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Representations of Solids

Martin Madaras

FMFI UK

madaras@fmph.uniba.sk

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Motivation

Data structures for model representation What is the most suitable representation to choose ?

Memory efficiency

Algorithms

Modification Rendering

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Overview

Primitive instancing Boundary representation (B-rep)

Constructive Solid Geometry (CSG )

Spatial-partitioning representation

Sweep Representation

Function representation (F-rep)

Point representation

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Primitive instancing

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B-rep

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B-rep (example)

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B-rep (2-manifold)

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B-rep (Euler’s formula for polyhedra)

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B-rep (generalized Euler’s formula)

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B-rep (Triangle mesh)

Break polygons into triangles Strip, Fan : n-2 triangles

List : n/3 triangles

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B-rep (winged edge)

Winged edge vertices of this edge

its left and right faces

predecessor and successor of this edge when traversing its left

face predecessor and successor of this edge when traversing its

right face

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B-rep (quad edge)

Quad edge

dual graph

single edge, in a closed polygonal mesh topology, sits between exactlytwo faces and exactly two vertices

edge structure

2 faces 2 vertices

4 edges

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B-rep (DCEL)

Double Connected Edge List Half edge

Starting vertex

Pointer to left face

Pointer to opposite half edge (Twin)

Pointer to half edge starting at the ending vertex, traveling

counterclockwise around the boundary

Limitation

Only orientable 2-manifols

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Beziér patches

Complicated theory, low data complexity Easy to modify

Intuitive control for GUI

Difficult to render directly

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CSG

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CSG (rendering)

Raycasting

Union :

Intersection: Difference (Obj2  – Obj1) :

)()()( 2121   LO LO LOO  

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CSG (example)

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Spatial partitioning

Solid is decomposed into a collection of nonintersecting solids thatare more primitive than the original solid

Primitives may vary in type, size, position, parameterization andorientation

Cell decomposition

Spatial-occupancy enumeration

Octrees Binary space-partitioning trees

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Spatial partitioning (Cell decomposition)

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Spatial partitioning

Spatial-occupancy enumeration

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Spatial partitioning (Quadtree)

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Spatial partitioning (Octree)

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Spatial partitioning (BSP)

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Sweep representation

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F-rep

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F-rep (Blobby objects)

Implicit form only Natural shape blending

Used to represent molecular shapes, liquid and

melting objects, human body

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F-rep (polygon to function)

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F-rep (P2F - 2D polygon)

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F-rep (P2F - example)

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F-rep (metamorphosis 1)

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F-rep (metamorphosis 2)

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F-rep (collision detection)

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Point representation

Oriented points (Surfels)

Point based rendering techniques

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Conclusion

Representations

Volumetric

describes solid (CSG, Octree, BSP, F-rep)

Boundary

describes surfaces of the solid (polygonal meshes, splines, F-rep)

Oriented points (Surfels)

Continues

Splines, F-rep

Discrete

polygonal meshes, Octree

(complexity of data and algorithms ???)