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    there are usually around one hundred product-independent parts
    in a mould set, and these parts are associated with each other
    with different kinds of constraints. It is time-consuming for
    the designer to orient and position the components in an
    assembly. Secondly, while mould designers, most of the time,
    think on the level of real-world objects, such as screws, plates,
    and pins, the CAD system uses a totally different level of
    geometrical objects. As a result, high-level object-oriented ideas
    have to be translated to low-level CAD entities such as lines,
    surfaces, or solids. Therefore, it is necessary to develop an
    automatic assembly modelling system for injection moulds to
    solve these two problems. In this paper, we address the following
    two key issues for automatic assembly modelling: representing
    a product-independent part and a mould assembly in
    a computer; and determining the position and orientation of a
    component part in an assembly.
    This paper gives a brief review of related research in
    assembly modelling, and presents an integrated representation
    for the injection mould assembly. A simplified geometric symbolic
    method is proposed to determine the position and orientation
    of a part in the mould assembly. An example of automatic
    assembly modelling of an injection mould is illustrated.
    2. Related Research
    Assembly modelling has been the subject of research in perse
    fields, such as, kinematics, AI, and geometric modelling. Libardi
    et al. [3] compiled a research review of assembly modelling.
    They reported that many researchers had used graph
    structures to model assembly topology. In this graph scheme,
    the components are represented by nodes, and transformation
    matrices are attached to arcs. However, the transformation
    matrices are not coupled together, which seriously affects the
    transformation procedure, i.e. if a subassembly is moved, all
    its constituent parts do not move correspondingly. Lee and
    Gossard [4] developed a system that supported a hierarchical
    assembly data structure containing more basic information
    about assemblies such as “mating feature” between the components.
    The transformation matrices are derived automatically
    from the associations of virtual links, but this hierarchical
    topology model represents only “part-of” relations effectively.
    Automatically inferring the configuration of components in
    an assembly means that designers can avoid specifying the
    transformation matrices directly. Moreover, the position of a
    component will change whenever the size and position of its
    reference component are modified. There exist three techniques
    to infer the position and orientation of a component in the
    assembly: iterative numerical technique, symbolic algebraic
    technique, and symbolic geometric technique. Lee and Gossard
    [5] proposed an iterative numerical technique to compute the
    location and orientation of each component from the spatial
    relationships. Their method consists of three steps: generation
    of the constraint equations, reducing the number of equations,
    and solving the equations. There are 16 equations for “against”
    condition, 18 equations for “fit” condition, 6 property equations
    for each matrix, and 2 additional equations for a rotational
    part. Usually the number of equations exceeds the number of
    variables, so a method must be devised to remove the redundant
    equations. The Newton–Raphson iteration algorithm is used to
    solve the equations. This technique has two disadvantages:
    first, the solution is heavily dependent on the initial solution;
    secondly, the iterative numerical technique cannot distinguish
    between different roots in the solution space. Therefore, it
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