Polygonal modeling

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. Alternate methods of representing 3D objects include NURBS surfaces, subdivision surfaces, and equation-based representations used in ray tracers. See polygon mesh for a description of how polygonal models are represented and stored. 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. Alternate methods of representing 3D objects include NURBS surfaces, subdivision surfaces, and equation-based representations used in ray tracers. See polygon mesh for a description of how polygonal models are represented and stored. 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 normals 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. Many modeling programs do not strictly enforce geometric theory; for example, it is possible for two vertices to have two distinct edges connecting them, occupying exactly the same spatial location. It is also possible for two vertices to exist at the same spatial coordinates, or two faces to exist at the same location. Situations such as these are usually not desired and many packages support an auto-cleanup function. If auto-cleanup is not present, however, they must be deleted manually. 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). Although it is possible to construct a mesh by manually specifying vertices and faces, it is much more common to build meshes using a variety of tools. A wide variety of 3D graphics software packages are available for use in constructing polygon meshes. One of the more popular methods of constructing meshes is box modeling, which uses two simple tools: A second common modeling method is sometimes referred to as inflation modeling or extrusion modeling. In this method, the user creates a 2D shape which traces the outline of an object from a photograph or a drawing. The user then uses a second image of the subject from a different angle and extrudes the 2D shape into 3D, again following the shape’s outline. This method is especially common for creating faces and heads. In general, the artist will model half of the head and then duplicate the vertices, invert their location relative to some plane, and connect the two pieces together. This ensures that the model will be symmetrical. Another common method of creating a polygonal mesh is by connecting together various primitives, which are predefined polygonal meshes created by the modeling environment. Common primitives include: