|
In
ACIS, a spline surface, also known as a parametric surface, is a mapping from a rectangle within a 2D real vector space (parameter space) into a 3D real vector space (object space), as shown in Figure 7-3. This mapping must be continuous and one-to-one, except possibly at the boundary of the rectangle in parameter space. It must be differentiable twice, and the normal direction must be continuous, though the derivatives need not be. The positive direction of the normal is in the sense of the cross product of the partial derivatives with respect to
u and v, in that order. At any point on the surface, the normal points toward the part of the neighborhood that is outside the surface.
|
|
|
|
|
Figure 7-3. Surface Mapping
|
|
|
A position in parameter space is represented in
ACIS by a structure called a
SPApar_pos, (parameter position) that contains two real coordinates, usually referred to as
u and
v. A displacement in parameter space is represented as a
SPApar_vec, (parameter vector). A unit direction is a
SPApar_dir (parameter direction). Equivalent constructs in object space are
SPAposition,
SPAvector, and
SPAunit_vector.
|
|
|
A pair of opposite sides of the rectangle may map into identical lines in object space (Figure 7-4). In this case, the surface is closed in the parameter direction normal to those boundaries. If the parameterization and derivatives also match at these boundaries, the surface is periodic in the parameter direction. The line in object space corresponding to the coincident boundaries is the seam of a periodic surface. A surface periodic in one direction but not in the other is a cylindrical surface. A surface that is periodic in both directions is a toroidal surface (Figure 7-5).
|
|
|
|
|
Figure 7-4. Surface Periodic in One Direction
|
|
|
|
|
Figure 7-5. Periodic in Both Directions
|
|
|
If a surface is periodic in one parameter direction, it is defined for all values of that parameter. A parameter value outside the domain rectangle is brought within the rectangle by adding a multiple of the rectangle's width in that parameter direction, and the surface evaluated at that value. If the surface is periodic in both parameters, it is defined for all parameter pairs (u,v), and both parameters are reduced to standard range.
|
|
|
One side of the rectangle may map into a single point in object space (Figure 7-6). This point is a parametric singularity of the surface. If the surface normal is not continuous at this point, it is a surface singularity.
|
|
|
|
|
Figure 7-6. One End Mapped to a Single Point
|
|
|
Classes
|
|
|
Within
ACIS, the
spline construction geometry class represents a sculptured surface that contains:
|
|
|
A pointer to an internal class description, called
spl_sur (spline surface).
|
|
|
A
reversed-sense bit that, if
TRUE, indicates that the sense of the
spl_sur is reversed.
|
|
|
spl_sur contains a
bs3_surface that is a pointer to a rational or nonrational nonuniform B-spline surface in the underlying sculptured surface package.
|
|
|
Permanent model geometry within an
ACIS model is represented by the
SPLINE class that contains a pointer to a
spline defining the sculptured surface geometry.
|
|
|
Restrictions
|
|
|
The evaluation of mass properties in
ACIS requires that the surface's first derivatives be continuous. Certain iterative processes use the derivatives for estimating the next test value, and may converge slowly, or not at all, if there are significant discontinuities in the derivatives, or if there are large and/or rapid variations in the magnitude of the derivatives. As a rule of thumb, avoid variations of more than an order of magnitude in the magnitude of first derivatives across the surface; however, such variations are tolerated if initial estimates are sufficiently accurate. Parametric singularities should be avoided, since they are likely to cause problems. Surface singularities that interfere with
ACIS topological entities must not exist.
|
|
|
ACIS correctly handles faces that span the seam(s) of a periodic surface, but not those that form a complete band around the (distorted) cylinder or torus.
|
|
|
The parameterization of a surface is unaffected by transformation in object space, and so (unlike earlier versions) is not required to be a distance measure. It is still advisable that the parameterization not be extreme, and particularly that the parameterization in the two directions be of similar scale.
|