Artistic Tessellations by Growing Curves Hua Li ∗ David Mould † Carleton University (a) Tiffany glass (b) Craquelure (c) Mosaics (d) Macaroni art Figure 1: From curves to artistic tessellations. Abstract In this paper we propose to tessellate a region by growing curves. We use a particle system, which flexibly provides good control over the final effects by variations of the initial placement, the place- ment order, curve direction, and curve properties. We also propose an automatic image-based mosaic method which has good texture indication, using a smoothed vector field to guide particle move- ment. The final irregular tessellation simulates stained glass where the elongated curved tiles suggest the content of highly textured areas. We give some additional applications, some of which resem- ble naturally occurring irregular patterns such as cracks and scales. We also notice that stacking a set of curves in a structured way can produce the illusion of a 3D shape. CR Categories: I.3.3 [Computer Graphics]: Picture/Image Generation—Line and curve generation; Keywords: Tessellation, particle system, natural patterns, mosaics 1 Introduction Human are very good at abstracting similar structure from natu- ral textures such as fish or snake scales, cracks, or bark. This is because the human brain contains mechanisms that rapidly define regions having common texture and color [Regan 2000]. We call these “natural patterns” if they have irregular tessellations but sig- nificantly similar other structural properties in abstraction. The existence of irregularities and randomness reminds us of the beauty of nature. Some artists also like to introduce unorganized elements into their work. Typical examples can be found in Tiffany glass or in stained-glass mosaics. Figure 2 (a) shows a Tiffany glass example that uses irregular tiles to represent objects. The curved ∗ e-mail: hli1@connect.carleton.ca † e-mail:mould@scs.carleton.ca boundaries are very jagged and rough. The artist Barbara Keith used a lot of uneven tiles of different shapes to express the hen in Figure 2 (b). In particular, the elongated tiles show the flow of the feathers beautifully. However, it is very difficult to automatically generate a tessellation that not only contains texture indication but also possesses organized irregularities. (a) Tiffany glass (b) Stained-glass mosaics Figure 2: Two typical art work showing the irregular tessellation. (a) By an anonymous artist from Flickr.com; (b) By Barbara Keith. There is a very long history of research on creating regular tessel- lations. The mainstream methods are region-based methods, typi- cally based on Voronoi diagrams [Okabe et al. 1992]. However, it is difficult to adapt such methods to represent natural patterns with elongated, irregular, or curved tiles; control over site placement is insufficient to create the desired tile shapes. It is common to see the use of regular tiles (e.g., square or hexagonal), especially in simulations of traditional mosaics [Hausner 2001; Elber and Wol- berg 2003; Faustino and de Figueiredo 2005; Smith et al. 2005; Liu et al. 2007; Liu et al. 2010]. High-frequency textures are not of- ten retained in earlier results due to tile shape constraints and style requirements. In non-photorealistic rendering (NPR), researchers dealt with im- ages either with region-based methods like mosaics or with stroke- based methods [Hertzmann 2003] separately. However, they rarely connect them together. The main contribution of this paper is an idea to tessellate a region by adding curves, either sequentially or in parallel, starting from a single point or from a distribution of points. The key is that, instead of tessellating with individual tiles (or re- gions), we build the boundary of each tile by the growth of curves. Our implementation employs a particle system where particle trails form curves. Control over the variations of the initial placement, the placement orders, curve directions, and curve properties provides