalist in the ACM Student Research Competition.}, doi = {10.1016/j.cag.2012.03.003}, url = {http://www.sciencedirect.com/science/article/pii/S0097849312000428} }
Circle patterns in Gothic architecture
Tiffany C. Inglis and Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture (2012). Pages 133–140
Preprint Publisher Page
Inspired by Gothic-influenced architectural styles, we analyze some of the circle patterns found in rose windows and semi-circular arches. We introduce a recursive circular ring structure that can be represented using a set-like notation, and determine which structures satisfy a set of tangency requirements. To fill in the gaps between tangent circles, we add Appollonian circles to each triplet of pairwise tangent circles. These ring structures provide the underlying structure for many designs, including rose windows, Celtic knots and spirals, and Islamic star patterns.
@inproceedings{inglis_kaplan_2012_b, author = {Inglis, Tiffany C. and Kaplan, Craig S.}, title = {Circle patterns in {G}othic architecture}, booktitle = {Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture}, editor = {Robert Bosch and Douglas McKenna and Reza Sarhangi}, year = {2012}, pages = {133--140}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, location = {Towson, Maryland}, isbn = {978-1-938664-00-7}, issn = {1099-6702} }
Pixelating vector line art
Tiffany C. Inglis and Craig S. Kaplan
Conference Paper 
Proceedings of the Symposium on Non-Photorealistic Animation and Rendering (2012). Pages 21–28
Publisher Page
Inspired by Gothic-influenced architectural styles, we analyze some of the circle patterns found in rose windows and semi-circular arches. We introduce a recursive circular ring structure that can be represented using a set-like notation, and determine which structures satisfy a set of tangency requirements. To fill in the gaps between tangent circles, we add Appollonian circles to each triplet of pairwise tangent circles. These ring structures provide the underlying structure for many designs, including rose windows, Celtic knots and spirals, and Islamic star patterns.
@inproceedings{inglis_kaplan_2012, author = {Inglis, Tiffany C. and Kaplan, Craig S.}, title = {Pixelating vector line art}, booktitle = {Proceedings of the Symposium on Non-Photorealistic Animation and Rendering}, series = {NPAR '12}, year = {2012}, pages = {21--28}, numpages = {8}, publisher = {Eurographics Association}, address = {Aire-la-Ville, Switzerland, Switzerland}, location = {Annecy, France}, isbn = {978-3-905673-90-6}, doi = {10.2312/PE/NPAR/NPAR12/021-028}, url = {http://dl.acm.org/citation.cfm?id=2330147.2330153} }
Consistent stylization and painterly rendering of stereoscopic 3D images
Lesley Northam, Paul Asente, and Craig S. Kaplan
Conference Paper 
Proceedings of the Symposium on Non-Photorealistic Animation and Rendering (2012). Pages 47–56
Publisher Page
We present a method for stylizing stereoscopic 3D images that guarantees consistency between the left and right views. Our method decomposes the left and right views of an input image into discretized disparity layers and merges the corresponding layers from the left and right views into a single layer where stylization takes place. We then construct new stylized left and right views by compositing portions of the stylized layers. Because the left and right views come from the same source layers, our method eliminates common artifacts that cause viewer discomfort. We also present a stereoscopic 3D painterly rendering algorithm tailored to our layer-based approach. This method uses disparity information to assist in stroke creation so that strokes follow surface geometry without ignoring painted surface patterns. Finally, we conduct a user study that demonstrates that our approach to stereoscopic 3D image stylization leads to images that are more comfortable to view than those created using other techniques.
@inproceedings{northam_asente_kaplan_2012, author = {Northam, Lesley and Asente, Paul and Kaplan, Craig S.}, title = {Consistent stylization and painterly rendering of stereoscopic 3{D} images}, booktitle = {Proceedings of the Symposium on Non-Photorealistic Animation and Rendering}, series = {NPAR '12}, year = {2012}, pages = {47--56}, numpages = {10}, publisher = {Eurographics Association}, address = {Aire-la-Ville, Switzerland, Switzerland}, location = {Annecy, France}, isbn = {978-3-905673-90-6}, doi = {10.2312/PE/NPAR/NPAR12/047-056}, url = {http://dl.acm.org/citation.cfm?id=2330147.2330158} }
Smooth self-similar curves
Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture (2011). Pages 209–216
Preprint Publisher Page
I present a technique for constructing self-similar curves from smooth base curves. The technique is similar to that used in Iterated Function Systems like the Koch curve, except that it does not require a piecewise linear path in order to induce a set of similarities. I explain the mathematical machinery behind the technique, describe a practical numerical approximation that can be implemented in software, and show some results.
@inproceedings{kaplan_2011, author = {Kaplan, Craig S.}, title = {Smooth self-similar curves}, booktitle = {Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture}, editor = {Reza Sarhangi and Carlo H. S\'{e}quin}, year = {2011}, pages = {209--216}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-0-9846042-6-5}, issn = {1099-6702} }
Sudoku art
Tiffany C. Inglis and Craig S. Kaplan
Conference Paper 
Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture (2011). Pages 187–194
Preprint Publisher Page
Sudoku, the popular logic puzzle, would have a greater artistic appeal if the final completed puzzle could be transformed into an image similar to nonogram outputs. To do this, we solve a mixed-integer nonlinear programming problem (MINLP) to find the Sudoku configuration that most closely represents a target image via an integer-colour mapping. This alone produces inadequate results, so we relax the problem by adding transparent regions to the target image and removing certain MINLP constraints. To produce puzzles that are feasible and interesting to players, additional components are added to avoid tedious recounting and unnecessary colouring.
@inproceedings{inglis_kaplan_2011_b, author = {Inglis, Tiffany C. and Kaplan, Craig S.}, title = {Sudoku art}, booktitle = {Proceedings of Bridges 2011: Mathematics, Music, Art, Architecture, Culture}, editor = {Reza Sarhangi and Carlo H. S\'{e}quin}, year = {2011}, pages = {187--194}, publisher = {Tessellations Publishing}, address = {Phoenix, Arizona}, isbn = {978-0-9846042-6-5}, issn = {1099-6702} }
Generating Op Art lines
Tiffany C. Inglis and Craig S. Kaplan
Conference Paper 
International Symposium on Computational Aesthetics in Graphics, Visualization, and Imaging (2011). Pages 25–32
Publisher Page
A common technique in Op Art is the use of parallel lines to depict simple shapes such as circles and squares. Some artists have attempted to create more complex images using this technique but faced the problem of producing undesirable artifacts such as line breaks and T-junctions within their artworks. To this end, we developed a novel algorithm that takes an arbitrary image and automatically generates the corresponding Op Art composition of this style. For 2-colour images, the algorithm produces artworks without any unwanted artifacts; for images with more colours, the basic algorithm cannot guarantee the removal of all artifacts, but we use a global optimization technique to minimize the number of artifacts. The results have applications in graphics design, data visualization, puzzle creation and line drawings.
