PK_LAW_sf_t

struct PK_LAW_sf_s
    {
    int                   degree;           --- The degree = order-1.      (0)
    int                   n_vertices;       --- The number of vertices.    (0)
    int                   vertex_dim;       --- The dimension of each vertex.
                                            --- (1)
    PK_LOGICAL_t          is_rational;      --- If the law is rational.
                                            --- (PK_LOGICAL_false)
    double               *vertex;           --- The vertices.           (NULL)
    int                   n_knots;          --- The number of distinct knot
                                            --- values. (0)
    int                  *knot_mult;        --- The multiplicity of each knot.
                                            --- (NULL)
    double               *knot;             --- The distinct knot values.
                                            --- (NULL)
    PK_knot_type_t        knot_type;        --- Enum describing the knot set.
                                            --- (PK_knot_unset_c)
    PK_LOGICAL_t          is_periodic;      --- If the law is periodic.
                                            --- (PK_LOGICAL_false)
    PK_LOGICAL_t          is_closed;        --- If the law is closed
                                            --- (PK_LOGICAL_false)
    };
typedef struct PK_LAW_sf_s PK_LAW_sf_t;




This data structure is the standard form for a law function.


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

Used in:

PK_blend_law_t
PK_blend_shape_t
PK_BODY_sweep_law_t

`vertex dim':

    This field gives the number of doubles per vertex in the array
    'vertex'.


'is_rational':

    This field should be set to 'PK_LOGICAL_true' if the law is rational.
    In this case the vertex_dim field must be one greater than the dimension
    of the law, and the weights must be included in the 'vertex' array.


'vertex':

    If the law is polynomial, i.e. non-rational, of dimension n, then the
    values in this filed represent the vertices explicitly, in the order
    [f1,1; f2,1; ...; fn,1; f1,2; f2,2; ...].  If it is rational then the
    cartesian points are multiplied by the weights, so that the values
    in this field represent [w1f1,1; w1f2,1; ...; w1fn,1; w1; w2f1,1; w2f2,2;
    ...].


'knot_mult' and 'knot':

    'knot_mult' is an array of length 'n_knots' giving 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.

    'knot' is an array of length 'n_knots'. The values must be distinct and
    form a strictly increasing set.

    See the documentation below for an explanation of the required total
    number of knots.


'knot_type':

    See the documentation for 'PK_knot_type_t'.


'is_periodic':

    If this field is set to 'PK_LOGICAL_true', the parametrisation of the
    law "wraps around".  It implies that the law has equal value and
    first derivative continuity at the seam.

    See the documentation below for additional requirements for the knot set
    of a periodic law.


'is_closed':

    This field being set to 'PK_LOGICAL_true' does not necessarily imply
    that the parametrisation is periodic.  It indicates that the law values at
    the ends of the parameter range are equal.



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'.

The law can the considered to be defined in terms of a NURBS curve of degree n
defined over k parameters (k-1 parameter intervals).  This 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 laws:

For a periodic law the "expanded knot set" must "wrap around" in the sense
that the intervals between the first n+1 knots (n "imaginary" knots and one
"real" knot) must 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) must be the same as those between the first n+1
"real" knots.

The "period" of such a law 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

In addition a periodic law must be so constructed as to preserve G1
continuity (continuity of tangent direction) at the join.