1. Overview of Modeling
김성남

2. Modeling
Main issue How do we … • Represent 3D objects in a computer? • Construct such represents quickly and/or automatically with a computer? • Manipulate 3D objects with a computer? ※Different methods for different object representations.

3. Examples
How can these objects be represneted in a computer?

4. Representations of Geometry
3D Representations provide the foundations • Computer Graphics • Computer-Aided Geometric Design • Visualization • Robotics …
They are languages for describing geometry Semantics Syntax values data structures operations algorithms

5. 3D Object Representations
Raw data • Point cloud,Range image,Polygon soup …
Surfaces • Mesh,Subdivision,Parametric,Implicit …
Solids • Voxels,BSP tree,Octree,CSG,Sweep …

6. Point Cloud
Unstructed set of 3D point samples • range finder,computer vision, etc

7. Range Image
Set of 3D points mapping to pixels of depth image • acquired from range scanner
Range Image
Tesselation
Range Surface

8. Polygon Soup
Unstructed set of polygons • Created with interactive modeling systems.

9. Mesh
Connected set of polygons (usually triangles) • may not be closed

10. Subdivision Surface
Coarse mesh & subdivision rule • Define smooth surface as limit of sequence of refinements

11. Parametric Surface
Tensor product spline patchs • Careful constraints to maintain continuity

12. Implicit Surface Points satisfying F(x,y,z) = 0 y X + Y – R = 0 x R2 2 2
(0,0) x

13. Voxels
Uniform grid of volumetric samples • acquired from CAT,MRI, etc

14. BSP tree
Binary Space Partition tree recursively divide space into pairs of subspaces, each separated by a plane of arbitrary orientation and position. ⇒ determine visible surfaces
Left child ⇒ inside, Right child ⇒ outside

15. Octree
Spatical coherence to reduce storage requirement for 3D objects. ⇒ until homogeneous ⇒ 공간분할 1 2 3 1 2 3 1 1
Data Element
In quadtree node 3 2 3 2 1 2 3
2D space
< Quadtree >

16. Octree 7 4 5 1 2 3 4 5 6 7 1 Data Element In octree node 6 3 21 2 3 4 5 6 7 1
Data Element
In octree node 6 3 2
3D space

17. CSG
Constructive Solid Geometry • Hierachy of boolean set operations - Union - Difference - Intersection • applied to simple shapes

18. Sweep
Useful for constructing 3D objects that possess - translational - rotational - other symmetries v u u v
P(u)
P(u,v)
P(u,v)

19. Taxonomy of 3D Representations
Discrete
Continuous
Voxel
Combinational
Functional
Topological
Set Membership
Parametric
Implicit
Mesh
BSP tree
Bezier
Algebraic
Subdivision
Octree
B - Spline