@inproceedings{inglis_kaplan_2011, author = {Inglis, Tiffany C. and Kaplan, Craig S.}, title = {Generating {O}p {A}rt lines}, booktitle = {International Symposium on Computational Aesthetics in Graphics, Visualization, and Imaging}, year = {2011}, pages = {25--32}, doi = {10.1145/2030441.2030446} }
Towards ground truth in geometric textures
Zainab AlMeraj, Craig S. Kaplan, Paul Asente, and Edward Lank
Conference Paper 
Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Non-Photorealistic Animation and Rendering (2011). Pages 17–26
Publisher Page
Two-dimensional geometric texture synthesis is the geometric analogue of raster-based texture synthesis. An absence of conventional evaluation procedures in recent synthesis attempts demands an inquiry into the visual significance of synthesized results. In this paper, we report on two psychophysical experiments that explore how people understand notions of similarity in geometric textures. We present perceptual metrics and human texture generation features that are crucial for future researchers when developing and assessing the success of their algorithms.
@inproceedings{almeraj_etal_2011, author = {AlMeraj, Zainab and Kaplan, Craig S. and Asente, Paul and Lank, Edward}, title = {Towards ground truth in geometric textures}, booktitle = {Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Non-Photorealistic Animation and Rendering}, series = {NPAR '11}, year = {2011}, pages = {17--26}, publisher = {ACM Press}, address = {New York}, doi = {10.1145/2024676.2024679} }
RTFX: on-set previs with UnrealEngine3
Lesley Northam, Joe Istead, and Craig S. Kaplan
Conference Paper 
Proceedings of the 10th International Conference on Entertainment Computing (ICEC 2011) (2011).
Publisher Page
This paper discusses on-set previsualization with distributed motion capture, virtual camera and asset control, and real-time rendering using a video game engine. Our test harness, RTFX, demonstrates the feasibility and usefulness of a system that couples Epic Games’ UnrealEngine3 with the Houdini 3D animation kit by Side Effects Software and a passive motion capture system by Vicon.
@inproceedings{northam_istead_kaplan_2011, author = {Northam, Lesley and Istead, Joe and Kaplan, Craig S.}, title = {R{TFX}: on-set previs with {U}nreal{E}ngine3}, booktitle = {Proceedings of the 10th International Conference on Entertainment Computing (ICEC 2011)}, year = {2011}, note = {4 pages}, doi = {10.1007/978-3-642-24500-8_59} }
Operations research in the visual arts
Craig S. Kaplan and Robert Bosch
Book Chapter 
Wiley Encyclopedia of Operations Research and Management Science (2010).
Publisher Page
This article explores applications of optimization algorithms to the computer‐based synthesis of art. The presentation covers title‐based methods, freeform arrangement of elements, and line art.
@incollection{kaplan_bosch_2010, author = {Kaplan, Craig S. and Bosch, Robert}, title = {Operations research in the visual arts}, booktitle = {Wiley Encyclopedia of Operations Research and Management Science}, year = {2010}, publisher = {John Wiley \& Sons, Inc.}, note = {15 pages}, doi = {10.1002/9780470400531.eorms0691} }
Computational symmetry in computer vision and computer graphics
Yanxi Liu, Hagit Hel-Or, Craig S. Kaplan, and Luc Van Gool
Journal Article 
Foundations and Trends in Computer Graphics and Vision, Volume 5 (June 2010). Pages 1–195
Publisher Page
In the arts and sciences, as well as in our daily lives, symmetry has made a profound and lasting impact. Likewise, a computational treatment of symmetry and group theory (the ultimate mathematical formalization of symmetry) has the potential to play an important role in computational sciences. Though the term computational symmetry was formally defined a decade ago by the first author, referring to algorithmic treatment of symmetries, seeking symmetry from digital data has been attempted for over four decades. Computational symmetry on real world data turns out to be challenging enough that, after decades of effort, a fully automated symmetry–savvy system remains elusive for real world applications. The recent resurging interests in computational symmetry for computer vision and computer graphics applications have shown promising results. Recognizing the fundamental relevance and potential power that computational symmetry affords, we offer this survey to the computer vision and computer graphics communities. This survey provides a succinct summary of the relevant mathematical theory, a historic perspective of some important symmetry-related ideas, a partial yet timely report on the state of the arts symmetry detection algorithms along with its first quantitative benchmark, a diverse set of real world applications, suggestions for future directions and a comprehensive reference list.
@article{liu_etal_2010, author = {Liu, Yanxi and Hel-Or, Hagit and Kaplan, Craig S. and Van Gool, Luc}, title = {Computational symmetry in computer vision and computer graphics}, journal = {Foundations and Trends in Computer Graphics and Vision}, volume = {5}, number = {1--2}, month = {June}, year = {2010}, pages = {1--195}, publisher = {Now Publishers}, doi = {10.1561/0600000008} }
Planned programming problem gotchas as lessons in requirements engineering
Daniel Berry and Craig S. Kaplan
Conference Paper 
Proceedings of the 5th International Workshop on Requirements Engineering Education and Training (REET 2010) (2010). Pages 20–25
Publisher Page
The term “gotcha” is used to describe an unforeseen exception or complexity in a programming assignment given to students. All too often, the students' discovering gotchas in a programming assignment embarrasses the instructor into revising the assignment. This paper argues that students' discovery of gotchas should be encouraged to promote experiential learning of the value and necessity of requirements engineering. Rather than viewing the discovery experience as a misfortune, an instructor should welcome the experience and even plan assignments with an abundance of gotchas to be discovered by students.
@inproceedings{berry_kaplan_2010, author = {Berry, Daniel and Kaplan, Craig S.}, title = {Planned programming problem gotchas as lessons in requirements engineering}, booktitle = {Proceedings of the 5th International Workshop on Requirements Engineering Education and Training (REET 2010)}, year = {2010}, pages = {20--25}, publisher = {IEEE}, doi = {10.1109/REET.2010.5633110} }
Curve evolution schemes for parquet deformations
Craig S. Kaplan
Conference Paper 
Bridges 2010: Mathematical Connections in Art, Music and Science (2010). Pages 95–102
Preprint Publisher Page Project Page
In this paper, I consider the question of how to carry out aesthetically pleasing evolution of the curves that make up the edges in a parquet deformation. Within the framework of simple arrangements of square tiles, I explore curve evolution models based on grids, iterated function systems, and organic labyrinthine paths.
