3D objects can be represented in several ways including point clouds, surfaces, and solids. Surfaces include meshes composed of polygons, subdivision surfaces defined by rules, and parametric or implicit surfaces with mathematical equations. Solids include voxel grids, binary space partition trees, constructive solid geometry using boolean operations, and swept volumes along trajectories. Each representation has advantages for different applications like rendering, modeling, or 3D scanning.

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