نيربس

نـِربس (NURBS:Non-uniform rational basis spline ) هو نموذج رياضي شاع استخدامة في رسومات الكمبيوتر لتوليد وتمثيل منحنيات وسطوح والذي يقدم قدر كبير من الدقة والمرونة في التعامل معه سواء بالطريقة التحليلية او بانشاء أشكال حرة (free form).

سطوح النيربس يمكن ان تأخذ أشكال عضوية معقدة. تحريك نقاط التحكم يؤثرعلى منحنى السطح

A NURBS curve.
Animated version

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 الاستخدام

 إنشاء دوال الأساس

From bottom to top: Linear basis functions $N_{1,1}$ (blue) and $N_{2,1}$ (green), their weight functions $f$ and $g$ and the resulting quadratic basis function. The knots are 0, 1, 2 and 2.5

$N_{i,n}=f_{i,n}N_{i,n-1}+g_{i+1,n}N_{i+1,n-1}$

$f_{i}$ rises linearly from zero to one on the interval where $N_{i,n-1}$ is non-zero, while $g_{i+1}$ falls from one to zero on the interval where $N_{i+1,n-1}$ is non-zero. As mentioned before, $N_{i,1}$ is a triangular function, nonzero over two knot spans rising from zero to one on the first, and falling to zero on the second knot span. Higher order basis functions are non-zero over corresponding more knot spans and have correspondingly higher degree. If $u$ is the parameter, and $k_{i}$ is the $i$-th knot, we can write the functions $f$ and $g$ as

$f_{i,n}(u)={{u-k_{i}} \over {k_{i+n}-k_{i}}}$

و

$g_{i,n}(u)={{k_{i+n}-u} \over {k_{i+n}-k_{i}}}$

The functions $f$ and $g$ are positive when the corresponding lower order basis functions are non-zero. By induction on n it follows that the basis functions are non-negative for all values of $n$ and $u$. This makes the computation of the basis functions numerically stable.

 الشكل العام لمنحنى نربس

Using the definitions of the basis functions $N_{i,n}$ from the previous paragraph, a NURBS curve takes the following form:

$C(u)=\sum _{i=1}^{k}{\frac {N_{i,n}w_{i}}{\sum _{j=1}^{k}N_{j,n}w_{j}}}{\mathbf {P}}_{i}={\frac {\sum _{i=1}^{k}{N_{i,n}w_{i}{\mathbf {P}}_{i}}}{\sum _{i=1}^{k}{N_{i,n}w_{i}}}}$

In this, $k$ is the number of control points ${\mathbf {P}}_{i}$ and $w_{i}$ are the corresponding weights. The denominator is a normalizing factor that evaluates to one if all weights are one. This can be seen from the partition of unity property of the basis functions. It is customary to write this as

$C(u)=\sum _{i=1}^{k}R_{i,n}{\mathbf {P}}_{i}$

in which the functions

$R_{i,n}={N_{i,n}w_{i} \over \sum _{j=1}^{k}N_{j,n}w_{j}}$

are known as the rational basis functions.

 انظر أيضاً

• Spline
• de Boor's algorithm
• Subdivision surface
• Triangle mesh
• Point cloud
• Rational motion
• Isogeometric analysis

 الهامش

هذه المقالة عبارة عن بذرة تحتاج للنمو والتحسين؛ فساهم في إثرائها بالمشاركة في تحريرها.