@inproceedings{kaplan_2010, author = {Kaplan, Craig S.}, title = {Curve evolution schemes for parquet deformations}, booktitle = {Bridges 2010: Mathematical Connections in Art, Music and Science}, year = {2010}, pages = {95--102} }
Procedural generation of surface detail for science fiction spaceships
Kate Kinnear and Craig S. Kaplan
Conference Paper 
Workshop on Computational Aesthetics (2010). Pages 83–90
Preprint Publisher Page
Procedural modelling can be used to generate digital content such as 3D digital models programmatically. In computer graphics, Procedural modelling has focused primarily on natural scenery and cityscapes. This paper considers the use of procedural modelling in a new domain: science fiction spaceships. We examine aesthetic principles as they relate to the beauty and visual interest of spaceships, especially surface details, and determine how these principles can be applied in a practical procedural modelling algorithm.We describe a prototype system that synthesizes and distributes surface details on large-scale spaceships. Given a surface representing the frame of a spaceship, we distribute geometry automatically in a coherent manner to achieve a characteristic science fiction aesthetic.
@inproceedings{kinnear_kaplan_2010, author = {Kinnear, Kate and Kaplan, Craig S.}, title = {Procedural generation of surface detail for science fiction spaceships}, booktitle = {Workshop on Computational Aesthetics}, editor = {Pauline Jepp and Oliver Deussen}, year = {2010}, pages = {83--90}, publisher = {Eurographics Association}, address = {London, UK}, isbn = {978-3-905674-24-8}, issn = {1816-0859}, doi = {10.2312/COMPAESTH/COMPAESTH10/083-090} }
Brush stroke ordering techniques for painterly rendering
Lesley Northam, Joe Istead, and Craig S. Kaplan
Conference Paper 
Workshop on Computational Aesthetics (2010). Pages 59–66
Publisher Page
Painterly rendering algorithms often mimic classical hand-painting techniques to automatically generate stylized paintings from input images. These algorithms use a combination of techniques to express a variety of styles and artistic properties (e.g., contrast, mood), but often restrict the user from controlling the rendering order of overlapping brush strokes. This paper illustrates the importance of brush stroke ordering in creating stylistic effects and presents a layer-based painterly rendering algorithm that allows the user to specify a brush stroke ordering. Several of the presented orderings enable the renderer to reduce detail obstruction, simulate handpainting techniques and enhance artistic styles.
@inproceedings{northam_istead_kaplan_2010, author = {Northam, Lesley and Istead, Joe and Kaplan, Craig S.}, title = {Brush stroke ordering techniques for painterly rendering}, booktitle = {Workshop on Computational Aesthetics}, editor = {Pauline Jepp and Oliver Deussen}, year = {2010}, pages = {59--66}, publisher = {Eurographics Association}, address = {London, UK}, isbn = {978-3-905674-24-8}, issn = {1816-0859}, doi = {10.2312/COMPAESTH/COMPAESTH10/059-066} }
Introductory Tiling Theory for Computer Graphics
Craig S. Kaplan
Book 
Morgan & Claypool (July 2009).
Publisher Page
@book{kaplan_2009_b, author = {Kaplan, Craig S.}, title = {Introductory {T}iling {T}heory for {C}omputer {G}raphics}, month = {July}, year = {2009}, publisher = {Morgan \& Claypool}, doi = {10.2200/S00207ED1V01Y200907CGR011} }
NPAR '09: Proceedings of the 7th international symposium on Non-photorealistic animation and rendering
Tobias Isenberg, Craig S. Kaplan, and Stephen N. Spencer
Editor 
ACM (2009).
Publisher Page
@proceedings{isenberg_kaplan_spencer_2009, title = {N{PAR} '09: {P}roceedings of the 7th international symposium on {N}on-photorealistic animation and rendering}, editor = {Isenberg, Tobias and Kaplan, Craig S. and Spencer, Stephen N.}, year = {2009}, publisher = {ACM}, address = {New York}, location = {New Orleans, Louisiana} }
Proceedings of Bridges 2009: Mathematics, Music, Art, Architecture, Culture
Craig S. Kaplan and Reza Sarhangi
Editor 
Tarquin Books (2009).
Publisher Page
@proceedings{kaplan_sarhangi_2009, title = {Proceedings of {B}ridges 2009: {M}athematics, {M}usic, {A}rt, {A}rchitecture, {C}ulture}, editor = {Kaplan, Craig S. and Sarhangi, Reza}, year = {2009}, publisher = {Tarquin Books}, address = {Hertfordshire, UK}, isbn = {978-0-96652-020-0}, issn = {1099-6702} }
Semiregular patterns on surfaces
Craig S. Kaplan
Conference Paper 
NPAR '09: Proceedings of the 7th international symposium on Non-photorealistic animation and rendering (2009). Pages 35–39
Preprint Publisher Page Project Page
Inspired by recent advances in high-quality mesh parameterization, I present a technique for decorating surfaces with seamless ornamental patterns. The patterns are transferred from planar drawings with wallpaper symmetry. I show that when the original drawing belongs to one of a few specific symmetry groups, then it can easily be rendered with low distortion on a suitably parameterized mesh. The result is not symmetric, but retains most of the structure of the original drawing.
@inproceedings{kaplan_2009, author = {Kaplan, Craig S.}, title = {Semiregular patterns on surfaces}, booktitle = {NPAR '09: Proceedings of the 7th international symposium on Non-photorealistic animation and rendering}, year = {2009}, pages = {35--39}, publisher = {ACM Press}, address = {New York}, doi = {10.1145/1572614.1572620} }
Automated landscape painting in the style of Bob Ross
Alex Kalaidjian, Craig S. Kaplan, and Stephen Mann
Conference Paper 
Computational Aesthetics in Graphics, Visualization, and Imaging (2009). Pages 115–122
Publisher Page
We present a system that can generate convincing synthetic landscape paintings with no user intervention whatsoever, nor any information about 3D geometry or lighting. The system is based on a direct implementation of the "wet-on-wet" oil painting technique taught by Bob Ross for many years on his show The Joy of Painting. We implement a canvas model and a set of brushes that correspond to the canvas and brushes that Bob Ross used on his show. We then compose brush strokes into landscape features that replicate his approach stroke by stroke. Finally, we develop an engine for automatic layout of these features in a painting. We demonstrate this automated system in the context of the Bob Ross painting Forest Hills.
