Freeform surface, or
freeform surfacing, is used in
CAD and other
computer graphics software to describe the skin of a 3D geometric element. Freeform surfaces do not have rigid radial dimensions, unlike regular surfaces such as
planes,
cylinders and
conic surfaces. They are used to describe forms such as
turbine blades, car bodies and boat
hulls. Initially developed for the automotive and
aerospace industries, freeform surfacing is now widely used in all
engineering design disciplines from consumer goods products to ships. Most systems today use
nonuniform rational B-spline (NURBS) mathematics to describe the surface forms; however, there are other methods such as
Gordon surfaces or
Coons surfaces . The forms of freeform surfaces (and curves) are not stored or defined in
CAD software in terms of
polynomial equations, but by their poles,
degree, and number of patches (segments with spline curves). The degree of a surface determines its mathematical properties, and can be seen as representing the shape by a polynomial with variables to the power of the degree value. For example, a surface with a degree of 1 would be a flat
cross section surface. A surface with degree 2 would be curved in one direction, while a degree 3 surface could (but does not necessarily) change once from
concave to
convex curvature. Some CAD systems use the term
order instead of
degree. The order of a polynomial is one greater than the degree, and gives the number of
coefficients rather than the greatest
exponent. The poles (sometimes known as
control points) of a surface define its shape. The natural surface edges are defined by the positions of the first and last poles. (Note that a surface can have trimmed boundaries.) The intermediate poles act like magnets drawing the surface in their direction. The surface does not, however, go through these points. The second and third poles as well as defining shape, respectively determine the start and
tangent angles and the
curvature. In a single patch surface (
Bézier surface), there is one more pole than the degree values of the surface. Surface patches can be merged into a single NURBS surface; at these points are knot lines. The number of knots will determine the influence of the poles on either side and how smooth the transition is. The smoothness between patches, known as
continuity, is often referred to in terms of a
C value: • C0: just touching, could have a nick • C1: tangent, but could have sudden change in curvature • C2: the patches are curvature continuous to one another Two more important aspects are the U and V parameters. These are values on the surface ranging from 0 to 1, used in the mathematical definition of the surface and for defining paths on the surface: for example, a trimmed boundary edge. Note that they are not proportionally spaced along the surface. A curve of constant U or constant V is known as an isoperimetric curve, or U (V) line. In CAD systems, surfaces are often displayed with their poles of constant U or constant V values connected together by lines; these are known as
control polygons. == Modelling ==