Special Section on Expressive Graphics Hamiltonian cycle art: Surface covering wire sculptures and duotone surfaces Ergun Akleman a,n , Qing Xing a , Pradeep Garigipati a , Gabriel Taubin b , Jianer Chen a , Shiyu Hu a a Texas A&M University, United States b Brown University, United States article info Article history: Received 5 September 2012 Received in revised form 30 December 2012 Accepted 7 January 2013 Available online 9 February 2013 Keywords: Shape modeling Computer aided sculpting Texture mapping Computational aesthetics Computational art abstract In this work, we present the concept of ‘‘Hamiltonian cycle art’’ that is based on the fact that any mesh surface can be converted to a single closed 3D curve. These curves are constructed by connecting the centers of every two neighboring triangles in the Hamiltonian triangle strips. We call these curves surface covering since they follow the shape of the mesh surface by meandering over it like a river. We show that these curves can be used to create wire sculptures and duotone (two-color painted) surfaces. To obtain surface covering wire sculptures we have developed two methods to construct corresponding 3D wires from surface covering curves. The first method constructs equal diameter wires. The second method creates wires with varying diameter and can produce wires that densely cover the mesh surface. For duotone surfaces, we have developed a method to obtain surface covering curves that can divide any given mesh surface into two regions that can be painted in two different colors. These curves serve as a boundary that define two visually interlocked regions in the surface. We have implemented this method by mapping appropriate textures to each face of the initial mesh. The resulting textured surfaces look aesthetically pleasing since they closely resemble planar TSP (traveling salesmen problem) art and Truchet-like curves. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction and motivation In this work, we introduce a simple approach that provides methods to create a variety of artworks. Our approach is based on converting any given mesh surface into a closed 3D curve that follows the shape of the given surface. Our work is based on Gabriel Taubin’s work on constructing Hamiltonian triangle strips on quadrilateral meshes [1–3]. In graph theory, a Hamiltonian path is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Note that not every graph has a Hamiltonian cycle. Hamiltonian triangle strips are defined in duals of triangular meshes. Taubin shows that it is always possible to construct a triangular mesh from any given quadrilateral mesh such that the dual of the triangular mesh has an Hamiltonian cycle. Moreover, he pre- sented simple linear time and space constructive algorithms to construct these triangle strips. His algorithms are based on splitting each quadrilateral face along one of its two diagonals and linking the resulting triangles along the original mesh edges. With these algorithms every connected manifold quadrilateral mesh without boundary can be represented as a single Hamilto- nian generalized triangle strip cycle. Using Taubin’s algorithms to construct a closed curve in 3D is straightforward. One can simply connect centers of triangles in the triangle strip to obtain a control polygon in 3D. The resulting control polygon can be turned into a smooth curve using a parametric curve such as B-spline as shown in Fig. 1 [2]. These curves can be used for creating artworks. Designers of these curves have significantly large number of aesthetic possibilities. There are two ways to control aesthetic possibilities for surface covering curves:  Designing mesh structures: The shape of any given surface can be approximated by a wide variety of meshes. Therefore, designers, by choosing different meshes, can obtain aestheti- cally different curves. Examples that show the effect of the structure of the underlying mesh on a spherical shape are shown in Fig. 1. In this figure, the control meshes are obtained using a variety of subdivision schemes available in TopMod3D such as honeycomb and pentagonal subdivisions [4–6]. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/cag Computers & Graphics 0097-8493/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cag.2013.01.004 n Corresponding author. Tel.: þ1 9797390094. E-mail addresses: ergun.akleman@gmail.com, ergun@tamu.edu (E. Akleman). Computers & Graphics 37 (2013) 316–332