OUPWPC ( bc, basis, coeffs, dim, order, nseg, ifail )
=====================================================
Output B-curve in piecewise form
Receives:
KI_tag_b_curve *bc --- B-curve
KI_cod_slba *basis --- representation method
Returns:
KI_tag_list_dbl *coeffs --- vectors defining the curve
KI_int_dimension *dim --- dimension of defining vectors
KI_int_order *order --- order of curve
KI_int_nitems *nseg --- number of segments in curve
KI_cod_error *ifail --- failure indicator
Specific errors:
KI_bad_order order must be four for Hermite basis
Description:
This function outputs a B-curve 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 (cubic only) ('basis' = SLBAHE)
o Taylor series ('basis' = SLBATA)
Coefficient data 'coeffs':
. Contains 'order'*'nseg' vectors of dimension 'dim'. If 'dim'=3, then the
vectors are 3-D vectors containing 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 coefficients are returned in order, segment by segment.
. The interpretation of the coefficients depends on the representation method
chosen; this is determined by the value of the argument 'basis'.
The expressions for each segment of the B-curve P(t) 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 curve segment is:
n n
-- / --
P(t) = > b (t) w V / > b (t) w
-- i i i / -- i i
i=0 i=0
where n = 'order'-1
V = Bezier vertex
i
w = weight for V
i i
b (t) = Bezier coefficient, defined by:
i
n! i n-i
b (t) = -------- * t * (1-t)
i i! (n-i)!
The Bezier vertices are returned V ,w ,...,V ,w for the rational form, or
0 0 n n
V ,...,V for the non-rational form.
0 n
. Polynomial coefficients SLBAPY:
The curve equation is given by a rational polynomial of order 'order':
n n
-- i / -- i
P(t) = > w A t / > w t
-- i i / -- i
i=0 i=0
where n = 'order'-1
A = polynomial coefficient
i
w = weight for A
i i
The polynomial coefficients are returned starting with the constant
term and ending with the term of highest degree.
. Hermite coefficients SLBAHE:
This method can only be used for cubics. The equation of the curve is:
f0(t) w0 P0 + g0(t) w1 P1 + f1(t) d0 D0 + g1(t) d1 D1
P(t) = -----------------------------------------------------
f0(t) w0 + g0(t) w1 + f1(t) d0 + g1(t) d1
2 3 2 3
where f0(t) = 1 - 31 + 2t g0(t) = 3t - 2t
2 3 2 3
f1(t) = t - 2t + t g1(t) = - t + t
P0, P1 = start and end points of segment
D0, D1 = derivatives at start and end
w0, w1 = weights at end points
d0, d1 = derivatives of weights at start and end
The coefficients are returned as P0,w0,P1,w1,D0,d0,D1,d1 for the rational
form, or P0,P1,D0,D1 for the non-rational form.
. Taylor series SLBATA:
This method stores the derivatives evaluated at the point start of each
segment, allowing the curve to be reconstructed as a Taylor series:
n
-- (i) (i) i
> w P t / i!
--
i=0
P(t) = -----------------------
n
-- (i) i
> w t / i!
--
i=0
where n = 'order'-1
(i)
P = i'th derivative at t = 0
(i)
w = i'th derivative of weight at t = 0
The point is returned first, followed by the 1st derivative and ending
with the derivative of order 'order'-1.
Dimension of coefficient vectors 'dim':
. For rational curves 'dim'=4.
. For non-rational curves 'dim'=3.
Order of each segment of the curve 'order':
. The order of the curve = degree + 1.
. The minimum order is 2.
. The Hermite basis ('basis' = SLBAHE) may only be chosen if the curve is
a cubic ('order' = 4).
