​NURBS Shape Function

In order to evaluate NURBS shape functions, the weighting function is defined:
In most engineering applications, weights have positive values. Unless otherwise stated, they will be considered positive for the scope of this thesis. W(ξ) is in fact the Z-coordinate of the projective B-SPLine curve. Projective transformation is applied by dividing the other two coordinates of the B-SPLine curve with the Z-coordinate. NURBS shape functions are calculated from
NURBS shape functions in multiple directions can be obtained as tensor products of one-directional basis functions:
Shape functions for two directions:
Similarly, we expand the equations in order to obtain tensor product 3D shape functions:
The weighting function is now defined as:

NURBS Shape Function Derivatives

Simple application of the quotient rule yields the derivatives of NURBS shape functions for one and multiple directions.
For bidirectional shape functions:
Derivatives of 3D shape functions per direction are evaluated as shown