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    CAD system.
    3.2.2 Relationships Between Parts
    In an assembly, the position and orientation of a part is usually
    associated with another part. Figure 3(b) shows a plate (PP1)
    and a screw (PP2). The relative placement of the screw is
    constrained by the counter-bore hole on the plate. The relationships
    between the screw and the plate are defined as follows:
     
    As we can see in Eqs. (8) and (9), it is essential to calculate
    the matrix Mp and Mr to determine the position and orientation
    of the screw with reference to the plate. Mp and Mr can be
    derived from the spatial constraints (SC). This derivation
    requires the task of inferring the configuration of a part in an
    assembly, which will be discussed in the next section.
    We have presented a representation of the injection mould
    assembly in a computer. At this stage, it might be worthwhile
    to summarise the benefits of this representation. Assemblies
    are represented as a collection of subassemblies that in turn
    may consist of subassemblies and/or component parts, and a
    component part can further be considered as the assembly of
    form features. Such hierarchical relationships imply an ordering
    on the assembly sequence and a parent–child link. The featurebased
    representation not only allows designers to work at a
    high-level of abstraction while designing inpidual parts, but
    also extends the feature paradigm to assembly modelling,
    because this representation allows a component to be changed
    parametrically with the other components consequently having
    their positions changed accordingly. The object-oriented representation
    can combine both the data structure and operation
    in an object. The encapsulated operational functions in an
    assembly object can help to automate the routine processes
    such as the pocketing and interference check.
         4. Inferring Part Configuration in the
         Assembly
    As we can see from Eqs (8) and (9), the positions and
    orientations of parts in an assembly are eventually represented
    by the transformation matrices. For the sake of convenience,
    the spatial relationships are usually specified by the high-level
    mating conditions such as “mate”, “align” and “parallel”. Thus,
    it is essential to derive automatically the explicit transformation
    matrices between parts from implicit constraint relationships.
    Three techniques to infer the configurations of parts in an
    assembly have been discussed in Section 2. Because the symbolic
    geometric approach can locate all solutions to constraint
    equations with polynomial time complexity, we use this
    approach to determine the positions and orientations of parts
    in an assembly. To implement this approach in assembly
    modelling software, a large amount of programming is required.
    Therefore, a simplified geometric approach is proposed to
    determine the positions and orientations of parts in an assembly.
    In the symbolic geometric approach, determining positions
    and orientations of parts is performed symbolically by generating
    a sequence of actions to satisfy each constraint incrementally.
    The information required to satisfy each constraint
    incrementally is stored in a table of “plan fragments”. Each
    plan fragment is a procedure that specifies a sequence of
    measurements and actions that move parts in such a way as
    to satisfy the corresponding constraint. The plan fragment also
    records the object’s new degrees of freedom (DOFs) and
    associated geometric invariants. Conceptually, Kramer’s plan
    fragment table is a 3D dispatch table. We use TDOF to
    represent translational degrees of freedom and RDOF to represent
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