Most polygonal meshes are built from triangular or quadrilateral polygons (quads). Triangles and triangle meshes are the most widely use form of polygonal surface because they are accepted by nearly all modeling software. Triangles are, by definition, always planar . Polygons with four or more vertices can be planar (2D) or not (3D) depending on the vertices' position. Some lower end and older modeling software cannot use non-planar polygons. This problem can be resolved by converting all polygons to triangles.

A mathematical smoothing operation can be performed on the sharp edges between adjacent polygons. Smoothing occurs across an edge when the angle formed across the edge between the two polygons is less than a specified maximum angle. This results in smoothing occurring in regions where the change in polygon orientation between adjacent polygons is low and no smoothing occurring in regions where the change in polygon orientation between adjacent polygons is high, such as the the right angled edge of a cube.

The effect of smoothing

When polygonal meshes are mapped with digital Images, 2D (U,V) coordinates corresponding to locations on the image are assigned to each of the mesh's vertices. The concept of 'UV space' is useful to separate'world space', where objects exist and animation takes place, from the individual self-contained 2D 'world' of each polygon mesh UV's. While polygon meshes describe shapes in 3D world space, the coordinates in UV space that constitute the image mapping instructions for the polygon mesh are in 2D.

Parametric geometric elements such as widely used cubic B-spline curves and surfaces are the output results of mathematical functions called polynomial functions. Polynomial functions take as their inputs the (X,Y, Z) coordinate values that describe locations in space. These points, represented by their coordinate values, are called 'control vertices' (CV's).

For cubic B-spline curves, the input values of at least four CV locations are required before the polynomial function will output a curve. This primitive simple curve is called a span. When more than four CV locations are entered into the function, for each additional CV, the function will create both a special point entity on the path of the curve called a knot, and a new span.

Parametric Curve Elements

Knots smoothly 'tie together' multiple spans. When curves are used to build surfaces, the knot positions establish surface patch subdivision boundaries called isoparms. Because each span shares CV's with its neighboring spans, multi-span curves have the property of appearing to be one long continuous curve. Curves composed of multiple spans are called 'piecewise curves'.

Piecewise curves are able to describe extremely complex 3D shapes. The shape and location of the curve is altered by changing the positions of the curve's CV's or knots.

The locations on cubic B-spline curves are defined by a single axis system with the axis, 'U', assigning numerical values along the curve from its first CV to its last. The method used to apply these values to the curve is called the curve's parameterization.

Surfaces are generated from curves. A simple cubic B-spline surface is a four sided shape bounded by four single span, four CV, edge curves joined at the corners. These primitive simple B-spline surfaces are called patches. Each simple patch has sixteen CV's; the twelve CV's along the edges and four CV's in the interior of the patch. These sixteen CV's constitute a 2D row and column array that controls, through their locations, the shape of the patch. For each additional row or column of CV's incorporated into the surface, a new row or column of patches is generated.

Surfaces composed of two or more patches are called 'piecewise surfaces'. Like piecewise curves, piecewise surfaces are able to represent extremely complex 3D shapes.

The surface curves used to define patch boundaries are called isoparms. The word 'isoparm' is short for isoparametic flow line. Isoparms form a 2 axis set of grid lines across a piecewise surface just as longitude and latitude grid the earth. In both cases the grid line are used to make precise addressing of the piecewise surface or our planet possible.

The surface's two axes are called'U' and 'V'. This surface based 2D coordinate system is called 'Parameter space'. The concept of parameter space is useful to separate'world space' where objects exist and animation takes place from the individual self-contained 2D 'world' of each piecewise parametric surface Within this 2D universe, like longitude and latitude on a map, each isoparametic flow line represents a constant numeric value along its corresponding, U or V axis. When cubic B-spline surfaces are mapped with digital Images, 2D (U,V) coordinates corresponding to locations on the image are mapped to 2D (U,V) coordinates on the surface.