Modeling 3D Objects: Part 2

Transcription

1 Modeling 3D Objects: Part 2 Patches, NURBS, Solids Modeling, Spatial Subdivisioning, and Implicit Functions 3D Computer Graphics by Alan Watt Third Edition, Pearson Education Limited, 2000

2 General Modeling Methods 1. Polygon Mesh 2. Bi-cubic Parametric Patches (patches) and curved Quadri-laterals (NURBS) 3. Constructive Solids Geometry (CSG) 4. Spatial Subdivision Techniques (voxels) 5. Implicit Representation 2

3 Bi-Cubic Parametric Patch Polyognal Mesh model Flat quadrilateral Four corners (vertices) joined by straight edges Surface is defined by the four vertices Bi-Cubic Parametric Patch Curvilinear Quadrilateral Four corners (vertices) joined by four cubic curves. Multiple patches are connected to create a net Addison & Wesley Publishing Ltd. 3

4 Bi-Cubic Parametric Patch Advantages over Polygonal Mesh More exact analytical representation Three-dimensional shape editing More economical representation. Modeling Complex Curvilinear Alternative to Polygonal Mesh CAD-CAM Computer Numerically Controlled (CNC) Machining and Milling Figure 1 - Polygonal Mesh Figure 2 - Bi-Cubic Parametric Patch Addison & Wesley Publishing Ltd. 4

5 Bezier Curves Three-dimensional space curve Pierre Bezier Quadratic Curve Cubic curve Non-Uniform Ration B-Splines (NURBs) Most common Greatest curve range Video Clip Video Clip Video Clip Quadratic Curve Cubic Curve Fourth-Order Quadratic Curve Addison & Wesley Publishing Ltd. 5

6 From Curves to Surfaces NURBS CV Curve NURBS CV Surface NURBS Point Curve NURBS Point Surface Images from 3D Studio Max Help Addison & Wesley Publishing Ltd. 6

7 Patches & Curved Quadri-laterals Bi-cubic Parametric Patches (Polygonal Patches) Curved Quadri-laterals (NURBS) Common Characteristics Meshes of curved surfaces. Better representation of curved surfaces Takes more memory to represent. Expensive to visualize patches. (rendering time) Must convert to polygonal mesh to render. We actually have the shape mathematically defined. They are 3d representations of a curvilinear surface. Can be exact or approximate depending on how close the curve matches the actual shape of the object. (automobile fender) Addison & Wesley Publishing Ltd. 7

8 Patches in 3D Max Video Clip Addison & Wesley Publishing Ltd. 8

9 Constructive Solid Geometry Solids Modeling Shapes treated as a collection of very fine cloud of vertices. Volumetric Representation shape represented by elementary surface volumes or primitives VS. Surfaces. CSG User Representation Objects that can be represented by simple primitive objects combined by boolean set operators. Accuracy is limited to allowable geometric shapes and operators. Geometric Primitives Boolean Set Operators Linear Transformations Modeling procedure stored as a attribute tree. (see figure) Addison & Wesley Publishing Ltd. 9

10 CSG Boolean Example a) Cylinder is added Cuboid using boolean union. b) A cylinder is subtracted from the cuboid+cylinder object using boolean difference. c) The above object is combined with another object using boolean intersection. Addison & Wesley Publishing Ltd. 10

11 CSG Parametric Modeling Parametric easy to make changes to a given component Constraints Addison & Wesley Publishing Ltd. 11

12 CSG Modeling Geometric limitations Geometric Primitives Boolean Set Operations Rendering images Converted to Polygonal Mesh to render Special rendering techniques CSG Modelers Solidworks Pro-Engineer Form-Z Addison & Wesley Publishing Ltd. 12

13 Boolean with Polygon Boolean Operations w/ Polygons Polygons instead of points Boolean operations leave artifacts Video Clip Addison & Wesley Publishing Ltd. 13

14 Spatial Sub-Division Techniques Voxels Volumetric Pixels Define image space with close packed cubes Addison & Wesley Publishing Ltd. 14

15 Implicit Representations Whole object by a single implicit formula Representing Bounding Volumes Creates Isocontours Addison & Wesley Publishing Ltd. 15

16 Example Implicit Modeling Sphere generators Dark spheres negative generators Interpretation of isosurfaces Addison & Wesley Publishing Ltd. 16

17 Issues with Implicit Modeling Machine Representation, not a User Representation. Hard to control the blending. Difficult to animate. Ease of collision detection. Difficult to render. Computational Fluid Dynamics and Medical Imaging Addison & Wesley Publishing Ltd. 17

18 End Addison & Wesley Publishing Ltd. 18

19 Suggestions from Kerlow Sketch your ideas first Use multiple camera views while modeling Keep a Project Journal. Polygons or Curves? Will the model be used for CADAM? NURBS modeling Shell Constructive Solid Geometry Volume Check the Preferences File AutoSave Determine what is saved with the file or the application? Save your work often. Addison & Wesley Publishing Ltd. 19

20 Summary Creation of Object / Representation Varying types of modeling Laser scanning Sweeping, lathing Nature of the primitive elements Boundary/Surface representations Polygonal Mesh Patches Volume representations Solids Voxels Accuracy Either exact or approximate. Polygon meshes are approximate increased with degree of expense Bi-cubic patches can be exact or approximate depending on the context. CSG representations are exact, but are only as accurate as the subset of primitives provided. Accuracy vs. data volume The continuing trade off and context dependent Data Volume vs. Complexity DV increases with complexity Ease of creation/editing / animation / rendering Editing static object CSG Patches are easy or difficult depending on the type of shape you are trying to create No method is easy for shapechanging animation such as the human face. NURBS are less difficult. Methods with Higher degree of accuracy more difficult to animate. Addison & Wesley Publishing Ltd. 20