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07object3d

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  • 1. 3D Object Representation 고려대학교 컴퓨터 그래픽스 연구실
  • 2. Overview
    • 3D Geometric Primitives
      • Point
      • Line
      • Plane
      • Polygon
      • Sphere
    • 3D Object Representations
      • Raw data
      • Surfaces
      • Solids
      • High-Level Structure
  • 3. 3D Geometric Primitives
    • Point
    • Line Segment
    • Polygon
    • Polyhedron
    • Curved Surface
    • Solid Object
    • Etc.
  • 4. 3D Point
    • Specifies a Location
      • Represented by three coordinates
      • Infinitely small
    typedef struct{ Coordinate x; Coordinate y; Coordinate z; } Point; (x, y, z)
  • 5. 3D Vector
    • Specifies a Direction and a Magnitude
      • Represented by three coordinates
      • Magnitude ||v||=sqrt(d x d x + d y d y +d z d z )
      • Has no location
    • Dot product of two 3D vector
      • V 1 V 2 = d x1 d x2 + d y1 d y2 + d z1 d z2
      • V 1 V 2 = ||V 1 ||||V 2 || cos( )
    typedef struct{ Coordinate x; Coordinate y; Coordinate z; } Vector; (d x 1 , d y 1 , d z 1 ) (d x 2 , d y 2 , d z 2 )
  • 6. 3D Line
    • Line Segment with Both Endpoints at Infinity
      • Parametric representation
        • P=P 1 + t V, ( )
    typedef struct{ Point P 1 ; Vector V; } Line; P 1 V
  • 7. 3D Ray
    • Line Segment with One Endpoints at Infinity
      • Parametric representation
        • P=P 1 + t V, ( )
    typedef struct{ Point P 1 ; Vector V; } Ray; P 1 V
  • 8. 3D Line Segment
    • Specifies a Linear Combination of Two Points
      • Parametric representation
        • P=P 1 + t (P 2 - P 1 ), ( )
    typedef struct{ Point P 1 ; Point P 2 ; } Segment; P 1 P 2
  • 9. 3D Plane
    • Specifies a Linear Combination of Three Points
      • Implicit representation
        • P N + d = 0, or
        • a x + b y + c z + d = 0
    typedef struct{ Vector N; Distance d; } Plane; N=(a, b, c) P 3 P 2 P 1 Origin d
  • 10. 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
  • 11. 3D Sphere
    • All Points at Distance “r” from Point (c x , c y , c z )
      • Implicit representation
        • (x-c x ) 2 + (y-c y ) 2 + (z-c z ) 2 = r 2
      • Parametric representation
        • x= r sin( ) cos( )
        • y= r sin( ) sin( )
        • z= r cos( )
    r (c x , c y , c z )
  • 12. 3D Object Representations
    • Raw Data
      • Point cloud
      • Range image
      • Polygon soup
    • Surfaces
      • Mesh, Subdivision, Parametric, Implicit
    • Solids
      • Voxel, BSP tree, CSG, Sweep
  • 13. Point Cloud
    • Unstructured Set of 3D Point Samples
      • Acquired from range finder, computer vision, etc
  • 14. Range Image
    • Set of 3D Points Mapping to Pixels of Depth Image
      • Acquired from range scanner
    Range Image Tessellation Range Surface
  • 15. Polygon Soup
    • Unstructured Set of Polygons
      • Created with interactive modeling systems
  • 16. 3D Object Representations
    • Raw Data
      • Point cloud, Range image, Polygon soup
    • Surfaces
      • Mesh
      • Subdivision
      • Parametric
      • Implicit
    • Solids
      • Voxel, BSP tree, CSG, Sweep
  • 17. Mesh
    • Connected Set of Polygons (Usually Triangles)
      • May not be closed
  • 18. Subdivision Surfaces
    • Coarse Mesh & Subdivision Rule
      • Define smooth surface as limit of sequence of refinements
  • 19. Parametric Surfaces
    • Tensor Product Spline Patches
      • Careful constraints to maintain continuity
  • 20. Implicit Surface
    • Points satisfying: F(x,y,z) = 0
    Polygonal Model Implicit Model
  • 21. 3D Object Representations
    • Raw Data
      • Point cloud, Range image, Polygon soup
    • Surfaces
      • Mesh, Subdivision, Parametric, Implicit
    • Solids
      • Voxel
      • BSP tree
      • CSG
      • Sweep
  • 22. Voxels
    • Uniform Grid of Volumetric Samples
      • Acquired from CAT, MRI, etc.
  • 23. BSP Tree
    • Binary Space Partition with Solid Cells Labeled
      • Constructed from polygonal representations
    a b c d e f g Object a b c d e f g Binary Spatial Partition 1 2 3 4 5 6 7 1 2 a 3 b c 4 d 5 6 e 7 f BSP Tree
  • 24. CSG
    • Hierarchy of Boolean Set Operations (Union, Difference, Intersect) Applied to Simple Shapes
  • 25. Sweep
    • Solid Swept by Curve Along Trajectory
    Constructing a Torus using Rotational Sweep
  • 26. Summary
    • Taxonomy of 3D Object Representations
    Voxel Discrete Continuous Combinational Functional Mesh Subdivision BSP Tree Bezier B-Spline Algebraic Topological Set Membership Parametric Implicit