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Freeform surface modelling is a technique for engineering freeform surfaces with a
CAD Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve comm ...
or CAID system. The technology has encompassed two main fields. Either creating aesthetic surfaces (
class A surfaces In automotive design, a class A surface is any of a set of freeform surface modelling, freeform surfaces of high efficiency and quality. Although, strictly, it is nothing more than saying the surfaces have curvature and tangency alignment – to ide ...
) that also perform a function; for example, car bodies and consumer product outer forms, or technical surfaces for components such as gas turbine blades and other fluid dynamic engineering components. CAD software packages use two basic methods for the creation of
surfaces A surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a ph ...
. The first begins with construction curves ( splines) from which the 3D surface is then swept (section along guide rail) or meshed (lofted) through. The second method is direct creation of the surface with manipulation of the surface poles/control points. From these initially created surfaces, other surfaces are constructed using either derived methods such as offset or angled extensions from surfaces; or via bridging and blending between groups of surfaces.


Surfaces

Freeform surface, or freeform surfacing, is used in
CAD Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve comm ...
and other
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great dea ...

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
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
s,
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...

cylinder
s and
conic In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the Conical surface, surface of a cone (geometry), cone with a plane (mathematics), plane. The three types of conic section are the hyperbola, the par ...

conic
surfaces. They are used to describe forms such as
turbine A turbine ( or ) (from the Greek , ''tyrbē'', or Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', ...

turbine
blades, car bodies and boat
hull Hull may refer to: Structures * Chassis, of an armored fighting vehicle * Fuselage, of an aircraft * Hull (botany), the outer covering of seeds * Hull (watercraft), the body or frame of a ship * Submarine hull Mathematics * Affine hull, in affin ...
s. Initially developed for the automotive and
aerospace Aerospace is a term used to collectively refer to the atmosphere and outer space. Aerospace activity is very diverse, with a multitude of commercial, industrial and military applications. Aerospace engineering consists of aeronautics and astronau ...

aerospace
industries, freeform surfacing is now widely used in all
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
design disciplines from consumer goods products to ships. Most systems today use
nonuniform rational B-spline Non-uniform rational basis spline (NURBS) is a mathematical model using basis splines (B-splines) that is commonly used in computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics ...
(NURBS) mathematics to describe the surface forms; however, there are other methods such as
Gordon surface Gordon may refer to: People * Gordon (given name), a masculine given name, including list of persons and fictional characters * Gordon (surname), the surname * Gordon (slave), escaped to a Union Army camp during the U.S. Civil War * Clan Gordon, a ...
s or
Coons surface In mathematics, a Coons patch, is a type of surface patch or manifold Parametrization (geometry), parametrization used in computer graphics to smoothly join other Surface (topology), surfaces together, and in computational mechanics applications, ...
s . The forms of freeform surfaces (and curves) are not stored or defined in
CAD Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve comm ...
software in terms of
polynomial equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s, but by their poles,
degree Degree may refer to: As a unit of measurement * Degree (angle) Image:Degree diagram.svg, One degree (shown in red) andeighty nine degrees (shown in blue) A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° ( ...
, 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 Cross section may refer to: * Cross section (geometry) **Multiview orthographic projection#Section, Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of det ...
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 Concave means curving in or hollowed inward, as opposed to convex. Concave may refer to: * Concave function In , a concave function is the of a . A concave function is also ously called concave downwards, concave down, convex upwards, convex c ...
to
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...
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
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s rather than the greatest
exponent Exponentiation is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
. 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
angles and the
curvature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

curvature
. In a single patch surface (
Bézier surface Bézier surfaces are a species of mathematical spline used in computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games ...
), 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

When defining a form, an important factor is the continuity between surfaces - how smoothly they connect to one another. One example of where surfacing excels is automotive body panels. Just blending two curved areas of the panel with different radii of curvature together, maintaining tangential continuity (meaning that the blended surface doesn't change direction suddenly, but smoothly) won't be enough. They need to have a continuous rate of curvature change between the two sections, or else their reflections will appear disconnected. The continuity is defined using the terms *G0 – position (touching) *G1 – tangent (angle) *G2 – curvature (radius) *G3 – acceleration (rate of change of curvature) To achieve a high quality
NURBS Non-uniform rational basis spline (NURBS) is a mathematical model using basis splines (B-splines) that is commonly used in computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics ...
or Bézier surface, degrees of 5 or greater are generally used.


History of terms

The term lofting originally came from the shipbuilding industry where loftsmen worked on "barn loft" type structures to create the keel and bulkhead forms out of wood. This was then passed on to the aircraft then automotive industries who also required streamline shapes. The term spline also has nautical origins coming from East Anglian dialect word for a thin long strip of wood (probably from old English and Germanic word splint).


Freeform surface modelling software


See also

*
Bézier surface Bézier surfaces are a species of mathematical spline used in computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games ...
*
Solid modelling Solid modeling (or modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional solid (mathematics), solids. Solid modeling is distinguished from related areas of geometric modeling and computer graphic ...
*
Computer representation of surfaces In technical applications of 3D computer graphics 3D computer graphics, sometimes called CGI, 3DCG or three-dimensional computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graph ...
*
Steven Anson Coons Steven Anson Coons (March 7, 1912 – August 1979) was an early pioneer in the field of computer graphical methods. He was a professor at the Massachusetts Institute of Technology Massachusetts Institute of Technology (MIT) is a private ...
*
NURBS Non-uniform rational basis spline (NURBS) is a mathematical model using basis splines (B-splines) that is commonly used in computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics ...
*
Parametric surface A parametric surface is a surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the s ...
* Freeform surface machining


References

{{reflist Computer-aided design 3D computer graphics Surfaces