@inproceedings{kalaidjian_kaplan_mann_2009, author = {Kalaidjian, Alex and Kaplan, Craig S. and Mann, Stephen}, title = {Automated landscape painting in the style of {B}ob {R}oss}, booktitle = {Computational Aesthetics in Graphics, Visualization, and Imaging}, editor = {Oliver Deussen and Peter Hall}, year = {2009}, pages = {115--122}, publisher = {Eurographics Association}, address = {Victoria, British Columbia, Canada}, isbn = {978-3-905674-17-0}, issn = {1816-0859}, doi = {10.2312/COMPAESTH/COMPAESTH09/115-122} }
Cut-out image mosaics
Jeff Orchard and Craig S. Kaplan
Conference Paper 
NPAR '08: Proceedings of the 6th international symposium on Non-photorealistic animation and rendering (2008). Pages 79–87
Publisher Page Project Page
An image mosaic is a rendering of a large target image by arranging a collection of small source images, often in an array, each chosen specifically to fit a particular block of the target image. Most mosaicking methods are simplistic in the sense that they break the target image into regular tiles (e.g., squares or hexagons) and take extreme shortcuts when evaluating the similarity between target tiles and source images. In this paper, we propose an efficient method to obtain higher quality mosaics that incorporate a number of process improvements. The Fast Fourier Transform (FFT) is used to compute a more fine-grained image similarity metric, allowing for optimal colour correction and arbitrarily shaped target tiles. In addition, the framework can find the optimal sub-image within a source image, further improving the quality of the matching. The similarity scores generated by these high-order cost computations are fed into a matching algorithm to find the globally-optimal assignment of source images to target tiles. Experiments show that each improvement, by itself, yields a more accurate mosaic. Combined, the innovations produce very high quality image mosaics, even with only a few hundred source images.
@inproceedings{orchard_kaplan_2008, author = {Orchard, Jeff and Kaplan, Craig S.}, title = {Cut-out image mosaics}, booktitle = {NPAR '08: Proceedings of the 6th international symposium on Non-photorealistic animation and rendering}, year = {2008}, pages = {79--87}, publisher = {ACM Press}, address = {New York}, location = {Annecy, France}, isbn = {978-1-60558-150-7}, doi = {10.1145/1377980.1377997} }
Artistic thresholding
Jie Xu and Craig S. Kaplan
Conference Paper 
NPAR '08: Proceedings of the 6th International Symposium on Non-Photorealistic Animation and Rendering (2008). Pages 39–47
Preprint Publisher Page Project Page
We consider the problem of depicting continuous-tone images using only black and white. Traditional solutions to this problem include halftoning, which approximates tones, and line drawing, which approximates edges. We introduce “artistic thresholding” as a technique that attempts to depict forms in an image. We apply segmentation to a source image and construct a planar subdivision that captures segment connectivity. Our artistic thresholding algorithm is a combinatorial optimization over this graph. The optimization is controlled by parameters that can be tuned to achieve different artistic styles.
@inproceedings{xu_kaplan_2008, author = {Xu, Jie and Kaplan, Craig S.}, title = {Artistic thresholding}, booktitle = {NPAR '08: Proceedings of the 6th International Symposium on Non-Photorealistic Animation and Rendering}, year = {2008}, pages = {39--47}, publisher = {ACM Press}, address = {New York}, doi = {10.1145/1377980.1377990} }
Precise vector textures for real-time 3D rendering
Zheng Qin, Michael D. McCool, and Craig S. Kaplan
Conference Paper 
SI3D '08: Proceedings of the 2008 symposium on Interactive 3D graphics and games (2008). Pages 199–206
Publisher Page

Vector graphics representations of images are resolution independent. Direct use of vector images for real-time texture mapping would be desirable to avoid sampling artifacts such as blurring common with raster images. Scalable Vector Graphics (SVG) files are typical of vector graphics image representations. Such representations composite images from layers of paths and strokes defined with lines, elliptical arcs, and quadratic and cubic parametric splines.

High-quality texture mapping requires both random access and anisotropic antialiasing. For vector images, these goals can be achieved by computing the distance to the closest primitives from a sample point and then mapping this distance through a soft threshold function. Representing transparency masks in this way is especially useful, since vector mattes can be used to render complex curvilinear geometry as textures on simple geometric primitives.

Unfortunately, computing the exact minimum distance to the parametric curves used in vector images is difficult. Previous work has used approximations, but an accurate minimum distance is desirable in order to enable wide strokes and special effects such as embossing. In this paper, a simple algorithm is presented that can efficiently and accurately compute the minimum distance to a parametric curve when the sample point is within its radius of curvature and the curve can be segmented into monotonic regions. This technique can be used in a GPU shader to render antialiased vector images exactly as defined by SVG files.

@inproceedings{qin_mccool_kaplan_2008, author = {Qin, Zheng and McCool, Michael D. and Kaplan, Craig S.}, title = {Precise vector textures for real-time 3{D} rendering}, booktitle = {SI3D '08: Proceedings of the 2008 symposium on Interactive 3D graphics and games}, year = {2008}, pages = {199--206}, publisher = {ACM Press}, doi = {10.1145/1342250.1342281} }
Metamorphosis in Escher's art
Craig S. Kaplan
Conference Paper 
Bridges 2008: Mathematical Connections in Art, Music and Science (2008). Pages 39–46
Preprint Publisher Page Project Page
M.C. Escher returned often to the themes of metamorphosis and deformation in his art, using a small set of pictorial devices to express this theme. I classify Escher's various approaches to metamorphosis, and relate them to the works in which they appear. I also discuss the mathematical challenges that arise in attempting to formalize one of these devices so that it can be applied reliably.
@inproceedings{kaplan_2008_b, author = {Kaplan, Craig S.}, title = {Metamorphosis in {E}scher's art}, booktitle = {Bridges 2008: Mathematical Connections in Art, Music and Science}, year = {2008}, pages = {39--46} }
Book review: Processing: A Programming Handbook for Visual Designers and Artists
Craig S. Kaplan
Book Review 
Journal of Mathematics and the Arts, Volume 2 (December 2008). Pages 211–214
Publisher Page
@article{kaplan_2008, author = {Kaplan, Craig S.}, title = {Book review: {P}rocessing: {A} {P}rogramming {H}andbook for {V}isual {D}esigners and {A}rtists}, journal = {Journal of Mathematics and the Arts}, volume = {2}, number = {4}, month = {December}, year = {2008}, pages = {211--214}, doi = {10.1080/17513470802701119} }
Image-guided maze construction
Jie Xu and Craig S. Kaplan
Journal Article 
ACM Transactions on Graphics, Volume 26 (July 2007). 29
Preprint Publisher Page Project Page
We present a set of graphical and combinatorial algorithms for designing mazes based on images. The designer traces regions of interest in an image and annotates the regions with style parameters. They can optionally specify a solution path, which provides a rough guide for laying out the maze's actual solution. The system uses novel extensions to well-known maze construction algorithms to build mazes that approximate the tone of the source image, express the desired style in each region, and conform to the user's solution path.
