3D Object Representation

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3D Object Representation

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Overview

• 3D Geometric Primitives
• Point
• Line
• Plane
• Polygon
• Sphere
• 3D Object Representations
• Raw data
• Surfaces
• Solids
• High-Level Structure

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3D Geometric Primitives

• Point
• Line Segment
• Polygon
• Polyhedron
• Curved Surface
• Solid Object
• Etc.

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3D Point

• Specifies a Location
• Represented by three coordinates
• Infinitely small

typedef struct Coordinate x Coordinate
y Coordinate z Point
(x, y, z)
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3D Vector

• Specifies a Direction and a Magnitude
• Represented by three coordinates
• Magnitude vsqrt(dxdx dydy dzdz)
• Has no location
• Dot product of two 3D vector
• V1 V2 dx1dx2 dy1dy2 dz1dz2
• V1 V2 V1V2 cos( )

typedef struct Coordinate x Coordinate
y Coordinate z Vector
(dx1, dy1, dz1)
(dx2, dy2, dz2)
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3D Line

• Line Segment with Both Endpoints at Infinity
• Parametric representation
• PP1tV, ( )

typedef struct Point P1 Vector V
Line
V
P1
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3D Ray

• Line Segment with One Endpoints at Infinity
• Parametric representation
• PP1tV, ( )

typedef struct Point P1 Vector V Ray
V
P1
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3D Line Segment

• Specifies a Linear Combination of Two Points
• Parametric representation
• PP1 t(P2 - P1), ( )

P2
typedef struct Point P1 Point P2
Segment
P1
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3D Plane

• Specifies a Linear Combination of Three Points
• Implicit representation
• P N d 0, or
• ax by cz d 0

N(a, b, c)
P1
P3
typedef struct Vector N Distance d
Plane
P2
d
Origin
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3D Polygon

• Area Inside a Sequence of Coplanar Points
• Triangle
• Quadrilateral
• Convex
• Star-shaped
• Concave
• Self-Intersecting
• Hole

typedef struct Point Points int
npoints Polygon
Points are in counter-clockwise order
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3D Sphere

• All Points at Distance r from Point (cx, cy,
  cz)
• Implicit representation
• (x-cx)2 (y-cy)2 (z-cz) 2 r 2
• Parametric representation
• x r sin( ) cos( )
• y r sin( ) sin( )
• z r cos( )

(cx, cy, cz)
r
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3D Object Representations

• Raw Data
• Point cloud
• Range image
• Polygon soup
• Surfaces
• Mesh, Subdivision, Parametric, Implicit
• Solids
• Voxel, BSP tree, CSG, Sweep

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Point Cloud

• Unstructured Set of 3D Point Samples
• Acquired from range finder, computer vision, etc

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Range Image

• Set of 3D Points Mapping to Pixels of Depth Image
• Acquired from range scanner

Range Image
Tessellation
Range Surface
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Polygon Soup

• Unstructured Set of Polygons
• Created with interactive modeling systems

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3D Object Representations

• Raw Data
• Point cloud, Range image, Polygon soup
• Surfaces
• Mesh
• Subdivision
• Parametric
• Implicit
• Solids
• Voxel, BSP tree, CSG, Sweep

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Mesh

• Connected Set of Polygons (Usually Triangles)
• May not be closed

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Subdivision Surfaces

• Coarse Mesh Subdivision Rule
• Define smooth surface as limit of sequence of
  refinements

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Parametric Surfaces

• Tensor Product Spline Patches
• Careful constraints to maintain continuity

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Implicit Surface

• Points satisfying F(x,y,z) 0

Polygonal Model
Implicit Model
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3D Object Representations

• Raw Data
• Point cloud, Range image, Polygon soup
• Surfaces
• Mesh, Subdivision, Parametric, Implicit
• Solids
• Voxel
• BSP tree
• CSG
• Sweep

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Voxels

• Uniform Grid of Volumetric Samples
• Acquired from CAT, MRI, etc.

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BSP Tree

• Binary Space Partition with Solid Cells Labeled
• Constructed from polygonal representations

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Object
Binary Spatial Partition
BSP Tree
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CSG

• Hierarchy of Boolean Set Operations (Union,
  Difference, Intersect) Applied to Simple Shapes

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Sweep

• Solid Swept by Curve Along Trajectory

Constructing a Torus using Rotational Sweep
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Summary

• Taxonomy of 3D Object Representations

Discrete
Continuous
Voxel
Combinational
Functional
Topological
Set Membership
Parametric
Implicit
Mesh Subdivision
BSP Tree
Bezier B-Spline
Algebraic