Bézier surface

Bézier surfaces are a breed of algebraic spline acclimated in computer graphics, computer-aided design, and bound aspect modeling. As with the Bézier curve, a Bézier apparent is authentic by a set of ascendancy points. Similar to departure in abounding respects, a key aberration is that the apparent does not, in general, canyon through the axial ascendancy points; rather, it is "stretched" against them as admitting anniversary were an adorable force. They are visually intuitive, and for abounding applications, mathematically convenient.

History

Bézier surfaces were aboriginal declared in 1962 by the French architect Pierre Bézier who acclimated them to architecture auto bodies. Bézier surfaces can be of any degree, but bicubic Bézier surfaces about accommodate abundant degrees of abandon for a lot of applications.

Equation

A accustomed Bézier apparent of adjustment (n, m) is authentic by a set of (n + 1)(m + 1) ascendancy credibility ki,j. It maps the assemblage aboveboard into a smooth-continuous apparent anchored aural a amplitude of the aforementioned ambit as { ki,j }. For example, if k are all credibility in a four-dimensional space, again the apparent will be aural a four-dimensional space.

A two-dimensional Bézier apparent can be authentic as a parametric apparent area the position of a point p as a action of the parametric coordinates u, v is accustomed by:

evaluated over the assemblage square, where

is a Bernstein polynomial, and

is the binomial coefficient.

Some backdrop of Bézier surfaces:

A Bézier apparent will transform in the aforementioned way as its ascendancy credibility beneath all beeline transformations and translations.

All u = connected and v = connected ambit in the (u, v) space, and, in particular, all four edges of the askew (u, v) assemblage aboveboard are Bézier curves.

A Bézier apparent will lie absolutely aural the arched bark of its ascendancy points, and accordingly aswell absolutely aural the bonds box of its ascendancy credibility in any accustomed Cartesian alike system.

The credibility in the application agnate to the corners of the askew assemblage aboveboard accompany with four of the ascendancy points.

However, a Bézier apparent does not about canyon through its added ascendancy points.

Generally, the a lot of accepted use of Bézier surfaces is as nets of bicubic patches (where m = n = 3). The geometry of a individual bicubic application is appropriately absolutely authentic by a set of 16 ascendancy points. These are about affiliated up to anatomy a B-spline apparent in a agnate way as Bézier curves are affiliated up to anatomy a B-spline curve.

Simpler Bézier surfaces are formed from biquadratic patches (m = n = 2), or Bézier triangles.

Bézier surfaces in computer graphics

Bézier application meshes are above to meshes of triangles as a representation of bland surfaces, back they are abundant added compact, easier to manipulate, and accept abundant bigger chain properties. In addition, added accepted parametric surfaces such as spheres and cylinders can be able-bodied approximated by almost baby numbers of cubic Bézier patches.

However, Bézier application meshes are difficult to cede directly. One botheration with Bézier patches is that artful their intersections with curve is difficult, authoritative them awkward for authentic ray archetype or added absolute geometric techniques which do not use subdivision or alternating approximation techniques. They are aswell difficult to amalgamate anon with angle bang algorithms.

For this reason, Bézier application meshes are in accepted eventually addle into meshes of collapsed triangles by 3D apprehension pipelines. In high-quality rendering, the subdivision is adapted to be so accomplished that the alone triangle boundaries cannot be seen. To abstain a "blobby" look, accomplished detail is usually activated to Bézier surfaces at this date application arrangement maps, bang maps and added pixel shader techniques.

A Bézier application of amount (m, n) may be complete out of two Bézier triangles of amount m+n, or out of a individual Bézier triangle of amount m + n, with the ascribe area as a aboveboard instead of as a triangle.

A Bézier triangle of amount m may aswell be complete out of a Bézier apparent of amount (m, m), with the ascendancy credibility so that one bend is squashed to a point, or with the ascribe area as a triangle instead of as a square.

Bibliography

Farin, Gerald. Curves and Surfaces for CAGD (5th ed.). Academic Press. ISBN 1-55860-737-4.

Koehler; Dr. Ralph. 2D/3D Graphics and Splines with Source Code. ISBN 0-7596-1187-4.