OUPWPS ( bs, basis, coeffs, dim, uorder, vorder, ncol, nrow, sense, ifail )
===========================================================================
Output B-surface in piecewise form
Receives:
KI_tag_b_surface *bs --- B-surface
KI_cod_slba *basis --- representation method
Returns:
KI_tag_list_dbl *coeffs --- vectors defining the surface
KI_int_dimension *dim --- dimension of defining vectors
KI_int_order *uorder --- order of surface in u
KI_int_order *vorder --- order of surface in v
KI_int_nitems *ncol --- number of columns of patches
KI_int_nitems *nrow --- number of rows of patches
KI_cod_logical *sense --- surface sense
KI_cod_error *ifail --- failure indicator
Specific errors:
KI_bad_order order must be four for Hermite basis
Description:
This function outputs a B-surface in a piecewise form chosen by the user. The
following methods of representing the data are available:
o Bezier ('basis' = SLBABZ)
o Polynomial ('basis' = SLBAPY)
o Hermite (bicubic only) ('basis' = SLBAHE)
o Taylor series ('basis' = SLBATA)
Coefficient data 'coeffs':
. Contains ('uorder' * 'vorder' * 'ncol' * 'nrow') vectors of dimension 'dim'.
If 'dim'=3, then the vectors are 3-D vectors giving the x, y and z
components. If 'dim'=4, then each vector has a weight (w) associated with
it, and x, y, z and w components are returned for each vector.
. The data is returned patch by patch, row by row.
. The interpretation of the patch data depends on the representation method
chosen; this is determined by the value of the argument 'basis'.
The expressions for each patch of the B-surface P(u,v) in the various
representations are given below. For generality, the rational form is
given. The simplification to the non-rational form can be obtained by
setting the weights equal.
. Bezier vertices SLBABZ:
The equation of a rational Bezier surface patch is:
nu nv nu nv
-- -- / -- --
P(u,v) = > > b (u) b (v) w V / > > b (u) b (v) w
-- -- i j ij ij / -- -- i j ij
i=0 j=0 i=0 j=0
where nu = 'uorder'-1, nv = 'vorder'-1
V = Bezier vertex
ij
w = weight for V
ij ij
b (u), b (v) = Bezier coefficients.
i j
For the rational form the Bezier vertices and weights are returned:
V ,w ,V ,w ,...,V ,w ,V ,w ,...,V ,w ,...,V ,w ,...,V ,w .
00 00 10 10 m0 m0 01 01 m1 m1 0n 0n mn mn
For the non-rational form the w's are missed out.
. Polynomial coefficients SLBAPY:
The surface equation is given by a rational bipolynomial of orders
'uorder', 'vorder':
nu nv nu nv
-- -- i j / -- -- i j
P(u,v) = > > w A u v / > > w u v
-- -- ij ij / -- -- ij
i=0 j=0 i=0 j=0
where nu = 'uorder'-1, nv = 'vorder'-1
For the rational form the polynomial coefficients Aij are returned:
A ,w ,A ,w ,...,A ,w ,A ,w ,...,A ,w ,...,A ,w ,...,A ,w
00 00 10 10 m0 m0 01 01 m1 m1 0n 0n mn mn
starting with the constant term and ending with the term of highest degree.
For the non-rational form the w's are missed out.
. Hermite coefficients SLBAHE
This method can only be used for bicubics.
The Hermite equation for the patch in matrix form is:
2 3 T 2 3 T
( 1 u u u ) M A M ( 1 v v v )
P(u,v) = --------------------------------
2 3 T 2 3 T
( 1 u u u ) M W M ( 1 v v v )
where M = ( 1 0 0 0 )
( 0 0 1 0 )
( -3 3 -2 -1 )
( 2 -2 1 1 )
A = ( w00*P00 w01*P01 wv00*Pv00 wv01*Pv01 )
( w10*P10 w11*P11 wv10*Pv10 wv11*Pv11 )
( wu00*Pu00 wu01*Pu01 wuv00*Puv00 wuv01*Puv01 )
( wu10*Pu10 wu11*Pu11 wuv10*Puv10 wuv11*Puv11 )
W = ( w00 w01 wv00 wv01 )
( w10 w11 wv10 wv11 )
( wu00 wu01 wuv00 wuv01 )
( wu10 wu11 wuv10 wuv11 )
and the superscript T denotes the transpose.
In the matrices A and W, the coefficients P, Pu, Pv and Puv
are the points at the corners and their derivatives. The w's are the
corresponding weights and their derivatives. P00 denotes P(0,0), etc.
For the rational form the coefficients are returned:
P00, w00, P10, w10, P01, w01, P11, w11
Pu00, wu00, Pu10, wu10, Pu01, wu01, Pu11, wu11
Pv00, wv00, Pv10, wv10, Pv01, wv01, Pv11, wv11
Puv00, wuv00, Puv10, wuv10, Puv01, wuv01, Puv11, wuw11
For the non-rational form, the w's are missed out.
. Taylor series SLBATA:
This method stores the derivatives evaluated u=0, v=0 corner of each patch,
allowing the surface to be reconstructed as a Taylor series:
nu nv
-- -- (i)(j) (i)(j) i j /
> > w P u v / i! j!
-- --
i=0 j=0
P(u,v) = ------------------------------------------
nu nv
-- -- (i)(j) i j /
> > w u v / i! j!
-- --
i=0 j=0
where nu = 'uorder'-1, nv = 'vorder'-1
i+j
(i)(j) d P
P = ------ (0,0)
i j
du dv
i+j
(i)(j) d w
w = ------ (0,0)
i j
du dv
The point is returned first, followed by the U derivatives in order and
ending with the derivative of order 'uorder'-1 in U, 'vorder'-1 in V.
Dimension of coefficient vectors 'dim':
. For rational surfaces 'dim'=4.
. For non-rational surfaces 'dim'=3.
Order of each patch of the surface in u 'uorder', and in v, 'vorder':
. The order = degree + 1.
. The minimum order is 2.
. If the Hermite basis is chosen ('basis' = SLBAHE) then the surface has
to be bicubic ('uorder' = 'vorder' = 4).
Surface sense 'sense'
. If 'sense' is KI_true the surface normal is given by the cross product of
the first derivatives of the surface with respect the u and v parameters.
i.e.
normal = Pu X Pv
. If 'sense' is KI_false the surface normal is in opposite to that described
for 'sense' == KI_true.
