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Entity-Entity Blends

The ability to specify completely arbitrary entity-entity blends is available only with ABL. Entity-entity blending can be used for blends other than face-face blends or for face-face blends between two faces that are not adjacent to one another (they may even be in disjoint shells). The standard blending of the Blending Component provides some support for entity-entity blends to resolve certain difficult blending configurations. However, it does not provide the functionality for defining arbitrary entity-entity blends.

These blends do not follow a sequence of blended edges. Instead, a pair of entities is given as a starting point and then the blend propagates itself as the ball rolls. The blend determines how to propagate itself by reading instructions from the entities which the spring curves intercept, rather than from a sequence of blended edges. entity-entity blends may switch arbitrarily between face-face, edge-face, face-edge, edge-edge and even vertex-face and vertex-edge blends. The blend implicitly follows a sequence of entity pairs in the sense that new entity pairs are blended together as the blend propagates. These pairs are not known in advance, but are computed as the blend progresses.

The default behavior when the contact point encounters another topological entity is described in the following table:

Table. Default Behaviors

Currently On Encountered Default Action
Face Smooth edge Roll over the edge onto the next face
Face Sharp edge Cap the blend with a side or end cap
Face Vertex with smooth face Roll onto the smooth face
Face Vertex without smooth face Cap the blend
Edge Face Roll onto the face
Edge Vertex with tangent edge Roll onto the tangent edge
Edge Vertex without tangent edge Cap the blend

Instruction attributes may be explicitly placed on entities to force certain behavior. Because an entity-entity blend has no sequence of blended edges, the only place from which the instructions can be read is the entities that the contact points of the rolling ball actually encounter. The instruction used is either roll on or cap, depending on which behavior is to be forced. For example, a blend on a face that runs into a sharp edge may be forced to roll on to that edge rather than to form a cap. The roll on instruction would actually be placed on the encountered edge.

Instructions may have positions associated with them to allow different behaviors at different points along the blended edge. In this case, the instruction nearest to the point of intercept applies.

Both constant and variable radius entity-entity blends may be defined. Variable radius blends require a single smooth defining curve and may have arbitrary cross sections.

These blends do not interact with edge sequence-following blends. Only a single entity-entity sequence may be blended at a time, as entity-entity blends do not occur in miters or vertex blends.

Blend Transitions

The transitions from one entity to another allowed with entity-entity blending depend on whether the blend is running along a face, along an edge, or around a vertex. In general, the blend comes off or away from the entity it is on, and it must compute what entity it will be blending next. The following sections describe the transition cases. In each case, the possible transitions are listed along with a description of how to obtain such a transition using instruction attributes. The default transitions do not need an instruction attribute.

Blends on Faces

Transitions onto new entities are only allowed when the spring curve of the blend leaves the face, having been inside it. Therefore, transitions are not supported at places at which the face boundary is merely grazed, either from the inside or outside. In some cases, an edge or vertex exists in a face only to support modeling functions, although it does not really need to be there in the pure boundary representation. This includes an edge in a periodic face that exists only to break the periodicity, such as the edge that splits the periodic face in a cylinder from one end cap to the other. In such a case, the edge is embedded in the face; that is, the same face exists on both sides of the edge. Transitions onto new entities are not supported at such edges or vertices.

Table. Blends on Faces

Run Into Transition How to Obtain
smooth edge roll onto next face place rollon on face (default)
smooth edge start capping place cap on face
sharp edge start capping place cap on edge (default)
sharp edge roll onto edge place rollon on edge
smooth vertex roll onto smooth face place rollon on face (default)
smooth vertex start capping place cap on face
sharp vertex start capping place cap on vertex (default)

A smooth vertex is a vertex at which there is a smoothly joined face into which the blend will definitely be able to travel. This excludes vertices at which there is a smooth face and the direction of travel of the blend lies along one of that faces boundaries at the vertex.

A sharp vertex is a vertex at which there is definitely no smoothly connected face, or if there is, the direction of travel of the blend definitely excludes it from going into this face. This excludes vertices at which there is a smooth face and the direction of travel of the blend lies along one of that faces boundaries at the vertex.

Any blend leaving the face at a vertex where there is a smoothly connected face, but the direction of travel of the blend lies along one of that faces boundaries at that vertex, is not currently supported.

Blends on Edges

The majority of blends running against edges are moving either forward or backward along the edge, without doubling back on that edge. Such a blend may leave the edge either by reaching one of its vertices or by coming away from the edge onto one of the adjoining faces. In this case, the blend always has an interior face, which is the face on the side that will be swallowed when the blend is fixed, and an exterior face, which is the face on the side that will remain.

Table. Blends on Edges

Run Into Transition How to Obtain
smooth vertex roll onto next edge place rollon on edge (default)
smooth vertex start capping place cap on next edge
sharp vertex start capping place cap on vertex (default)
sharp vertex roll onto vertex place rollon on vertex
interior face roll onto face no other choice
exterior face roll onto face place rollon on face (default)
exterior face stay on edge place cap on face

A smooth vertex is one at which there is another tangent edge incident that can be followed. A sharp vertex is one at which there is no other tangent edge to follow.

Transitions directly onto a face that occur at one of the ends of the edge are not supported.

A blend is disallowed from following the edge when it has the option of rolling onto the interior face. Not rolling onto the interior face would violate the rule that concave blends add material and that convex ones remove it.

Some blends do not move along the edge, but remain wholly stationary at a point on the edge. In such cases the faces cannot be distinguished as either interior or exterior and the blend is forced to roll onto both. The only such blend allowed is the osculating torus produced by blending a plane against a straight line normal to it.

Blends on Vertices

For blends on vertices, the only transitions allowed are onto the edges of the vertex. Transitions directly from a vertex onto a face are not supported. Transitions onto edges are forced, and they cannot be avoided.

Capping Restrictions

Entity-entity blending uses the same capping algorithms as standard blending, so all the standard blending capping restrictions apply to entity-entity blending. The entity-entity blending algorithms also impose some additional restrictions.

In order to process caps successfully and to determine the correct extent of an entity-entity blend sequence, it is necessary to distinguish side caps (which do not define the end of the sequence) from end caps (which do). The algorithms for doing this rely on the fact that the end points of any caps will lie on the boundaries of the underlying blended entities. If this does not hold for any particular situation, then the cap may be identified incorrectly. Even if these stipulations do hold, but the number of body edges involved between the start and end points of the cap is large (typically around half a dozen or more), then the cap may be identified incorrectly. Periodic blend surfaces pose a problem since end and side caps may be particularly difficult to distinguish. It may sometimes be apparent that a cap has been identified incorrectly if an error message mentions a particular kind of cap (end or side) when the other type is actually desired.

Entity-Entity Blending Limitations

Entity-entity blends are designed to be used in conjunction with ordinary edge sequence following blends, and to provide a number of topologically new blends (such as edge-face blends). Although they will blend faces against one another, they are not intended as a substitute to standard blending, as they have some limitations. These include:

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