struct PK_BCURVE_sf_s
{
int degree; --- The degree = order-1.
int n_vertices; --- The number of vertices.
int vertex_dim; --- The dimension of each vertex.
PK_LOGICAL_t is_rational; --- True if the curve is rational.
double *vertex; --- The vertices.
PK_BCURVE_form_t form; --- The curve shape.
int n_knots; --- The number of knot values.
int *knot_mult; --- The multiplicity of each knot.
double *knot; --- The distinct knot values.
PK_knot_type_t knot_type; --- Enum describing the knot set.
PK_LOGICAL_t is_periodic; --- True if the curve is periodic.
PK_LOGICAL_t is_closed; --- True if the curve is closed.
PK_self_intersect_t self_intersecting; --- Yes no or maybe.
};
typedef struct PK_BCURVE_sf_s PK_BCURVE_sf_t;
This data structure is the standard form for a b-curve. See the documentation
below for the relationship between this data structure and STEP.
Specific Errors:
PK_ERROR_bad_value unreadable value
PK_ERROR_bad_knots illegal knot multiplicity
or knot set not monotonic increasing
PK_ERROR_wrong_number_knots knots inconsistent with vertices
PK_ERROR_bad_dimension illegal vertex_dim
PK_ERROR_weight_le_0 one of the weights is zero or negative
Converting between PK_BCURVE_sf_t and STEP:
This form is easily convertible to the STEP standard, but there are a
number of important differences, designed to make its use simpler and more
efficient.
`vertex dim':
This field gives the number of doubles per vertex in the array
vertex. Allowable values are 2, 3, 4. A value of 2 implies a
parameter space (SP) curve. A value of 3 implies a 3-d polynomial curve or
a rational parameter space curve. A value of 4 implies a 3-d rational
curve. The STEP "cartesian_point" entity allows 1, 2 or 3.
is_rational:
This field is set to true if the curve is rational. In this case the
vertex_dim field must be one greater than the dimensionality of the curve,
and the weights must be included in the vertex array. The STEP standard
uses a distinct entity subtype "rational_b_spline_curve".
vertex:
If the curve is polynomial, dimension 2 or 3, then the values in this field
represent the vertices explicitly, in the order [u1, v1, u2, v2, ...], or
[x1, y1, z1, x2, y2, ...] respectively. If it is rational, dimension 4,
then the cartesian points are multiplied by the weights, so that the values
in this field represent [x1w1, y1w1, z1w1, w1, x2w2, y2w2, ...]. The STEP
standard uses a separate array of weights, and does not multiply the
cartesian points by the weights.
form:
See the documentation of PK_BCURVE_form_t.
This field corresponds to the field of the same name in the STEP
"b_spline_curve" entity, except for the PK_BCURVE_form_unset_c value
which does not exist in the STEP standard.
knot_mult and knot:
The values in knot_mult give the number of times each knot is to be
repeated. The minimum multiplicity allowed for any knot is 1.
The maximum multiplicity allowed other than for the first and last knot is
degree. The maximum allowed for the first or last knot is degree+1.
The values in knot must be distinct and form a monotone increasing set.
See the documentation below for an explanation of the required total
number of knots.
These arrays correspond exactly to the fields "knot_multiplicities" and
"knots" in the STEP entity sub-type "b_spline_curve_with_knots".
knot_type:
See the documentation for PK_knot_type_t.
The STEP standard does not contain a field corresponding to this. The
information described is conveyed by the use of entity subtypes, such as
"bezier_curve". The values PK_knot_bezier_ends_c and
PK_knot_non_uniform_c have no corresponding subtype. A curve of one of
these types would be translated to STEP as a "bspline_curve_with_knots".
is_periodic:
If this field is set to true, the parametrisation of the curve "wraps
around". It implies that the curve is closed, with G1 continuity at the
seam. This field does not exist in the STEP standard.
See the documentation below for additional requirements for the knot set
of a periodic b-curve.
is_closed:
This field being set to true does not necessarily imply that the
parametrisation is periodic. It corresponds to the field of the
same name in the STEP entity "b_spline_curve".
self_intersecting:
See the documentation of PK_self_intersect_t.
This corresponds to the field "self_intersect" in the STEP entity
"b_spline_curve", except that the STEP standard does not provide for an
unset value.
The number of knots:
The knot set referred to here is the "expanded knot set" obtained by repeating
each value in the array knot the number of times given by the corresponding
element in knot_mult.
A NURBS curve of degree n defined over k parameters (k-1 parameter intervals)
requires a further n parameters at either end of its knot set in order for the
necessary k+n-1 b-spline basis functions to be fully defined. These
additional parameters are commonly known as the "imaginary knots". Thus the
total number of knots p is related to the total number of basis functions and
hence the number m of control vertices by the expression p = m+n+1.
Periodic Curves:
A periodic curve must be so constructed as to preserve G1 continuity
(of tangent direction) at the join.
In addition, it is recommended that a periodic curve be constructed
with C(n-1) continuity at the join (where n is the degree). This
would give C2 continuity for a cubic. This most easily achieved by
making the "expanded knot set" "wrap around" in the sense that the
intervals between the first n+1 knots (n "imaginary" knots and one
"real" knot) should be the same as the corresponding intervals between
the last n+1 "real" knots, and the intervals between the last n+1
knots (one "real" knot and n "imaginary" knots) should be the same
as those between the first n+1 "real" knots.
The "period" of such a curve is the parameter interval separating the
first and last "real" knots.
if indices begin at 0:
period = t - t
k+n-1 n
for i = 0 to n:
t - period = t
k+n+i-1 n+i
t + period = t
i k+i-1
If the knot_type "PK_knot_smooth_seam_c" is set, the "expanded knot set"
is generated to achieve the conditions stated above. If is_periodic
is set to PK_LOGICAL_false, "PK_knot_smooth_seam_c" is equivalent to
"PK_knot_unset_c".
