CRPWPS ( dim, uorder, vorder, ncol, nrow, coeffs, basis, bs, ifail )
====================================================================
Create B-surface from piecewise data
Receives:
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_dbl_coefficients coeffs[dim * uorder * vorder * ncol * nrow]
--- vectors defining the surface
KI_cod_slba *basis --- representation method
Returns:
KI_tag_b_surface *bs --- B-surface
KI_cod_error *ifail --- failure indicator
Specific errors:
KI_discontinuous_surface adjacent patches must meet
KI_weight_le_0 weights must be greater than zero
KI_bad_order 'order' must be four for Hermite basis
Description:
This function creates a B-surface from piecewise data. The following
methods of representing the data are available:
o Bezier ('basis' = SLBABZ)
o Polynomial ('basis' = SLBAPY)
o Hermite (cubic only) ('basis' = SLBAHE)
o Taylor series ('basis' = SLBATA)
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 of the surface = degree + 1.
. The minimum order is 2.
. If the Hermite basis is used ('basis' = SLBAHE) then the surface has
to be bicubic ('uorder' = 'vorder' = 4).
Number of columns 'ncol':
. There must be at least one column.
Number of rows 'nrow':
. There must be at least one row.
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 supplied for each vector. The weights
supplied must be greater than zero.
. The data is supplied 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'.
. Adjacent patches (in the u and v directions) must meet all along the
corresponding boundary; i.e. the surface must be continuous.
Representation method '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 both the weights and the denominator equal to 1.0.
. 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 supplied:
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 supplied:
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 supplied:
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 supplied first, followed by the U derivatives in order and
ending with the derivative of order 'uorder'-1 in U, 'vorder'-1 in V.
