3D Object Representation cgvr.korea.ac.kr

Overview 3D Geometric Primitives 3D Object Representations Point Line Plane Polygon Sphere 3D Object Representations Raw data Surfaces Solids High-Level Structure cgvr.korea.ac.kr

3D Geometric Primitives Point Line Segment Polygon Polyhedron Curved Surface Solid Object Etc. cgvr.korea.ac.kr

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

3D Vector Specifies a Direction and a Magnitude Represented by three coordinates Magnitude ||v||=sqrt(dxdx + dydy +dzdz) Has no location Dot product of two 3D vector V1 V2 = dx1dx2 + dy1dy2 + dz1dz2 V1 V2 = ||V1||||V2|| cos( ) (dx1, dy1, dz1) typedef struct{ Coordinate x; Coordinate y; Coordinate z; } Vector; (dx2, dy2, dz2) cgvr.korea.ac.kr

3D Line Line Segment with Both Endpoints at Infinity Parametric representation P=P1+tV, ( ) V typedef struct{ Point P1; Vector V; } Line; P1 cgvr.korea.ac.kr

3D Ray Line Segment with One Endpoints at Infinity Parametric representation P=P1+tV, ( ) V typedef struct{ Point P1; Vector V; } Ray; P1 cgvr.korea.ac.kr

3D Line Segment Specifies a Linear Combination of Two Points Parametric representation P=P1 + t(P2 - P1), ( ) P2 typedef struct{ Point P1; Point P2; } Segment; P1 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 N=(a, b, c) P1 P3 typedef struct{ Vector N; Distance d; } Plane; P2 d Origin 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; Points are in counter-clockwise order cgvr.korea.ac.kr

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 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 cgvr.korea.ac.kr

Point Cloud Unstructured Set of 3D Point Samples Acquired from range finder, computer vision, etc cgvr.korea.ac.kr

Range Image Set of 3D Points Mapping to Pixels of Depth Image Acquired from range scanner Range Image Tessellation Range Surface cgvr.korea.ac.kr

Polygon Soup Unstructured Set of Polygons Created with interactive modeling systems 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 cgvr.korea.ac.kr

Mesh Connected Set of Polygons (Usually Triangles) May not be closed cgvr.korea.ac.kr

Subdivision Surfaces Coarse Mesh & Subdivision Rule Define smooth surface as limit of sequence of refinements cgvr.korea.ac.kr

Parametric Surfaces Tensor Product Spline Patches Careful constraints to maintain continuity cgvr.korea.ac.kr

Implicit Surface Points satisfying: F(x,y,z) = 0 Polygonal Model Implicit Model 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 cgvr.korea.ac.kr

Voxels Uniform Grid of Volumetric Samples Acquired from CAT, MRI, etc. cgvr.korea.ac.kr

Binary Spatial Partition BSP Tree Binary Space Partition with Solid Cells Labeled Constructed from polygonal representations 1 a b 1 g g a a 6 c 2 f f 3 d 3 5 e e 7 e 4 d c 4 d c b b f 5 2 6 7 Object Binary Spatial Partition BSP Tree cgvr.korea.ac.kr

CSG Hierarchy of Boolean Set Operations (Union, Difference, Intersect) Applied to Simple Shapes cgvr.korea.ac.kr

using Rotational Sweep Solid Swept by Curve Along Trajectory Constructing a Torus using Rotational Sweep cgvr.korea.ac.kr

Summary Taxonomy of 3D Object Representations Discrete Continuous Voxel Combinational Functional Topological Set Membership Parametric Implicit Mesh Subdivision BSP Tree Bezier B-Spline Algebraic cgvr.korea.ac.kr

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