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Geometry of virtual world (Geometric models)

Virtual world consists of geometric models which is represented in a 3D Euclidean space with Cartesian coordinates.

Geometric models are of surfaces or solid regions in R3 and contain infinite number of points. Representation of models must be finite, so these are defined in terms of Primitives in which each represents an infinite set of points.

The simplest and most useful primitives is a 3D triangle. A planar surface patch that corresponds to all points inside and so on the boundary of the triangle is specified by the coordinates of the triangle vertices.

To model a complicated object or body in the virtual world, numerous triangle can be arranged into a mesh.

 Data Structures

Geometric models are usually encoded in efficient data structures. The choice of the data structures.The choice of the data structure depend on which operations will be preformed on the model. One of the most common data structure is doubly connected edge list, also known as half edge data structure.

Three elements of this data structure are faces, edges, and vertices. Each face represents a triangle, each edge represents the border of one or two triangles, without duplication. Each vertex is shared between one or more triangles, without duplication.

The data structure contains pointers between adjacent faces, edges, and vertices so that algorithms can quickly traverse the model components in a way that corresponds to how they are connected together. 

Inside vs. outside

If mesh triangles fit together perfectly so that every edge boarders exactly two triangle and no triangles intersect unless they are adjacent along the surface, the model forms a barrier between inside and outside of the object. These models are called as coherent models.

Coherent models are required if the notion inside or outside is critical to VWG.

High level design tool like Blender or Maya automatically built coherent models.

Stationary vs. movable models

Stationary models keep the same coordinates forever. (Example : streets, floors, and buildings)

Movable models can be transformed into various positions and orientations. (Example : vehicles, avatars)

Choosing coordinate axes

Coordinates for the models are chosen in terms of their placement scale and scale.

For movable models, the location of the original and axis directions become extremely important because they affect how the model is rotated.

Viewing the models

Transformation helps in determining where the points in the virtual world appear on the display.

Rendering takes into account that how each part of this model should appear after taking into account lighting sources and material properties.

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