1. Graphics Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Object Representation 고려대학교 컴퓨터 그래픽스 연구실

2. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Overview 3D Geometric Primitives Point Line Plane Polygon Sphere 3D Object Representations Raw data Surfaces Solids High-Level Structure

3. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Geometric Primitives Point Line Segment Polygon Polyhedron Curved Surface Solid Object Etc.

4. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Point Specifies a Location Represented by three coordinates Infinitely small typedef struct{ Coordinate x; Coordinate y; Coordinate z; } Point; typedef struct{ Coordinate x; Coordinate y; Coordinate z; } Point; (x, y, z)

5. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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; 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. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Line Line Segment with Both Endpoints at Infinity Parametric representation  P=P 1 +tV, ( ) typedef struct{ Point P 1 ; Vector V; } Line; typedef struct{ Point P 1 ; Vector V; } Line; P1P1 V

7. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Ray Line Segment with One Endpoints at Infinity Parametric representation  P=P 1 +tV, ( ) typedef struct{ Point P 1 ; Vector V; } Ray; typedef struct{ Point P 1 ; Vector V; } Ray; P1P1 V

8. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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; typedef struct{ Point P 1 ; Point P 2 ; } Segment; P1P1 P2P2

9. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Plane Specifies a Linear Combination of Three Points Implicit representation  P N + d = 0, or  ax + by + cz + d = 0 typedef struct{ Vector N; Distance d; } Plane; typedef struct{ Vector N; Distance d; } Plane; N=(a, b, c) P3P3 P2P2 P1P1 Origin d

10. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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; typedef struct{ Point *Points; int npoints; } Polygon; Points are in counter-clockwise order

11. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 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. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Object Representations Raw Data Point cloud Range image Polygon soup Surfaces Mesh, Subdivision, Parametric, Implicit Solids Voxel, BSP tree, CSG, Sweep

13. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Point Cloud Unstructured Set of 3D Point Samples Acquired from range finder, computer vision, etc

14. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Range Image Set of 3D Points Mapping to Pixels of Depth Image Acquired from range scanner Range ImageTessellationRange Surface

15. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Polygon Soup Unstructured Set of Polygons Created with interactive modeling systems

16. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Object Representations Raw Data Point cloud, Range image, Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxel, BSP tree, CSG, Sweep

17. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Mesh Connected Set of Polygons (Usually Triangles) May not be closed

18. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Subdivision Surfaces Coarse Mesh & Subdivision Rule Define smooth surface as limit of sequence of refinements

19. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Parametric Surfaces Tensor Product Spline Patches Careful constraints to maintain continuity

20. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Implicit Surface Points satisfying: F(x,y,z) = 0 Polygonal ModelImplicit Model

21. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr 3D Object Representations Raw Data Point cloud, Range image, Polygon soup Surfaces Mesh, Subdivision, Parametric, Implicit Solids Voxel BSP tree CSG Sweep

22. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Voxels Uniform Grid of Volumetric Samples Acquired from CAT, MRI, etc.

23. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr BSP Tree Binary Space Partition with Solid Cells Labeled Constructed from polygonal representations a b cd e f g Object a b cd 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. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr CSG Hierarchy of Boolean Set Operations (Union, Difference, Intersect) Applied to Simple Shapes

25. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Sweep Solid Swept by Curve Along Trajectory Constructing a Torus using Rotational Sweep

26. CGVR Graphics Lab @ Korea University cgvr.korea.ac.kr Summary Taxonomy of 3D Object Representations Voxel Discrete Continuous Combinational Functional Mesh Subdivision Mesh Subdivision BSP Tree Bezier B-Spline Bezier B-Spline Algebraic TopologicalSet Membership ParametricImplicit