@article{xu_kaplan_2007_c, author = {Xu, Jie and Kaplan, Craig S.}, title = {Image-guided maze construction}, journal = {ACM Transactions on Graphics}, volume = {26}, number = {3}, month = {July}, year = {2007}, pages = {29}, publisher = {ACM}, address = {New York}, issn = {0730-0301}, note = {Proceedings of SIGGRAPH 2007}, doi = {10.1145/1276377.1276414} }
Vortex maze construction
Jie Xu and Craig S. Kaplan
Journal Article 
Journal of Mathematics and the Arts, Volume 1 (March 2007). Pages 7–20
Preprint Publisher Page Project Page
The creation of engaging mazes requires both mathematical and aesthetic considerations. We present an algorithm for producing mazes that we feel are difficult to solve. This algorithm is based on constructing and connecting obfuscating maze devices called vortices. A vortex is a collection of passages that wind around each other in a spiral and converge on a central junction. We explore variations on vortex construction that enable a range of aesthetic opportunities in the design of mazes, and demonstrate our technique with several of our own examples.
@article{xu_kaplan_2007_b, author = {Xu, Jie and Kaplan, Craig S.}, title = {Vortex maze construction}, journal = {Journal of Mathematics and the Arts}, volume = {1}, number = {1}, month = {March}, year = {2007}, pages = {7--20}, publisher = {Taylor \& Francis}, doi = {10.1080/17513470701225236} }
Computer-generated papercutting
Jie Xu, Craig S. Kaplan, and Xiaofeng Mi
Conference Paper 
PG '07: 15th Pacific Conference on Computer Graphics and Applications (2007). Pages 343–350
Preprint Publisher Page Project Page
The craft of papercutting is part of the folk art traditions of cultures all over the world. From the point of view of computer graphics, papercutting can be seen as a method of composing bi-level images under a set of geometric connectivity constraints. In this paper, we present a technique for composing digital paper-cut designs. The elements of a design may be images, which are processed via a multilayer thresholding operation, or they may be procedurallygenerated arrangements of shapes. Elements are composed using a set of boolean operators that preserve connectivity. The resulting designs are well suited to being cut by a new generation of inexpensive computer peripherals.
@inproceedings{xu_kaplan_mi_2007, author = {Xu, Jie and Kaplan, Craig S. and Mi, Xiaofeng}, title = {Computer-generated papercutting}, booktitle = {PG '07: 15th Pacific Conference on Computer Graphics and Applications}, year = {2007}, pages = {343--350}, doi = {10.1109/PG.2007.10} }
Calligraphic packing
Jie Xu and Craig S. Kaplan
Conference Paper 
GI '07: Proceedings of the 2007 conference on Graphics interface (2007). Pages 43–50
Preprint Publisher Page Project Page
There are many algorithms in non-photorealistic rendering for representing an image as a composition of small objects. In this paper, we focus on the specific case where the objects to be assembled into a composition are letters rather than images or abstract geometric forms. We develop a solution to the “calligraphic packing” problem based on dividing up a target region into pieces and warping a letter into each piece. We define an energy function that chooses a warp that best represents the original letter. We discuss variations in rendering style and show results produced by our system.
@inproceedings{xu_kaplan_2007, author = {Xu, Jie and Kaplan, Craig S.}, title = {Calligraphic packing}, booktitle = {GI '07: Proceedings of the 2007 conference on Graphics interface}, year = {2007}, pages = {43--50}, publisher = {Canadian Human-Computer Communications Society}, doi = {10.1145/1268517.1268527} }
The trouble with five
Craig S. Kaplan
Journal Article 
Plus Magazine (December 2007).
Publisher Page
@article{kaplan_2007, author = {Kaplan, Craig S.}, title = {The trouble with five}, journal = {Plus Magazine}, month = {December}, year = {2007}, note = {15 pages. Invited article on five-fold tilings.} }
Vortex maze construction
Xu Jie and Craig S. Kaplan
Conference Paper 
Bridges 2006: Mathematical Connections in Art, Music and Science (2006). Pages 301–308
Preprint Publisher Page Project Page
Labyrinths and mazes have existed in our world for thousands of years. Spirals and vortices are important elements in maze generation. In this paper, we describe an algorithm for constructing spiral and vortex mazes using concentric offset curves. We join vortices into networks, leading to mazes that are difficult to solve. We also show some results generated with our techniques.
@inproceedings{xu_kaplan_2006, author = {Xu Jie and Kaplan, Craig S.}, title = {Vortex maze construction}, booktitle = {Bridges 2006: Mathematical Connections in Art, Music and Science}, year = {2006}, pages = {301--308} }
symTone: two-handed manipulation of tone reproduction curves
Celine Latulipe, Ian Bell, Charles L. A. Clarke, and Craig S. Kaplan
Conference Paper 
GI '06: Proceedings of the 2006 conference on Graphics interface (2006). Pages 9–16
Publisher Page
We present symTone, a dual-mouse, symmetric image manipulation application. symTone includes a symmetric method for manipulating a tone reproduction curve using two standard USB mice. The symTone technique is an important contribution because the two mice are manipulating a geometric object as a tool to improve the underlying digital image, thus a spatial object (the curve) is being used to manipulate non-spatial data (the image tones). Our empirical evaluation of the technique shows that symmetric interaction can be effective for manipulating non-spatial data. This novel technique offers a significant improvement in ease of use and is a precursor to more advanced symmetric tone-mapping applications.
@inproceedings{latulipe_etal_2006_b, author = {Latulipe, Celine and Bell, Ian and Clarke, Charles L. A. and Kaplan, Craig S.}, title = {sym{T}one: two-handed manipulation of tone reproduction curves}, booktitle = {GI '06: Proceedings of the 2006 conference on Graphics interface}, year = {2006}, pages = {9--16}, publisher = {Canadian Information Processing Society} }
A meditation on Kepler's Aa
Craig S. Kaplan
Conference Paper 
Bridges 2006: Mathematical Connections in Art, Music and Science (2006). Pages 465–472
Preprint Publisher Page
Kepler's Harmonice Mundi includes a mysterious arrangement of polygons labeled Aa, in which many of the polygons have fivefold symmetry. In the twentieth century, solutions were proposed for how Aa might be continued in a natural way to tile the whole plane. I present a collection of variations on Aa, and show how it forms one step in a sequence of derivations starting from a simpler tiling. I present alternate arrangements of the tilings based on spirals and substitution systems. Finally, I show some Islamic star patterns that can be derived from Kepler-like tilings.
@inproceedings{kaplan_2006, author = {Kaplan, Craig S.}, title = {A meditation on {K}epler's {A}a}, booktitle = {Bridges 2006: Mathematical Connections in Art, Music and Science}, year = {2006}, pages = {465--472} }
symSpline: symmetric two-handed spline manipulation
Celine Latulipe, Stephen Mann, Craig S. Kaplan, and Charles L. A. Clarke
Conference Paper 
CHI '06: Proceedings of the SIGCHI conference on Human Factors in computing systems (2006). Pages 349–358
Publisher Page
We introduce symSpline: a symmetric, dual-mouse technique for the manipulation of spline curves. In symSpline, two cursors control the positions of the ends of the tangent to an edit point. By moving the tangent with both mice, the tangent and the edit point can be translated while the curvature of the spline is adjusted simultaneously, according to the length and angle of the tangent. We compare the symSpline technique to two asymmetric dual-mouse spline manipulation techniques and to a standard single-mouse technique. In a spline matching experiment, symSpline outperformed the two asymmetric dual-mouse techniques and all three dual-mouse techniques proved to be faster than the single-mouse technique. Additionally, symSpline was the technique most preferred by test participants.
