Bézier surfaces are a species of mathematical spline In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees used in computer graphics Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer, computer-aided design Computer-aided design is the use of computer technology for the design of objects, real or virtual. CAD often involves more than just shapes. As in the manual drafting of technical and engineering drawings, the output of CAD often must convey also symbolic information such as materials, processes, dimensions, and tolerances, according to, and finite element modelling. As with the Bézier curve In the mathematical field of numerical analysis, a Bézier curve is a parametric curve important in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case, a Bézier surface is defined by a set 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.

Contents

History

Bézier surfaces were first described in 1962 by the French France is a founding member state of the European Union and is the largest one by area. France has been a major power for several centuries with strong cultural, economic, military and political influence in Europe and in the world. During the 17th and 18th centuries, France colonised great parts of North America; during the 19th and early 20th engineer Pierre Bézier Pierre Étienne Bézier (IPA: [ˈpjɛʁ eˈtjɛn beˈzje]) was a French engineer and patentor (but not the inventor) of the Bézier curves and Bézier surfaces that are now used in most computer-aided design and computer graphics systems who used them to design automobile An automobile, motor car or car is a wheeled motor vehicle used for transporting passengers, which also carries its own engine or motor. Most definitions of the term specify that automobiles are designed to run primarily on roads, to have seating for one to eight people, to typically have four wheels, and to be constructed principally for the bodies. Bézier surfaces can be of any degree, but bicubic Bézier surfaces generally provide enough degrees of freedom Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. In mathematical terms, the degrees of freedom are the dimensions of a phase space for most applications.

Equation

A given Bézier surface of order (n, m) is defined by a set of (n + 1)(m + 1) control points ki,j. It maps the unit square into a smooth-continuous surface embedded within a space of the same dimensionality In mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two as { ki,j }. For example, if k 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 Bernstein polynomial, and

is the binomial coefficient In mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial power n.

Some properties of Bézier surfaces:

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 to the way Bézier curves are linked up to form a B-spline curve.

Bézier surfaces in computer graphics

Bézier patch meshes are superior to meshes of triangles as a representation of smooth surfaces, since they are much more compact, easier to manipulate, and have much better continuity In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is properties. In addition, other common parametric surfaces such as spheres A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The and cylinders A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since 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 In computer graphics, ray tracing is a technique for generating an image by tracing the path of light through pixels in an image plane and simulating the effects of its encounters with virtual objects. The technique is capable of producing a very high degree of visual realism, usually higher than that of typical scanline rendering methods, but at 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 pipelines. 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 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 quadrilateral. This means that it has four equal sides and four equal angles . A square with vertices ABCD would be denoted ABCD instead of as a triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ABC.

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 as a square.

See also

Bibliography

External links

Categories: Surfaces Categories: Differential geometry | Algebraic geometry | Manifolds | Topological graph theory | Geometric shapes | Multivariate interpolation

 

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