CMPUT 498 Solid Modeling

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CMPUT 498Solid Modeling

• Lecturer Sherif Ghali
• Department of Computing Science
• University of Alberta

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Representations

• Constructive Solid Geometry (CSG)
• interior as well as surfaces are defined
• special rendering algorithms
• solid is always valid
• Boundary representation (B-rep)
• (oriented) surfaces only are defined
• raster scan algorithms suffice for rendering
• solid validity should be verified
• A modeler could use one or both

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Sweeping/extrusion
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Modeling operations
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CSG primitives

• examples are parallelepiped, sphere, cylinder,
  cone, torus, triangular prism
• defined in a local coordinate system ? require a
  transformation to the world coordinate system

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Regularized boolean operations

• examples in 2D
• set-theoretic boolean operations vs. regularized
  boolean operations
• a point p is an interior point in a solid if a
  sufficiently small ball centered at p is wholly
  inside the solid
• set-theoretic boolean ops followed by closure of
  interior result in a regularized boolean op

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CSG Tree
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CSG treeparameters vs. structure

• The shape of the object can vary depending on the
  parameters and not only on the structure of the
  tree

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Nodes in a CSG tree

• Leaf nodes
• instantiated solid primitives
• Interior nodes
• transformation nodes
• regularized boolean operations nodes
• Material information stored outside the tree
  (possibly also at root node)

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Point/solid classification

• Determine whether a given point is inside, on the
  surface, or outside a CSG solid
• Method
• Propagate query down the tree
• determine answer at leaves
• propagate answer up the tree
• Practice in lower dimensions
• 1D primitive is an interval
• 2D primitive is a polygon

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Propagation

• Downward propagation
• node is a boolean operation
• pass on to child nodes
• node is a transformation node
• apply the inverse of the transformation
• node is a solid (leaf node)
• classify point w.r.t solid

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Propagation

• Upward propagation
• node is a boolean operation
• node is a transformation node
• pass to parent

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Neighborhood

• The neighborhood of a point p w.r.t. a solid S is
  the intersection with S of an open ball of
  infinitesimal radius centered at p

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Neighborhood at faces, edges, and vertices
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Refined upward propagation

• Maintain neighborhood information
• face plane equation
• edge wedges pairs of plane equations
• vertices sets of edges

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Edge neighborhood

• Merging edge neighborhood can produce
• two wedges
• a single wedge
• a face

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Vertex neighborhood

• Merging vertex neighborhood can produce
• a vertex
• an edge
• a face

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Face neighborhood

• Merging face neighborhoods produces an edge,
  unless the two faces are coplanar, resulting in a
  full ball, a face, or an empty ball

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CSG Rendering

• Ray tracing
• illumination model relying on recursion to
  determine reflection, refraction, and shadows,
  and using ray casting to compute visibility
• Ray casting
• determining the visible surface given a ray (a
  ray is a half line)
• Phong shading
• determining rendering colour using material
  properties, the location of the viewer, light
  sources, and the surface normal vector

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CSG Ray Casting
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CSG Ray Casting
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Redundancy
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Boundary representations

• B-reps are described by
• Topological information
• vertex, edge, and face adjacency
• Geometric information
• embedding in space location of vertices, edges,
  faces, and normal vectors

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Boundary representations

• We study
• closed, oriented manifolds embedded in 3-space
• A manifold surface
• each point is homeomorphic to a disc
• A manifold surface is oriented if
• any path on the manifold maintains the
  orientation of the normal
• An oriented manifold surface is closed if
• it partitions 3-space into points inside, on, and
  outside the surface
• A closed, oriented manifold is embedded in
  3-space if
• Geometric (and not just topological) information
  is known

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Boundary representations

• Non-manifold surfaces
• Non-oriented Manifolds
• Non-closed oriented manifolds

Moebius strip
Klein bottle
(unbounded) cone, (unbounded) cylinder, (unbounded
) paraboloid,

paraboloid
hyperbolic paraboloid
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Euler- Poincaré equation

• V E F 2 0
• no holes or interior voids in one solid
• V E F 2(1 G) 0
• any number of handles
• V E F (L F) 2(S G) 0
• any number of loops and shells

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Euler-Poincaré equation

• V E F 2 0
• V E F 2(1 G) 0
• V E F (L F) 2(S G) 0
• V of vertices
• E of edges
• F of faces
• G genus
• L of loops (a.k.a. ring)
• S of shells

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Leonhard Euler (1707 1783)Henri Poincaré (1854
1912)
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Genus

• Genus zero
• Genus one
• Genus two

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Genus two solids
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(extended) Euler- Poincaré example

• V E F (L F) 2(S G) 0
• V 24
• E 36
• F 16
• G 1
• L 18
• S 2
• 24 36 16 (2) 2(2 1) 0

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Euler operators

• mvfs
• mev

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Euler operators

• mef
• kemr

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Euler operators
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Euler operators
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Euler operatorskfmrh
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Euler operators
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Convex combinations and Simplicial complexes
for two points
for three points
for a set of linearly independent points
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Reference