# 3D Modeling: Solid Models

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1 CS 430/536 Computer Graphics I 3D Modeling: Solid Models Week 9, Lecture 18 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University 1

2 Overview 3D solid model representations Implicit models Super/quadrics Blobbies Swept objects Boundary representations Spatial enumerations Distance fields Quadtrees/octrees Stochastic models 2

3 Implicit Solid Modeling Computer Algebra meets CAD Idea: Represents solid as the set of points where an implicit global function takes on certain value F(x,y,z) < val Primitive solids are combined using CSG Composition operations are implemented by functionals which provide an implicit function for the resulting solid From M.Ganter, D. Storti, G. 3 UW

4 Quadratic Surfaces Sphere x 2 + y 2 + z 2 = r 2 Ellipsoid Torus È Í r - Í Î Ê x ˆ Á Ë General form r x 2 Ê x ˆ Á Ë r x Ê + y ˆ Á Ë r y 2 2 Ê + y ˆ Á Ë r y Ê + z ˆ Á Ë r z =1 Ê + z ˆ Á Ë r z 2 =1 a x 2 + b y 2 + c z f yz + 2g xz + 2h xy + 2p x + 2q y + 2r z + d = 0 4

5 Superellipsoid Surfaces Generalization of ellipsoid s 2 Control parameters s 1 and s 2 È 2 / s Ê x ˆ 2 Ê Á + y ˆ Í Í Ë r Á x Ë r Î y 2 / s 2 s 2 / s 1 + z 2 / s Ê ˆ 1 Á =1 r Ë z s 1 If s 1 = s 2 =1 then regular ellipsoid Has an implicit and parametric form! 5

7 CSG with Superellipsoids 7

8 Blobby Objects Do not maintain shape, topology Water drops Molecules Force fields But can maintain other properties, like volume 8

9 Gaussian Bumps Model object as a sum of Gaussian bumps/blobs f (x, y,z) = Âb k e -a 2 kr k - T = 0 k Where r k 2 = x k 2 + y k 2 + z k 2 and T is a threshold. 9

10 Metaballs (Blinn Blobbies) 10

11 Ray-traced Metaballs 11

12 Implicit Modeling System U. of Calgary Combine primitives Points, lines, planes, polygons, cylinders, ellipsoids Calculate field around primitives View Iso-surface of implicit function 12

13 Implicit Modeling System U. of Calgary Can apply blends and warps 13

14 Sweep Representations An alternative way to represent a 3D object Idea Given a primitive (e.g. polygon,sphere ) And a sweep (e.g. vector, curve ) Define solid as space swept out by primitive 14 Foley/VanDam, 1990/1994

15 Sweep Representations Issues: How to generate resulting surface? What about selfintersections? How to define intersection? 15 Foley/VanDam, 1990/1994

16 Approximate Representations Idea: discretize the world! Surface Models Mesh, facet and polygon representations Volume Models spatial enumeration voxelization 16

17 Examples From exact to facets. 17 Pics/Math courtesy of Dave UMD-CP

18 Boundary Representation Solid Modeling The de facto standard for CAD since ~1987 BReps integrated into CAGD surfaces + analytic surfaces + boolean modeling Models are defined by their boundaries Topological and geometric integrity constraints are enforced for the boundaries Faces meet at shared edges, vertices are shared, etc. 18

19 Let s Start Simple: Polyhedral Solid Modeling Definition Solid bounded by polygons whose edges are each a member of an even number of polygons A 2-manifold: edges members of 2 polygons 19

20 Properties of 2-Manifolds For any point on the boundary, its neighborhood is a topological 2D disk If not a 2-manifold, neighborhood not a disk 20 Foley/VanDam, 1990/1994

21 Euler s Formula For simple polyhedra (no holes): #Vertices - #Edges + #Faces = 2 21 Foley/VanDam, 1990/1994

22 Euler s Formula (Generalized) #Vertices - #Edges + #Faces - #Holes_in_faces = 2 (#Components Genus) Genus is the # holes through the object Euler Operators have been the basis of several modeling systems (Mantyla et al.) 22 Foley/VanDam, 1990/1994