@inproceedings{latulipe_etal_2006, author = {Latulipe, Celine and Mann, Stephen and Kaplan, Craig S. and Clarke, Charles L. A.}, title = {sym{S}pline: symmetric two-handed spline manipulation}, booktitle = {CHI '06: Proceedings of the SIGCHI conference on Human Factors in computing systems}, year = {2006}, pages = {349--358}, publisher = {ACM Press}, doi = {10.1145/1124772.1124825} }
Real-time texture-mapped vector glyphs
Zheng Qin, Michael D. McCool, and Craig S. Kaplan
Conference Paper 
I3D '06: Proceedings of the 2006 symposium on Interactive 3D graphics and games (2006). Pages 125–132
Publisher Page

We present a vector graphics representation suitable for real-time rendering on GPUs. Our representation can be used in place of a texture map, and renders precise antialiased edges at any magnification. A combination of texture data and procedural computation is used to evaluate an exact signed distance to a contour and its gradient. An optimized uniform grid accelerator is created using Voronoi analysis and redundancy elimination, so only the distances to a small constant number of features need be computed at every access. Contours and sharp features can be exactly reconstructed using a constant amount of computation per pixel. Our representation supports inexpensive high-quality anisotropic antialiasing as well as special effects such as outlining (with both rounded and sharp miters) and embossing.

We have applied our representation to the important application of glyph rendering. Variations in glyph complexity are handled by storing different glyphs at different grid resolutions. Large blocks of glyphs can be rendered efficiently with a single indirection through an index texture.

@inproceedings{qin_mccool_kaplan_2006, author = {Qin, Zheng and McCool, Michael D. and Kaplan, Craig S.}, title = {Real-time texture-mapped vector glyphs}, booktitle = {I3D '06: Proceedings of the 2006 symposium on Interactive 3D graphics and games}, year = {2006}, pages = {125--132}, publisher = {ACM Press}, doi = {10.1145/1111411.1111433} }
The design space of Islamic star patterns
Craig S. Kaplan
Book Chapter 
Enlightenment through creativity: collaboration of the arts and sciences (2006). Pages 3–7
Preprint
@incollection{kaplan_2006_b, author = {Kaplan, Craig S.}, title = {The design space of {I}slamic star patterns}, booktitle = {Enlightenment through creativity: collaboration of the arts and sciences}, editor = {Amir Esfahani}, year = {2006}, pages = {3--7}, note = {Invited essay to accompany an exhibition of Esfahani's art.} }
Bimanual and unimanual image alignment: an evaluation of mouse-based techniques
Celine Latulipe, Craig S. Kaplan, and Charles L. A. Clarke
Conference Paper 
UIST '05: Proceedings of the 18th annual ACM symposium on User interface software and technology (2005). Pages 123–131
Publisher Page
We present an evaluation of three mouse-based techniques for aligning digital images. We investigate the physical image alignment task and discuss the implications for interacting with virtual images. In a formal evaluation we show that a symmetric bimanual technique outperforms an asymmetric bimanual technique which in turn outperforms a unimanual technique. We show that even after mode switching times are removed, the symmetric technique outperforms the single mouse technique. Subjects also exhibited more parallel interaction using the symmetric technique than when using the asymmetric technique.
@inproceedings{latulipe_kaplan_clarke_2005_b, author = {Latulipe, Celine and Kaplan, Craig S. and Clarke, Charles L. A.}, title = {Bimanual and unimanual image alignment: an evaluation of mouse-based techniques}, booktitle = {UIST '05: Proceedings of the 18th annual ACM symposium on User interface software and technology}, year = {2005}, pages = {123--131}, publisher = {ACM Press}, doi = {10.1145/1095034.1095057} }
Simultaneous rotation and translation
Celine Latulipe, Craig S. Kaplan, and Charles L. A. Clarke
Conference Paper 
HCI 2005 Proceedings, Volume 2 (2005). Pages 63–67
@inproceedings{latulipe_kaplan_clarke_2005, author = {Latulipe, Celine and Kaplan, Craig S. and Clarke, Charles L. A.}, title = {Simultaneous rotation and translation}, booktitle = {HCI 2005 Proceedings, Volume 2}, year = {2005}, pages = {63--67}, publisher = {The British Computer Society} }
TSP art
Craig S. Kaplan and Robert Bosch
Conference Paper 
Bridges 2005: Mathematical Connections in Art, Music and Science (2005). Pages 301–308
Preprint Publisher Page Project Page
Bosch and Herman recently described how to use the traveling salesman problem (TSP) to construct a continuous line drawing based on a user-supplied image. They create a distribution of “cities” hat approximates the darkness of the source image, and pass the cities to a heuristic TSP solver. We discuss their method and present alternative algorithms for city distribution that yield more attractive line drawings.
@inproceedings{kaplan_bosch_2005, author = {Kaplan, Craig S. and Bosch, Robert}, title = {T{SP} art}, booktitle = {Bridges 2005: Mathematical Connections in Art, Music and Science}, year = {2005}, pages = {301--308} }
Aliasing artifacts and accidental algorithmic art
Craig S. Kaplan
Conference Paper 
Bridges 2005: Mathematical Connections in Art, Music and Science (2005). Pages 349–356
Preprint Publisher Page Project Page
While developing a program to render Voronoi diagrams, I accidentally produced a strange and surprising image. The unexpected behaviour turned out to be caused by a combination of reasons from signal processing and computer architecture. I describe the process that led to the pattern, explain its structure, and display many of the wonderful designs that can be produced from this and related techniques.
@inproceedings{kaplan_2005_b, author = {Kaplan, Craig S.}, title = {Aliasing artifacts and accidental algorithmic art}, booktitle = {Bridges 2005: Mathematical Connections in Art, Music and Science}, year = {2005}, pages = {349--356} }
Islamic star patterns from polygons in contact
Craig S. Kaplan
Conference Paper 
GI '05: Proceedings of the 2005 conference on Graphics interface (2005). Pages 177–185
Preprint Publisher Page
We present a simple method for rendering Islamic star patterns based on Hankin's “polygons-in-contact” technique. The method builds star patterns from a tiling of the plane and a small number of intuitive parameters. We show how this method can be adapted to construct Islamic designs reminiscent of Huff's parquet deformations. Finally, we introduce a geometric transformation on tilings that expands the range of patterns accessible using our method. This transformation simplifies construction techniques given in previous work, and clarifies previously unexplained relationships between certain classes of star patterns.
