Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales.
Catesian Coordinate System
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. The three-dimensional Cartesian coordinate system is a natural extension of the two-dimensional version formed by the addition of a third "in and out" axis mutually perpendicular to the x- and y-axes defined above. This new axis is conventionally referred to as the z-axis and the coordinate z may lie anywhere in the interval (-infty,infty). An ordered triple (x,y,z) in three-dimensional Cartesian coordinates is often called a point or a 3-vector.
Geometric Theory and Polygons
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. Four sided polygons 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.
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.
Primitives
Primitives are the building blocks of 3D—basic geometric forms that you can use as is or modify with transforms and Booleans. Although it's possible to create most of these objects by lathing or extruding 2D shapes, most software packages build them in for speed and convenience.The most common 3D primitives are cubes, pyramids, cones, spheres, and tori. Like 2D shapes, these primitives can have a resolution level assigned to them so that you can make them look smoother by boosting the number of sides and steps used to define them.
Surfaces
http://www.onlinedesignteacher.com/images/subdivision%20surface.png
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.
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