For simplicity, consider first a D-NURBS space curve. The D-NURBS curve is defined as in (1), but it is also a function of the spatial parameter u and time t:
The control points 
and weights 
, which are now
functions of time, comprise the generalized coordinates of D-NURBS. To
simplify notation, we concatenate the generalized
coordinates into the following vectors:
where 
denotes transposition. Note that we can express the
curve 
as 
in order to emphasize its
dependence on the vector of generalized coordinates 
whose
components are functions of time. The velocity of the kinematic
spline is
where an overstruck dot denotes a time derivative and 
is the Jacobian matrix. Because 
is a 3-component
vector-valued function and is an 4(n+1) dimensional vector,
is a 
matrix which is the concatenation of
the vectors 
and 
, 
. Let us investigate the
contents of . For , let 
be a
diagonal matrix whose diagonal entries are the rational
basis functions
and let the 3-vector
We collect the 
into 
and the 
into 
as follows
The Jacobian matrix may then be written as
Using the foregoing notation, we can express
Appendix A shows that
so that we can express the D-NURBS as the product of the Jacobian matrix and the generalized coordinate vector:
Another interesting relationship is 
, and
it will enable us to simplify the discretized version of the D-NURBS
differential equations and arrive at an efficient numerical
implementation.
