43 Polygonal Modelling

In 3D computer graphics, polygonal modeling is an approach for modeling objects by representing or approximating their surfaces using polygons. Polygonal modeling is well suited to scanline rendering and is therefore the method of choice for real-time computer graphics.

The basic object used in mesh modeling is a vertex, a point in three-dimensional space. Two vertices connected by a straight line become an edge. Three vertices, connected to each other by three edges, define a triangle, which is the simplest polygon in Euclidean space. More complex polygons can be created out of multiple triangles, or as a single object with more than 3 vertices. Four sided polygons (generally referred to as quads) and triangles are the most common shapes used in polygonal modeling. A group of polygons, connected to each other by shared vertices, is generally referred to as an element. Each of the polygons making up an element is called a face.

In Euclidean geometry, any three non-collinear points determine a plane. For this reason, triangles always inhabit a single plane. This is not necessarily true of more complex polygons, however. The flat nature of triangles makes it simple to determine their surface normal, a three-dimensional vector perpendicular to the triangle’s surface. Surface normal are useful for determining light transport in ray tracing, and are a key component of the popular Phong shading model. Some rendering systems use vertex normals instead of face normals to create a better-looking lighting system at the cost of more processing. Note that every triangle has two face normals, which point to opposite directions from each other. In many systems only one of these normals is considered valid – the other side of the polygon is referred to as a backface  , and can be made visible or invisible depending on the programmer’s desires.

A group of polygons which are connected by shared vertices is referred to as a mesh. In order for a mesh to appear attractive when rendered, it is desirable that it be non-self-intersecting, meaning that no edge passes through a polygon. Another way of looking at this is that the mesh cannot pierce itself. It is also desirable that the mesh not contain any errors such as doubled vertices, edges, or faces. For some purposes it is important that the mesh be a manifold – that is, that it does not contain holes or singularities (locations where two distinct sections of the mesh are connected by a single vertex).

A polygon mesh is a collection of vertices, edges and faces that defines the shape of a polyhedral object in 3D computer graphics and solid modeling. The faces usually consist of triangles (triangle mesh), quadrilaterals, or other simple convex polygons, since this simplifies rendering, but may also be composed of more general concave polygons, or polygons with holes.

The study of polygon meshes is a large sub-field of computer graphics and geometric modeling. Different representations of polygon meshes are used for different applications and goals. The variety of operations performed on meshes may include Boolean logic, smoothing, simplification, and many others. Volumetric meshes are distinct from polygon meshes in that they explicitly represent both the surface and volume of a structure, while polygon meshes only explicitly represent the surface (the volume is implicit). As polygonal meshes are extensively used in computer graphics, algorithms also exist for ray tracing, collision detection, and rigid-body dynamics of polygon meshes.

Objects created with polygon meshes must store different types of elements. These include vertices, edges, faces, polygons and surfaces. In many applications, only vertices, edges and either faces or polygons are stored. A renderer may support only 3-sided faces, so polygons must be constructed of many of these, as shown above. However, many renderers either support quads and higher-sided polygons, or are able to convert polygons to triangles on the fly, making it unnecessary to store a mesh in a triangulated form. Also, in certain applications like head modeling, it is desirable to be able to create both 3- and 4-sided polygons.

Polygon models are divided in two main categories

Polygons are used in computer graphics to compose images that are three-dimensional in appearance. Usually (but not always) triangular, polygons arise when an object’s surface is modeled, vertices are selected, and the object is rendered in a wire frame model. This is quicker to display than a shaded model; thus the polygons are a stage in computer animation. The polygon count refers to the number of polygons being rendered per frame.


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