cad/graphics 2013, hong kong footpoint distance as a measure of distance computation between curves...
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CAD/Graphics 2013, Hong Kong
Footpoint distance as a measure of distance computation between curves and surfaces
Bharath Ram Sundar*, Abhijit Chunduru*, Rajat Tiwari*, Ashish Gupta^ and Ramanathan Muthuganapathy*.
*Department of Engineering Design Indian Institute of Technology Madras Chennai, India
^Renishaw, PuneIndia (Formerly worked in India Science lab, General Motors, India)
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Overview• Introduction • Statement • Curve-curve case• Distance query- Surface-Surface • Distance query- Curve-Surface
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Introduction• Most of CAD design requirements can be
modeled as geometric queries, such as distance to edge, planarity, gap, interference and parallelism.
• Typically done in discrete domain, thus there is need to solve in continuous domain.
• Should be scalable efficiently for a larger domain.
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Motivation• Commercial CAD
packages offer elementary computations, difficult to scale and generally discretely computed.
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Disadvantages with discrete computation
– Approximation- Queries made are approximate as faceted models are approximate representation of geometry.
– Computational complexity-Computational expense increases with densely faceted model.
– Result remapping- Mapping back to original geometry further adds to approximation.
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Problem Statement• Surface-Surface– Given two freeform surfaces, compute regions on each
surface, such that, for any point (P) in a region on one surface there lies a corresponding point (P’) on the other surface at a distance less than a threshold value.
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• Curve-Surface– Given a freeform curve and a set of freeform surfaces,
compute segments of the curve where the minimum distance between the curve and any of the surfaces is more than a threshold value.
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Existing Distance functions• Typical minimum distance computation is
performed.• Hausdorff distance.
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Contributions• To the best of our knowledge, no work seems to exist that
compute corresponding patches of curves/surfaces satisfying above or below a certain distance value, which is the focus of this work. Our major contributions are: – Footpoint distance measure has been proposed as a measure for
distance computation. – Established points of correspondence through footpoints was
explored in the case of curve-curve case and found to be an useful tool.
– Corresponding surface patches for the surface surface case are identified using footpoint distance. Alpha shape has been used to detect boundaries including island regions.
– A lower-envelope based approach has been proposed and demonstrated for the distance query between a curve and a set of surfaces. CAD/Graphics 2013 Hong Kong
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Curve-curve
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• It is desired to find the exact distance bounds for curves C1(t) and C2(r) that correspond.
• Let d1 be Minimum of the antipodal distances. Let d2 be the subsequent minima.
• Distance for the shown segments of is bound by the distances d1 and d2.
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Surface-Surface
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Let S1(u1, v1) and S2(u2, v2) be the surfaces and D1(u1, v1, u2, v2) be the distance function. The basic partial differential equations for extremum are
• Symbolic representation of bisector surface is possible for curve-curve case.
• Such a representation for the bisector of a pair of surfaces and subsequently for D1(u1, v1, u2, v2) has not been shown to be possible yet.
• Using antipodal points as the start looks infeasible and this motivated us to directly work on the footpoint distance, given a query distance.
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Surface-Surface Distance query• Footpoint distance and α-shape • Solving distance query• Boundary detection using α-shape • Boundary identification for islands
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Boundary detection using α-shape
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Patches in the form of point sets on both surfaces for Dq =0.8
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Boundary detection using α-shape
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Points Sets in parametric space Dq =0.8
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α-shape
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α-shapes for points in parametric space for Dq =0.8
α-shape for S2α-shape for S1
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Boundary identification for islands
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Island boundaries identified in parametric space.
Regions on the surfaces for the identified boundaries.
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Curve and set of free-form Surfaces• We initially find all the bisector points B between a curve C(t)
and a surface S = S(u,v), which can be identified by solving the following equations
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Min footpoint distance function• For a point on the curve, there are several footpoints
on the surface.• We take the minimum distance footpoint (MinF)..
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Curve-Surface Lower envelope• Given a distance query valu Dq, Lower
envelope technique is then computed about Dq .
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Conclusions• Algorithms for computing distance between
curves and surfaces that satisfy a distance input value has been proposed and implemented.
• Footpoint distance has been shown to be an appropriate distance measure for the intended problems.
• Implementation Results have been provided.
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