**Real-Time Rendering of Massive Objects**

*General problem:* This problem concerns the visual representation of large, 3-dimensional objects. Objects must be represented by a computer program in such a way as to allow them to be moved and manipulated in real time (20 or more frames/sec.)

*Geometric methods:* Triangulation is necessary for methods that divide the polyhedral object into smaller, simple ones that approximate the shape. Weighted graphs might also be used in other methods, which use lines and curves instead of polyhedra, and which evaluate the amount of light falling on each edge.

**Interactive Boundary Computation of Boolean Combinations of Sculptured Solids**

*General problem:* Visual models of complex solids can be created by considering Boolean operations performed on the boundary curves. This allows real-time placement of models to occur without collisions, and also allows complex solids to be built quickly by performing Boolean operations on simple solids.

*Geometric methods:* Finding the intersections of lines and curves; finding edge intersections of several graphs.

**Metamorphosis of the Cube**

*General problem:* A simple polyhedron (like a cube) can be "unfolded" in such a way so as to create a complex polyhedron upon refolding. Likewise, a complex polyhedron can be unfolded (a process called star unfolding) and refolded into a simple polyhedron. The questions of exactly how to do this remain open.

*Geometric methods:* Division of the plane

**Silhouette Modelling**

(this is not the real title, which I missed because it flashed up so quickly)

*General problem: * When complex solids need to be represented as 3D models in real time, an efficient method is to use only a silhouette of the model. The silhouette can be found by representing only those edges which touch both a face that points toward the viewer and one that points away. These "silhouette edges" can be found using the dual of the graph representing the solid.

*Geometric methods: * Nonplanar graphs; finding the dual of nonplanar graphs

**Feature-Based Surface Composition for Metamorphosis of Polyhedra**

*General Problem: *This is an algorithm used to merge two polyhedra, or to "morph" one polyhedron into another. The user of the program specifies some verticies that will be involved in the transformation; the program finds the rest using a shortest-path algorithm. The information on the edges and verticies of the polyhedra is mapped to a straight-line graph in order to perform the transformation.

*Geometric methods: * Mapping of polyhedra to nonplanar graphs; finding the shortest path on a graph

**Mesh Collapse Compression**

*General problem:* Solid models can be created and removed quickly via a "mesh compression" method which collapses a solid into a single vertex. The solid is represented by a graph, or "mesh," of lines and vertices. The graph is compressed one edge at a time, and each withdrawn edge is encoded by an integer. These integers have to do with the degree of vertices that remain inside the loop created by the compressed edge. The integers, put together into a numeric code, are used to "decompress" the solid.

*Geometric methods:*Creating nonplanar graphs, finding the degrees of edges

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