VIZA 657 / CSCE 648
The course introduces the mathematical and artistic principles of 3D modeling and sculpting. The topics include but are not limited to parametric curves and surfaces (including B-splines and NURBs), implicit surfaces, constructive solid geometry, subdivision surfaces, proportions in construction, expressions and postures.

This course,conceptually, is built upon Topological Mesh Modeling, which is an umbrella term that covers all the work based on computer graphics extensions the theory of graph rotation systems. It includes (1) Orientable 2-manifol mesh modeling using graph rotation systems and its computer graphics applications, (2) Knot modeling with immersions of non-orientable manifold meshes and (3) Topological constructions that is based on geometric and physical constraints with graph rotation systems. The course will also include immersions of 3-manifolds as a representation to develop shape modeling systems.

Orientable Mesh Modeling: provides a solid foundation for orientable 2-manifold mesh modeling using graph rotation systems. Based on this theory, TopMod , which is is an orientable 2-manifold mesh modeling system is developed. TopMod is also a by-product of this class. TopMod provides a wide vriety of High Genus Modeling tools, Remeshings & Subdivisions, and Extrusions & Replacements. Using TopMod, one can find a wide variety of ways to create high genus shapes; almost all subdivision algorithms, wide variety of ways to remeshing shapes and new extrusions. These tools are also useful for Architectural applications, Design and Sculpting and Sketch Based Modeling.

(a1) (a2) (a3) (b1) (b2) (b3) (c1) (c2) (c3) (d1)
Figure 1. Topmod has motivated people around the world to create unusual shapes. These are shapes produced by the users of TopMod. Shapes (a1) through (a3) were created by Torolf Sauermann, a sculptor from Germany. Shapes (b1) through (b3) were created by Oliver Miller, a designer from Great Britain. Shapes (c1) through (c3) were created by Jonathan Johanson (sjoo), a sculpture fromFrance. Shape (d1) was created by Vladimir Alexeev, an artist from Russia.

Figure 2. Example that shows how non-orientable 2-manifold meshes can be turned into woven objects.
Figure 3. A topological construction of Bunny using graph rotation systems.
Knots Modeling provides a solid foundation for knot, link and cyclic woven object modeling using extended graph rotation systems. If we twist an arbitrary subset of edges of a mesh on an orientable surface, we can obtain non-orientable surfaces. The resulting extended graph rotation system can be used to induce a cyclic weaving on the original surface, that corresponds a 3-space immedding of a non-orientable surface.

Topological Construction: Discrete Gaussian-Bonnet theorem and Gaussian curvatures relate mesh topologic concepts to geometry. Using this relationship, it is possible to developed methods to physically construct shapes. This relationship is particularly useful to turn mesh data structures such as quad edges or winged edges into physical data structures such that sculptures themselves become the data structures.

Immersions of 3-Manifolds: Using an extension of graph rotation systems it is also possible to represent 3-space immersions of 3-manifolds by employing a topological graph theory concept called 3D thickening.

Figure 4. Gears; a class project by Cem Yuksel
Final/Research projects: The main goal of the class is to employ these computational topological and geometrical modeling techniques to develop new design and construction methodologies as final projects. Students are organized into interdisciplinary teams that consists of Engineering, Computer Science, Architecture and Visualization students. These teams work on to come up new ideas in areas such as (1) Gears; (2) D-Forms; (3) Linked structures using High Genus Surfaces; (4) Interesting decompositions of 3-Manifold structures; (5) Woven objects; (6) Relief cuts to obtain flexible materials; (7) Unfolding techniques; (8) Dynamic Linked Structures; (9) Origami structures.