23 Euler Operators Loop L H, Shell S C 23

24 Steps to Creating a Polyhedral Representation Solid Modeler Points, Lines/Edges, Polygons Modeling Generalization of 3D clipping to nonconvex polyhedra, enables implementation of booleans 24

25 State of the Art: BRep Solid Modeling but much more than polyhedra Two main (commercial) alternatives All NURBS, all the time Pro/E, SDRC, Analytic surfaces + parametric surfaces + NURBS +. all stitched together at edges Parasolid, ACIS, 25

26 Issues in Boundary Representation Solid Modeling Very complex data structures NURBS-based winged-edges, etc Complex algorithms manipulation, booleans, collision detection Robustness Integrity Translation Features Constraints and Parametrics 26

27 Other Issues: Non-Manifold Solids There are cases where you may need to model entities that are not entirely 3D 27 Pics/Math courtesy of Dave UMD-CP

28 Cell Decomposition Set of primitive cells Parameterized Often curved Compose complex objects by gluing cells together Used in finiteelement analysis 28 Foley/VanDam, 1990/1994

29 Brute force A grid Pixels Spatial Occupancy Enumeration Picture elements Voxels Volume elements Quadtrees 2D adaptive representation Octrees 3D adaptive representation Extension of quadtrees 29

30 Brute Force Spatial Occupancy Enumeration Impose a 2D/3D grid Like graph paper or sugar cubes Identify occupied cells Problems High fidelity requires many cells Modified Partial occupancy 30 Foley/VanDam, 1990/1994

31 Distance Volume Store signed distance to surface at each voxel Narrow-band representation Iso-surface at value 0 approximates the original surface. 31

32 Offset Surfaces from Distance Volumes 32

33 Quadtree Hierarchically represent spatial occupancy Tree with four regions NE, NW, SE, SW dark if occupied 33 Foley/VanDam, 1990/1994

34 Quadtree Data Structure F = full P = partially full E = empty 34 Foley/VanDam, 1990/1994

35 Octree 8 octants 3D space Left, Right, Up, Down, Front, Back 35 Foley/VanDam, 1990/1994

36 Boolean Operations on Octrees S T S» T S «T 36 Foley/VanDam, 1990/1994

38 Applications for Spatial Occupancy Enumeration Many different applications GIS Medical Engineering Simulation Volume Rendering Video Gaming Approximating real-world data. 38

39 Issues with Spatial Occupancy Approximate Enumeration Kind of like faceting a surface, discretizing 3D space Operationally, the combinatorics (as opposed to the numerics) can be challenging Not as good for applications wanting exact computation (e.g. tool path programming) 39

40 Binary Space Partition Trees (BSP Trees) Recursively divide space into subspaces Arbitrary orientation and position of planes Homogeneous regions are leafs called in/out cells 40 Foley/VanDam, 1990/1994

41 Statistical Representations Store density (material vs. void) Statistical description of geometry Goal describe the porosity without storing the geometry information 41 Generated by Termite Agents Simulation.

42 Stochastic Geometry Need some way of converting a solid into some representative statistical form From each material voxel, calculate the distance to the nearest voxel that is not material Repeat for void voxels Store distributions: one for empty space one for material density value Distance vs. Probability 42

43 Application: Biological Models Bone tissue MRI data Other biological data Solid modeling MRI scan of left shoulder Bone matrix from scanned data 43

44 Application: Surface Texture 44

45 Application: Surface Texture 45

46 Programming Assignment 5 Extend XPM to 60 different RGB colors Read 3 models and assign each a color Implement Z-buffer rendering Implement front & back cutting planes Only render parts of models between planes Implement linear depth-cueing Color = base_color*(z-far)/(near-far) Re-use and extend 2D polygon filling 46

47 End 47

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