@inproceedings{kaplan_2005, author = {Kaplan, Craig S.}, title = {Islamic star patterns from polygons in contact}, booktitle = {GI '05: Proceedings of the 2005 conference on Graphics interface}, year = {2005}, pages = {177--185}, publisher = {Canadian Human-Computer Communications Society} }
NPAR '04: Proceedings of the 3rd international symposium on Non-photorealistic animation and rendering
Aaron Hertzmann, Craig S. Kaplan, and Stephen N. Spencer
Editor 
ACM (2004).
Publisher Page
@proceedings{hertzmann_kaplan_spencer_2004, title = {N{PAR} '04: {P}roceedings of the 3rd international symposium on {N}on-photorealistic animation and rendering}, editor = {Hertzmann, Aaron and Kaplan, Craig S. and Spencer, Stephen N.}, year = {2004}, publisher = {ACM}, address = {New York}, location = {Annecy, France}, isbn = {1-58113-887-3} }
A new paradigm for woodworking with NC machines
Craig S. Kaplan, Sanjeev Bedi, Stephen Mann, Gilad Israeli, and Gilbert Poon
Journal Article 
The Journal of Computer-Aided Design and Applications, Volume 1 (2004). Pages 217–222
Publisher Page
We present a paradigm that makes the flexibility of NC machining available to the non-technical woodworker. In this context, general-purpose CAD software and manufacturing systems are not appropriate due to prohibitive complexity and cost. We propose a machine architecture and suite of software tools that together offer a cost-effective and simple way of realizing art in wood. Designs can be tested in a 3D simulation before being realized. As a proof-of-concept of the new paradigm, we show a prototype NC milling lathe, a design tool for the special case of Islamic star patterns, and a decorative piece designed and cut using the system.
@article{kaplan_etal_2004, author = {Kaplan, Craig S. and Bedi, Sanjeev and Mann, Stephen and Israeli, Gilad and Poon, Gilbert}, title = {A new paradigm for woodworking with {NC} machines}, journal = {The Journal of Computer-Aided Design and Applications}, volume = {1}, number = {1--4}, year = {2004}, pages = {217--222}, note = {Also appeared in Proceedings of the CAD'04 conference.}, doi = {10.1080/16864360.2004.10738261} }
Islamic star patterns in absolute geometry
Craig S. Kaplan and David H. Salesin
Journal Article 
ACM Trans. Graph., Volume 23 (2004). Pages 97–119
Preprint Publisher Page
We present Najm, a set of tools built on the axioms of absolute geometry for exploring the design space of Islamic star patterns. Our approach makes use of a novel family of tilings, called “inflation tilings,” which are particularly well-suited as guides for creating star patterns. We describe a method for creating a parameterized set of motifs that can be used to fill the many regular polygons that comprise these tilings, as well as an algorithm to infer geometry for any irregular polygons that remain. Erasing the underlying tiling and joining together the inferred motifs produces the star patterns. Because Najm is built using only the axioms of absolute geometry, which makes no assumption about the behaviour of parallel lines, star patterns created by Najm can be designed equally well to fit the Euclidean plane, the hyperbolic plane, or the surface of a sphere.
@article{kaplan_salesin_2004_b, author = {Kaplan, Craig S. and Salesin, David H.}, title = {Islamic star patterns in absolute geometry}, journal = {ACM Trans. Graph.}, volume = {23}, number = {2}, year = {2004}, pages = {97--119}, publisher = {ACM Press}, doi = {10.1145/990002.990003} }
Dihedral Escherization
Craig S. Kaplan and David H. Salesin
Conference Paper 
GI '04: Proceedings of the 2004 conference on Graphics interface (2004). Pages 255–262
Preprint Publisher Page
“Escherization” is a process that finds an Escher-like tiling of the plane from tiles that resemble a user-supplied goal shape. We show how the original Escherization algorithm can be adapted to the dihedral case, producing tilings with two distinct shapes. We also use a form of the adapted algorithm to create drawings in the style of Escher's print Sky and Water. Finally, we develop an Escherization algorithm for the very different case of Penrose's aperiodic tilings.
@inproceedings{kaplan_salesin_2004, author = {Kaplan, Craig S. and Salesin, David H.}, title = {Dihedral {E}scherization}, booktitle = {GI '04: Proceedings of the 2004 conference on Graphics interface}, year = {2004}, pages = {255--262}, publisher = {Canadian Human-Computer Communications Society} }
Computer Graphics and Geometric Ornamental Design
Craig S. Kaplan
PhD Thesis 
(2002).
Project Page

Throughout history, geometric patterns have formed an important part of art and ornamental design. Today we have unprecedented ability to understand ornamental styles of the past, to recreate traditional designs, and to innovate with new interpretations of old styles and with new styles altogether.

The power to further the study and practice of ornament stems from three sources. We have new mathematical tools: a modern conception of geometry that enables us to describe with precision what designers of the past could only hint at. We have new algorithmic tools: computers and the abstract mathematical processing they enable allow us to perform calculations that were intractable in previous generations. Finally, we have technological tools: manufacturing devices that can turn a synthetic description provided by a computer into a real-world artifact. Taken together, these three sets of tools provide new opportunities for the application of computers to the analysis and creation of ornament.

In this dissertation, I present my research in the area of computer-generated geometric art and ornament. I focus on two projects in particular. First I develop a collection of tools and methods for producing traditional Islamic star patterns. Then I examine the tesselations of M. C. Escher, developing an “Escherization” algorithm that can derive novel Escher-like tesselations of the plane from arbitrary user-supplied shapes. Throughout, I show how modern mathematics, algorithms, and technology can be applied to the study of these ornamental styles.

@phdthesis{kaplan_2002, author = {Kaplan, Craig S.}, title = {Computer {G}raphics and {G}eometric {O}rnamental {D}esign}, year = {2002}, school = {Department of Computer Science \& Engineering, University of Washington} }
Polygons cuttable by a circular saw
Erik D. Demaine, Martin L. Demaine, and Craig S. Kaplan
Journal Article 
Comput. Geom. Theory Appl., Volume 20 (2001). Pages 69–84
Publisher Page
We introduce and characterize a new class of polygons that models wood, stone, glass and ceramic shapes that can be cut with a table saw, lapidary trim saw, or other circular saw. In this model, a circular saw is a line segment (in projection) that can move freely in empty space, but can only cut straight into a portion of material. Once a region of material is separated from the rest, it can be picked up and removed to allow the saw to move more freely. A polygon is called cuttable by a circular saw if it can be cut out of a convex shape of material by a sufficiently small circular saw. We prove that a polygon has this property precisely if it does not have two adjacent reflex vertices. As a consequence, every polygon can be modified slightly to make it cuttable by a circular saw. We give a linear-time algorithm to cut out such a polygon using a number of cuts and total length of cuts that are at most 2.5 times the optimal. We also study polygons cuttable by an arbitrarily large circular saw, which is equivalent to a ray, and give two linear-time recognition algorithms.
