Bézier surfaces are a type of
mathematical spline used in
computer graphics
Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
,
computer-aided design
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 c ...
, and
finite element modeling.
As with
Bézier curve
A Bézier curve ( , ) is a parametric equation, parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approxima ...
s, a Bézier surface is defined by a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of control points. Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central control points; rather, it is "stretched" toward them as though each were an attractive force. They are visually intuitive and, for many applications, mathematically convenient.
History
Bézier surfaces were first described in 1962 by the
French engineer
Pierre Bézier who used them to design
automobile
A car, or an automobile, is a motor vehicle with wheels. Most definitions of cars state that they run primarily on roads, Car seat, seat one to eight people, have four wheels, and mainly transport private transport#Personal transport, peopl ...
bodies. Bézier surfaces can be of any degree, but bicubic Bézier surfaces generally provide enough
degrees of freedom
In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
for most applications.
Equation
A given Bézier surface of degree (''n'', ''m'') is defined by a set of (''n'' + 1)(''m'' + 1)
control points k
''i'',''j'' where ''i'' = 0, ..., ''n'' and ''j'' = 0, ..., ''m''. It maps the
unit square
In mathematics, a unit square is a square whose sides have length . Often, ''the'' unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinat ...
into a smooth-continuous surface embedded within the space containing the k
''i'',''j'' s – for example, if the k
''i'',''j'' s are all points in a four-dimensional space, then the surface will be within a four-dimensional space.
A two-dimensional Bézier surface can be defined as a
parametric surface where the position of a point p as a
function of the parametric coordinates ''u'', ''v'' is given by:
[ ]
:
evaluated over the unit square, where
:
is a basis
Bernstein polynomial
In the mathematics, mathematical field of numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of #Bernstein basis polynomials, Bernstein basis polynomials. The idea is named after mathematician Sergei Nata ...
, and
:
is a
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
.
Some properties of Bézier surfaces:
* A Bézier surface will transform in the same way as its control points under all
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s and
translation
Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...
s.
* All ''u'' = constant and ''v'' = constant lines in the (''u'', ''v'') space, and – in particular – all four edges of the deformed (''u'', ''v'') unit square are Bézier curves.
* A Bézier surface will lie completely within the
convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of its control points, and therefore also completely within the
bounding box of its control points in any given
Cartesian coordinate system
In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
.
* The points in the patch corresponding to the corners of the deformed unit square coincide with four of the control points.
* However, a Bézier surface does not generally pass through its other control points.
Generally, the most common use of Bézier surfaces is as nets of bicubic patches (where ''m'' = ''n'' = 3). The geometry of a single bicubic patch is thus completely defined by a set of 16 control points. These are typically linked up to form a
B-spline surface in a similar way as Bézier curves are linked up to form a
B-spline
In numerical analysis, a B-spline (short for basis spline) is a type of Spline (mathematics), spline function designed to have minimal Support (mathematics), support (overlap) for a given Degree of a polynomial, degree, smoothness, and set of bre ...
curve.
Simpler Bézier surfaces are formed from biquadratic patches (''m'' = ''n'' = 2), or
Bézier triangles.
Bézier surfaces in computer graphics
Bézier patch meshes are superior to triangle meshes as a representation of smooth surfaces. They require fewer points (and thus less memory) to represent curved surfaces, are easier to manipulate, and have much better
continuity properties. In addition, other common parametric surfaces such as
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s and
cylinders can be well approximated by relatively small numbers of cubic Bézier patches.
However, Bézier patch meshes are difficult to render directly. One problem with Bézier patches is that calculating their intersections with lines is difficult, making them awkward for pure
ray tracing or other direct geometric techniques which do not use subdivision or successive approximation techniques.
They are also difficult to combine directly with perspective projection algorithms.
For this reason, Bézier patch meshes are in general eventually decomposed into meshes of flat triangles by 3D
rendering pipeline
The computer graphics pipeline, also known as the rendering pipeline, or graphics pipeline, is a framework within computer graphics that outlines the necessary procedures for transforming a 3D computer graphics, three-dimensional (3D) scene into ...
s. In high-quality rendering, the subdivision is adjusted to be so fine that the individual triangle boundaries cannot be seen. To avoid a "blobby" look, fine detail is usually applied to Bézier surfaces at this stage using
texture maps,
bump maps and other
pixel shader
In computer graphics, a shader is a computer program that calculates the appropriate levels of light, darkness, and color during the rendering of a 3D scene—a process known as '' shading''. Shaders have evolved to perform a variety of s ...
techniques.
A Bézier patch of degree (''m'', ''n'') may be constructed out of two
Bézier triangles of degree ''m'' + ''n'', or out of a single Bézier triangle of degree ''m'' + ''n'', with the input domain as a
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
instead of a
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
.
A Bézier triangle of degree ''m'' may also be constructed out of a Bézier surface of degree (''m'', ''m''), with the control points so that one edge is squashed to a point, or with the input domain as a triangle instead of a square.
See also
*
NURBS
*
Computational geometry
*
Bicubic interpolation
*
Bézier curve
A Bézier curve ( , ) is a parametric equation, parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approxima ...
*
Bézier triangle
*
Biharmonic Bézier surface
Bibliography
Surfaces
Multivariate interpolation
{{DEFAULTSORT:Bezier Surface