Representation and modeling of 3D objects (basic concepts)

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Representation and modeling of 3D objects(basic
concepts)
Computer Graphics 2
Marek Zimányi

• FMFI UK
• Katedra pocítacovej grafiky a spracovania obrazu

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Contents

• Polygonal representation
• Boundary rep.
• Sweep
• Constructive Solid Geometry
• Implicit representation
• F-rep
• Spatial subdivision techniques
• Procedural modeling

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Polygonal representation

• Boundary representation (B-Rep)
• Topological description
• Specifies vertices, edges, faces
• Geometrical description
• Specifies equation of the surface
• Sweep

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Boundary representation
Polygonal representation

• Polygon and Polygonal Mesh
• Polyhedron
• Normals
• Eulers formula
• DCEL

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Boundary representation
Polygonal representation

• Describing of boundary points set
• faces,edges, vertices
• region, loop, shell, hole,handle
• vertices
• V1 (x1, y1, z1), V2 (x2, y2, z2),
• edges
• h1 (1, 2), h2 (2, 3),
• faces
• P1 (1, 2, 3), P2 (4, 2, 1),

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Boundary representation - Mesh
Polygonal representation

• Collection of faces (polygons)
• Advance shading process
• Parameters
• Vertices, faces, normals
• Vertex normals versus face normals
• ! This it not the same !

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Boundary representation - Mesh

• Definition
• Set of vertices,
• Normals
• Faces,
• With Indexes of vertices
• With Indexes of normals
• Be careful -gt left or right hand CS

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Boundary representation - Mesh

• Normal computing

• Martin Newell method

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Boundary representation - Mesh

• Properties
• Solidity enclose a positive and finite amount
  of space
• Connectedness exist unbroken path between any
  two vertices
• Simplicity solid with no holes
• Planarity every face is a planar polygon
• Convexity line between any two point lies
  inside the object

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Boundary rep. - Polyhedron
Polygonal representation

• Def.
• A polyhedron is a connected mesh of simple planar
  polygons that enclose a finite amount of space

• Prop.
• Every edge is shared by exactly two faces
• At least 3 edges meets at each vertex
• Faces do not interpenetrate

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Manifolds
Polygonal representation - B-rep

• Unreal objects gt Non-Manifolds

• 2-manifolds ? point from manifold has small
  neighborhood of points around, topologically the
  same as a disk in the plane
• Orientable lt-gt NonOrientable manifold Mobius
  strip, Klein bottle
• Many b-rep systems support only solid whose
  boundaries are 2-manifold

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Manifolds
Polygonal representation - B-rep

• Klein bottle
• http//www.math.ohio-state.edu/fiedorow/math655/K
  lein2.html
• http//www.kleinbottle.com/
• Mobius strip
• http//www.mobiusproductsandservices.com/tms.html
• http//www.cut-the-knot.org/do_you_know/moebius.sh
  tml
• Another -)
• http//pbskids.org/zoom/phenom/index.html

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Non-Manifolds
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The Winged-Edge Representation
Polygonal representation - B-rep

• Double-connected Edge List (DCEL)
• For each edge is given info
• Incident vertices
• Left and right adjacent face
• Preceding and succeeding edge in clockwise order
• Preceding and succeeding edge in counterclockwise
  order

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B-Rep
Polygonal representation
Polygonal representation - B-rep

• Important facts in the B-Rep
• Classification of edges
• sharp and auxiliary
• Normals in vertexes
• Boundary of face
• find all edges of a face
• Position of a point in the space
• Point is IN or OUT of object

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Sweep
Polygonal representation

• Sweeping an object along a trajectory
• Translation sweep
• Rotational sweep
• Conical sweep
• Sphere sweep
• General Cylinder
• Polygonal rep ?

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Translational Sweep
Polygonal representation

• Method
• Define the object with the contour (2D) and the
  path.

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Rotational Sweep
Polygonal representation

• Method
• Define the object by rotating of the contour (2D)
  with the arbitrary axis.

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Conical Sweep
Polygonal representation

• Method
• Define the object by the contour (2D) and a
  3D-point (top of the pyramide).

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Sphere Sweep
Polygonal representation

• Method
• Define the object by the sphere with the varying
  radius and a path.

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General Cylinder
Polygonal representation

• Method
• Define the object by the set of
  control-contours and by the path.

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Polygonal rep ?
Polygonal representation

• General sweeps difficult to model efficiency
• General sweeps do not always generate solids
• It is a polygonal rep?

Trajectory and object shape may swept object
intersect ? Volume calculations complicated

• Sweeping a 2D area in its own plane
• ?
• another 2D area

Yes and No.
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Polygonal rep ?
Polygonal representation

• It is a difficult to apply regularized Boolean
  operations(after BOOL op. it wouldnt be SWEEP
  obj)
• BUT sweeps are a natural and intuitive way to
  construct of the objects
• Many systems allow user construct sweepsBUT

store the objects in other representation
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Constructive Solid Geometry

• CSG
• Composing of Primitives
• Sphere, Cone, Cube, Cylinder, ...
• Operations I, U, -, ...
• The leafs of CSG-tree are primitives and the
  operations are in the rest of the tree nodes. The
  CSG-tree represents the solid. (Scene graph
  idea!)

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Example
Polygonal representation - CSG
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Notes
Polygonal representation - CSG

• Pros
• Low memory consumption
• Simple combining
• Exact representation of complex surfaces
• 12 Mantylas operators and his proof
• Cons
• Slow and difficult processing
• ?
• Not unique representation

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CSG processing
Polygonal representation - CSG

• Rendering Algorithms
• ray casting (tracing)
• extended depth buffer algorithm
• Evaluation/Conversion Algorithms
• boundary representation
• octree
• constructive cubes

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Implicit representation

• Jim Blinn 1982
• Implicit surfaces
• Blobby objects
• Soft objects
• Meta balls

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Spreadsheet Rendering
Implicit representation
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Animation Path in t1t2 Plane
Implicit representation
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Spatial subdivision techniques

• Octrees
• Iterative space subdivision
• Nodes have property V, F or M
• Subdividing P-nodes until the desired
  Approximation Quality
• The dividing planes do not depend on objects in
  the scene (differs from BSP tree!)
• Analogy in 2D Quad Tree
• BSP trees

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Octree
Spatial subdivision techniques

• Sort form of space
• Disadvantage sampling

V void F full M mixed   (FFFFF(FFFVFVFV)VV)
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Octree
Spatial subdivision techniques

• Boolean set operations and transformations
• Traverse both trees top-down in parallel
• Rotations (90?), scaling (2)
• General transformations severe
  problem(aliasing)
• Neighbor finding
• Not trivial
• Linear notation

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Binary Space-Partitioning Trees (BSP trees)
Spatial subdivision techniques

• octrees divide space uniformly - drawback
• BSP divide space recursively into pairs of
  subspaces
• each subspace has the same object complexity (
  of objects)
• Usual division step
• sort the objects according to actual coordinate
  (x,y,z)
• Choose median
• Place halfplane (halfline in 2D) between median
  and next object
• Shift halfline if too many objects intersects
• the halfplane

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Procedural modeling

• Alternative to slow user modeling
• Useful for complex, hard-to-manipulate, organic,
  fuzzy, scalable, large, objects
• Software designer prepares a procedure,
  implementing object-production algorithm
• Software user specifies only the input parameters
  for the procedure
• Stochastic principle is often applied
• (no 2 trees are exactly identical)