@article{demaine_demaine_kaplan_2001, author = {Demaine, Erik D. and Demaine, Martin L. and Kaplan, Craig S.}, title = {Polygons cuttable by a circular saw}, journal = {Comput. Geom. Theory Appl.}, volume = {20}, number = {1--2}, year = {2001}, pages = {69--84}, publisher = {Elsevier Science Publishers B. V.}, note = {Special issue of selected papers from the 12th annual Canadian conference on computational geometry (CCCG 2000).}, doi = {http://dx.doi.org/10.1016/S0925-7721(01)00036-0} }
Symmetrohedra: polyhedra from symmetric placement of regular polygons
Craig S. Kaplan and George W. Hart
Conference Paper 
Bridges 2001: Mathematical Connections in Art, Music and Science (2001). Pages 21–28
Preprint Publisher Page
In the quest for new visually interesting polyhedra with regular faces, we define and present an infinite class of solids, constructed by placing regular polygons at the rotational axes of a polyhedral symmetry group. This new technique can be used to generate many existing polyhedra, including most of the Archimedean solids. It also yields novel families of attractive symmetric polyhedra.
@inproceedings{kaplan_hart_2001, author = {Kaplan, Craig S. and Hart, George W.}, title = {Symmetrohedra: polyhedra from symmetric placement of regular polygons}, booktitle = {Bridges 2001: Mathematical Connections in Art, Music and Science}, year = {2001}, pages = {21--28} }
A computer analysis of Boggle™
Craig S. Kaplan
Conference Paper 
POCSI433, Problems of Computer Science in Room 433 (2001).
Preprint
Boggle is a fast-paced word search game played on a five-by-five grid of letters. To remove any trace of fun from the game, I have conducted an extensive analysis of Boggle, using a newly-constructed software tool. I present the tool and the Boggle insights it provides.
@inproceedings{kaplan_2001, author = {Kaplan, Craig S.}, title = {A computer analysis of {B}oggle\texttrademark}, booktitle = {POCSI433, Problems of Computer Science in Room 433}, year = {2001}, note = {3 pages} }
Escherization
Craig S. Kaplan and David H. Salesin
Conference Paper 
SIGGRAPH '00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques (2000). Pages 499–510
Publisher Page
This paper introduces and presents a solution to the “Escherization” problem: given a closed figure in the plane, find a new closed figure that is similar to the original and tiles the plane. Our solution works by using a simulated annealer to optimize over a parameterization of the “isohedral” tilings, a class of tilings that is flexible enough to encompass nearly all of Escher's own tilings, and yet simple enough to be encoded and explored by a computer. We also describe a representation for isohedral tilings that allows for highly interactive viewing and rendering. We demonstrate the use of these tools – along with several additional techniques for adding decorations to tilings – with a variety of original ornamental designs.
@inproceedings{kaplan_salesin_2000, author = {Kaplan, Craig S. and Salesin, David H.}, title = {Escherization}, booktitle = {SIGGRAPH '00: Proceedings of the 27th annual conference on Computer graphics and interactive techniques}, year = {2000}, pages = {499--510}, publisher = {ACM Press/Addison-Wesley Publishing Co.}, doi = {10.1145/344779.345022} }
Computer generated Islamic star patterns
Craig S. Kaplan
Conference Paper 
Bridges 2000: Mathematical Connections in Art, Music and Science (2000). Pages 105–112
Preprint Publisher Page
Islamic star patterns are a beautiful and highly geometric art form whose original design techniques are lost in history. We describe one procedure for constructing them based on placing radially-symmetric motifs in a formation dictated by a tiling of the plane, and show some styles in which they can be rendered. We also show some results generated with a software implementation of the technique.
@inproceedings{kaplan_2000, author = {Kaplan, Craig S.}, title = {Computer generated {I}slamic star patterns}, booktitle = {Bridges 2000: Mathematical Connections in Art, Music and Science}, year = {2000}, pages = {105--112} }
Voronoi diagrams and ornamental design
Craig S. Kaplan
Conference Paper 
ISAMA'99: The first annual symposium of the International Society for the Arts, Mathematics, and Architecture (1999). Pages 277–283
Preprint Project Page
A set of points in the plane induces a Voronoi diagram, a division of the plane based on proximity to points in the set. Voronoi diagrams have been used extensively in engineering and scientific disciplines, but the possibility of using them for creating abstract ornamental designs is largely unexplored. I present some techniques for creating attractive ornamental designs using Voronoi diagrams. I focus on two features of Voronoi diagrams that make them particularly useful artistic tools: their conservation of symmetry, which be used to construct interesting tilings of tne plane, and their continuity with respect to changes in the generators, which makes possible smooth, organic animations of tilings.
@inproceedings{kaplan_1999, author = {Kaplan, Craig S.}, title = {Voronoi diagrams and ornamental design}, booktitle = {ISAMA'99: The first annual symposium of the International Society for the Arts, Mathematics, and Architecture}, year = {1999}, pages = {277--283} }
Predicate dispatching: a unified theory of dispatch
Michael Ernst, Craig S. Kaplan, and Craig Chambers
Conference Paper 
ECCOP '98: Proceedings of the 12th European Conference on Object-Oriented Programming (1998). Pages 186–211
Publisher Page Project Page
Predicate dispatching generalizes previous method dispatch mechanisms by permitting arbitrary predicates to control method applicability and by using logical implication between predicates as the overriding relationship. The method selected to handle a message send can depend not just on the classes of the arguments, as in ordinary ob ject-oriented dispatch, but also on the classes of subcomponents, on an argument's state, and on relationships between ob jects. This simple mechanism subsumes and extends ob ject-oriented single and multiple dispatch, ML-style pattern matching, predicate classes, and classiers, which can all be regarded as syntactic sugar for predicate dispatching. This paper introduces predicate dispatching, gives motivating examples, and presents its static and dynamic semantics. An implementation of predicate dispatching is available
@inproceedings{ernst_kaplan_chambers_1998, author = {Ernst, Michael and Kaplan, Craig S. and Chambers, Craig}, title = {Predicate dispatching: a unified theory of dispatch}, booktitle = {ECCOP '98: Proceedings of the 12th European Conference on Object-Oriented Programming}, year = {1998}, pages = {186--211}, publisher = {Springer-Verlag}, doi = {10.1007/BFb0054092} }