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'''Bézier surfaces''' are a species of [[spline (mathematics)|mathematical spline]] used in [[computer graphics]], [[computer-aided design]], and [[finite element]] modeling.  
As with the [[Bézier curve]], 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.


==History==
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Bézier surfaces were first described in 1962 by the [[France|French]] engineer [[Pierre Bézier]] who used them to design [[automobile]] bodies.  Bézier surfaces can be of any degree, but bicubic Bézier surfaces generally provide enough [[degrees of freedom (physics and chemistry)|degrees of freedom]] for most applications.
 
==Equation==
[[File:Bézier_surface_example.svg|300px|thumb|right|Sample Bézier surface; red - control points, blue - control grid, black - surface approximation]]
A given Bézier surface of degree (''n'',&nbsp;''m'') is defined by a set of (''n''&nbsp;+&nbsp;1)(''m''&nbsp;+&nbsp;1) [[Control point (mathematics)|control point]]s '''k'''<sub>''i,j''</sub>. It maps the [[unit square]] into a smooth-continuous surface embedded within a space of the same [[dimensionality]] as { '''k'''<sub>''i,j''</sub> }. 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: <ref> {{cite book |first=Gerald |last=Farin |title=Curves and Surfaces for CAGD |edition=5th |publisher=Academic Press |location= |isbn=1-55860-737-4 }} </ref>
 
:<math>\mathbf{p}(u, v) =
    \sum_{i=0}^n \sum_{j=0}^m
    B_i^n(u) \; B_j^m(v) \; \mathbf{k}_{i,j}
</math>
 
evaluated over the [[unit square]], where
 
:<math>
B_i^n(u) = {n \choose i} \; u^i (1-u)^{n-i}
</math>
 
is a [[Bernstein polynomial]], and
 
:<math> {n \choose i} = \frac{n!}{i! (n-i)!} </math>
 
is the [[binomial coefficient]].
 
Some properties of Bézier surfaces:
*  A Bézier surface will transform in the same way as its control points under all [[linear transformation]]s and [[translation]]s.
* All ''u'' = constant and ''v'' = constant lines in the (''u'',&nbsp;''v'') space, and, in particular, all four edges of the deformed (''u'',&nbsp;''v'') unit square are Bézier curves.
* A Bézier surface will lie completely within the [[convex hull]] of its control points, and therefore also completely within the [[bounding box]] of its control points in any given [[Cartesian coordinate system]].
* 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]] curve.
 
Simpler Bézier surfaces are formed from '''biquadratic patches''' (''m'' = ''n'' = 2), or [[Bézier triangle]]s.
 
==Bézier surfaces in computer graphics==
 
[[Image:Bicubic_Patches.png|right|thumb|250px|[[Ed Catmull]]'s "Gumbo" model, composed from patches]]
 
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 [[continuous function|continuity]] properties. In addition, other common parametric surfaces such as [[sphere]]s and [[cylinder (geometry)|cylinder]]s 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 (graphics)|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]]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 map]]s, [[bump map]]s and other [[pixel shader]] techniques.
 
<!-- The following two paragraphs are probably correct, can someone verify? -->
A Bézier patch of degree (''m'', ''n'') may be constructed out of two [[Bézier triangle]]s of degree m+n, or out of a single Bézier triangle of degree ''m''&nbsp;+&nbsp;''n'', with the input domain as a [[square (geometry)|square]] instead of as a [[triangle]].
 
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 ==
* [[NURBS]]
* [[Computational geometry]]
* [[Bicubic interpolation]]
* [[Bézier curve]]
* [[Bézier triangle]]
 
== Bibliography ==
{{Reflist}}
 
[[Category:Surfaces]]
[[Category:Multivariate interpolation]]
 
{{DEFAULTSORT:Bezier Surface}}

Revision as of 22:56, 21 February 2014

When you go to shop for your next new apparel, weigh up how much time, money, and energy you are squandering. Instead, try online clothes gift buying. It's a quick, easy, and fun way to check for your next apparel. There are few reasons should not try it.



One of things which actually turn anyone on are unexpected intimate touches from his lovely women. Next time more powerful and healthier to have his car keys, reach into his pocket all of them. If tend to be the club together, subtly back into him and also be there on a moment. If he opens the door for you rub you across his back once you walk according to. Sure these gestures are suggestive, and guys love them and interact with them.

On the other hand, by employing flares . isn't quite enough eliminate the stain completely, let the brisket sit for fifteen minutes longer before placing it into a washing machine cycle. Make use of a strong laundry detergent and add some Hydrogen Peroxide in the actual use of water too. Take it out, dry it and should not have any problems using a nice clean piece of cloth back again.

Some dogs, particularly small breeds, browse through the cold more acutely than most people realize. And small dogs tend to get cold faster than larger breeds. A sweater or coat on a cold day can make outdoor time more pleasant for the pup. If you're shivering on the cold, its likely that your dog feels it, too.

Full grain leather, sometimes referred to as top grain leather, is the popular and dear leathers. It's also the most basic form. It is created by removing about the hair for the epidermis, the outer layer, of the hide. It may be finished in aniline or semi-aniline shape. With aniline finishes, the leather is dipped best suited clear dye, which retains the natural effect while coloring it in a uniform trendy. With semi-aniline finished, the leather emerged a stain-resistant coating.

Check the weather forecast to guarantee that you can dress as necessary. If you are hiking in cool weather, always placed on layers. You could always take clothing if you get too very. Also, among essentially the most important hiking safety tips is to use an outer layer of clothing that is water wash and light fast. This will deter you from getting chilled should it start to rain. Possess a few extra articles of clothing with you, including an extra pair of socks.

Indoor Venue - If you've got decided to book a wedding event hall make sure around the globe well ventilated and has proper alternating current since having hundreds of guest within a hall, how so very big, helps it to be quite stuffy and imprisoning.

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