Geometric Aspects of the Casting Process

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Geometric Aspects of the Casting Process

• Hee-Kap Ahn
• KIST Image Media Research Center

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Manufacturing ?
Extrusion
Stereolithography
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Casting ?

• The Casting Process of old days

Zeus throwing lightnings, Bronze, ca. 470 BC
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Casting
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Casting
Materials Iron, Aluminum, Polymer, Zinc,
etc. Feeding systems Gravity, Pressure (injection
)
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Casting
Injection
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CAD/CAM
Geometry

• Design phase
• No Prototype

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Geometric Aspects
Concept
Mass Production
Reuse the same Cast as many times as possible
retraction
ejection
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Geometric Aspects
Separability
Given a 3D object, is there a cast for it whose
parts can be removed after the liquid has
sol-idified?
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Geometric Aspects
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Castability
H. Ahn, M. de Berg, P. Bose, S. Cheng, D.
Halperin, J. Matousek, O. Schwarzkopf Separating
an object from its cast, ACM Symposium on
Computational Geometry, 1997 Computer-Aided
Design, 34(8), 2002
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Castability
H. Ahn, O. Cheong, R. van Oostrum Casting an
object with directional Uncertainty, accepted to
Computational Geometry Theory and Applications
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Castability
H. Ahn, S. Cheng, O. Cheong Casting an object
with skewed ejection direction, Intl Symposium
on Algorithms and Computation, 1998
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Castability
H. Ahn, S. Cheng, O. Cheong, H. Edelsbrunner, J.
Snoeyink The Reflex-Free Hull, Canadian
Conference on Computational Geometry, 2001
Cavities
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Opposite cast removal

• Given removal direction d
• Necessary Sufficient conditions under which
  such a partition exists
• monotone in direction d

Every line with direction d intersects the
interior of the object in at most one connected
component
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Opposite cast removal
Every line with direction d intersects the
interior of the object in at most one connected
component
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Opposite cast removal
Given removal direction d Lamma An object Q is
castable d if and only if it is monotone in d
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Opposite cast removal
Given removal direction d Lamma An object Q is
castable in d if and only if it is monotone in d
Sweeping
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Opposite cast removal

• Given removal direction d
• Necessary Sufficient conditions under which
  such a partition exists.
• Simple way to verify the conditions in O(n log
  n) time.

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Opposite cast removal
reflex silhouette edge
d
Convex silhouette
reflex silhouette
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Opposite cast removal
Non-crossing shadow curves
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Opposite cast removal

• Given removal direction d
• Necessary Sufficient conditions under which
  such a partition exists.
• Simple way to verify the conditions in O(n log
  n) time.
• Algorithm that computes a partitioning of the
  cast into two removable parts

Triangulation in the projection
Parting Surface
Lifting-up the triangles
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Opposite cast removal
Triangulation in the projection
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Opposite cast removal
Theorem Let P be a vertically monotone
polyhedron with n vertices. It is possible to
construct a cast for P in O(n log n) time.
Theorem An object Q is vertically monotone,
and therefore castable, if and only if its
reflex silhouette is empty and its shadow curves
are non-crossing.
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Opposite cast removal

• Find all castable directions
• Combinatorially distinct directions in which the
  object is castable.
• Exact algorithm in O(n4) time.
• Experiments a sampling approach

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Experiments
Torus
Bat
Convex Polyhedron
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Experiments
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Experiments
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