Solid Constructive Geometry

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Solid ConstructiveGeometry
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Solid geometry

• Inherently 3D, geometric elements describe sets
  of spaces enclosed by 2D boundaries
• For example, a solid sphere is the simplest solid
  element. Other simple primitives include the
  cube, cylinder, cone, and torus

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Solid geometry

• Other objects are defined by combinations of
  primitives.
• An entire math has been explored related to the
  combination of solid primitives called
  constructive solid geometry (CSG)
• With CSG, complex shapes may be generated from
  operations formed on primitives

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Solid geometry

• Boolean operators are defined tools for combining
  solid geometry for CSG
• These perform group operations on the points
  included in the solid primitives
• They are
• Union
• Subtraction
• Intersection

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Solid geometry

• Boolean operator Union combines two elements
  into a single one

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Solid geometry

• Boolean operator Union combines two elements
  into a single one

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Solid geometry

• Boolean operator Subtract take the difference
  between two elements

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Solid geometry

• Boolean operator Subtract take the difference
  between two elements

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Solid geometry

• Boolean operator Subtract take the difference
  between two elements

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Solid geometry

• Boolean operator Subtract

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Solid geometry

• Boolean operator Intersection finds the common
  points in the given primitives

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Solid geometry
CSG Tree Graph for hierarchy of
Boolean operations Often used for CAD and Mech
Eng
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Other models?
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Boundary Representations (B-Rep)

• Solids as a set of vertices, edges and faces with
  topological relations among them
• Boolean operations implemented in the
  representation framework

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Boolean Operations in B-Rep

• Planar (Polyhedral) solids
• Calculating plane-plane intersection only
• Split generates single line cuts in B-Rep
• Free-form solids
• Intersection between free-form surfaces
• A high degree algebraic space curve

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BOOLE A Boundary Evaluation System for Boolean
Combinations of Sculptured Solids

• 1999 S. Krishnan, et al.

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BOOLE Algorithm - Stages 1 2
Stage1 Convert solid into B-rep using
patches Stage2 Couple Patches of solids to be
tested
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BOOLE Algorithm - Stages 3 4
Stage3 Intersect patch pairs Stage4
Merge Intersection curves
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BOOLE Algorithm - Stage 5
Stage 5 Combine boundary components
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BOOLE Algorithm - Stage 6 7
Stage 6 Classify components
Stage 7 Result
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B-rep vs. CSG

• Both have different inherent strengths and
  weaknesses.
• CSG object always valid
• its surface is closed and orientable and
  encloses a volume, provided the primitives are
  valid.
• A B-rep object is easily rendered on a graphic
  display system.

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CSG Subdivision Surfaces
A
B
A ? B
A ? B
A - B
B - A
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CSG Subdivision Surfaces
Surface evolution with loop subdivision
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Problematic Intersections 
General problem
Subdivision
Subdivision Mesh
Control Mesh
Subdivision
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Intersecting Subdivision Surfaces
Intersection is needed to deduce boolean
operations
Sphere ? Cube
Sphere ? Cube
Sphere - Cube
Cube - Sphere
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Approximate Boolean Operationson Free-form
Solids

• Henning Biermann, et al (Siggraph 2001)

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Approximate Boolean Operations

• Generate a refined control mesh for intersecting
  surfaces (approximating the result)
• Determine the parameterization of the new surface
  with respect to the original surfaces
• Optimize quality of new control mesh

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Approximate Operations Step 1

• Compute an approximate intersection curve,
  finding its projection in each of the two
  parametric domains of the original surfaces.

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Approximate Operations Step 2

• Construct the connectivity of the control mesh
  for the result

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Approximate Operations Step 3

• Optimize the parameterization of the result over
  the original domains

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Approximate Operations Step 4

• Determine geometric positions for the control
  points of the result

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Approximate Intersection

• Working from control mesh
• Triangle edge intersection
• using barycentric coordinate
• Speed up with bounding boxes
• Avoid problems with perturbation of points

?
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Surfaces splitting

• Split along the intersection curve
• Label each part of the object (inside/outside)

A
C
A
C ? A
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Example
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Intersection curve example
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Split and label operations
Interior faces
Exterior faces
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Reconstruction
Depending on boolean operation Merge operation
along the intersection curve
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Results
A - B
B - A
Union
Intersection
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Multiresolution Subdivision
Multiresolution Subdivision Detail Vectors
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Multiresolution Subdivision
Ideally, prefer one point / edge
subdivision
subdivision
Actual Intersection Curve
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Mesh updating
Multiresolution Subdivision
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Multiresolution Subdivision
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Surface pasting
Beirmann et al (Siggraph 02)
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Surface pasting
Method Overview
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