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3-D Object Modeling

Learn about the two ways to model objects in 3-D - space-partitioning (solid modeling) and boundary representations (B-reps). Understand the advantages and techniques for representing polygon surfaces and approximations. Explore the polygon tables and plane equations used in 3-D object processing.

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3-D Object Modeling

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  1. 3-D Object Modeling CS-321Dr. Mark L. Hornick

  2. There are basically two ways to model objects in 3-D • Space-partitioning (Solid modeling) • Describe interior as union of solids • Parametric modeling • what parameters would we use? • Boundary representations (B-reps) • Describe the set of surfaces • That define the exterior surface of an object • That separate object interior from exterior • What kinds of surfaces? CS-321Dr. Mark L. Hornick

  3. Most common is Boundary Representation • Advantage: surface equations are linear • Representing objects: • Polyhedrons - no problem • General shapes - approximation • Tessellate (subdivide) to polygon mesh CS-321Dr. Mark L. Hornick

  4. Polygon Meshes can be used to describe a general surface • Replaces it with an approximation • Common sets of polygons • Triangles • Advantage: 3 vertices determine plane • Quadrilaterals CS-321Dr. Mark L. Hornick

  5. Polygon Meshes CS-321Dr. Mark L. Hornick

  6. Polygon Description • Geometric data • Description of position and shape • Attribute data • Description of surface • Color • Transparency • Surface reflectivity • Texture CS-321Dr. Mark L. Hornick

  7. Polygon Description CS-321Dr. Mark L. Hornick

  8. How do you represent a polygon surface approximation? • List of vertices (in 3-D) could work • Sufficient description • But, polygons are joined • Shared vertices and edges • We need a more general description that • Reduces redundancy • Represents component polygons CS-321Dr. Mark L. Hornick

  9. Polygon Tables • Vertex table • For all polygons in composite object • Edge table • Each edge listed once • Even if part of more than one polygon • Polygon-surface table Pointers or other links between tables for easy access CS-321Dr. Mark L. Hornick

  10. E8 E1 E2 E7 E3 E4 E6 E5 Polygon Table Example V1 E1: V1,V3 E2: V1,V2 E3: V2,V3 E4: V1,V4 E5: V3,V4 E6: V4,V5 E7: V2,V5 E8: V1,V5 S1: E1, E2, E3 S2: E2, E7, E8 S3: E1, E4, E5 S4: E4, E6, E8 S5: E3, E5, E6 , E7 V2 V5 V3 CS-321Dr. Mark L. Hornick V4

  11. Polygon surfaces can be represented by Plane Equations • Often need information on • Spatial orientation of surfaces • Visible surface identification • Surface rendering (e.g. shading) • Processing a 3-D object involves • Equations of polygon planes • Coordinate transformations (3-D) CS-321Dr. Mark L. Hornick

  12. The Plane Equation Must be satisfied for any point (x,y,z) in the plane. We want to solve for coefficients (A, B, C, D). How many points define a plane? How many unknowns are there? CS-321Dr. Mark L. Hornick

  13. Plane Equation Solution (1) Use Cramer’s Rule CS-321Dr. Mark L. Hornick

  14. Plane Equation Solution (2) CS-321Dr. Mark L. Hornick

  15. Vector Formulation (1) For a plane described by A vector normal to the plane surface is N Vector N is also the cross product of two successive polygon edges. We go CCW around the polygon to find the “inside-to-outside” normal vector: CS-321Dr. Mark L. Hornick

  16. Vector Formulation (2) The normal vector: Gives us values for A, B, and C in the plane equation: But what about D? N D specifies which one of the planes normal to N we are talking about. Calculate D by substituting A, B, C, and one polygon vertex into plane equation. CS-321Dr. Mark L. Hornick

  17. Vector Formulation (3) Solving for D in the plane equation: N P CS-321Dr. Mark L. Hornick

  18. Testing Points If point is “inside” the plane, then: If point is “outside” the plane, then: Poutside This can be used to determine what surfaces are hidden N Pinside CS-321Dr. Mark L. Hornick

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