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Geometric Restrictions |
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Contents
The following assumptions are imposed on Foreign Geometry at the current release of Parasolid.
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A.1
Parameter range and derivatives
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FG curves are assumed to be parametrized over a finite interval [t0, t1]
(the default interval is [0, 1]).
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FG surfaces are assumed to be parametrized over a finite range [u0, u1] [v0, v1]
(the default range is [0, 1] [0, 1] ).
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FG evaluators must return evaluations up to and including second derivatives
Thus for a curve evaluator C(t) where t in [t0, t1] we have:
C, dC/dt, d2C/dt2 : [t0, t1] ! R3
and for a surface evaluator S(u, v) where (u, v) in [u0, u1] [v0, v1] we have:
S, dS/du, dS/dv, d2S/duduv, d2S/du2, d2S/dv2 : [u0, u1] [v0, v1] ! R3
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A.2 Continuity
A curve is said to be G1 continuous if it has a continuous unit tangent vector, and to be C1 continuous if position and first derivative are both continuous.
A surface is said to be G1 continuous if it has a continuous unit normal vector, and to be C1 continuous if position and first derivatives are all continuous. C1 continuity implies G1 continuity and is the stronger condition.
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FG curves are required to be G1 continuous.
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FG surfaces are required to be G1 continuous. In addition, Parasolid requires that constant parameter lines on the surface are G1 continuous.
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A.3 Geometric and parametric properties
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FG curves and surfaces are required to not have periodic parametrization.
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FG curves and surfaces may not be closed.
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No degeneracies or singularities on FG curves,
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The only allowed singularity on an FG surface is the case when, at a corner (i.e. parameter positions (u0, v0), (u0, v1), (u1, v0), (u1, v1) ) first partial derivatives are parallel. Therefore, local to that corner the surface degenerates to a line and the surface normal is not well defined.
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Edges of an FG surface are allowed to degenerate subject to:
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the whole edge must degenerate to a point,
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adjacent edges on an FG surface may not be degenerate (so that there are at most two degenerate edges).
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