5. Intersection Problems

• 5.1 Overview of intersection problems
• 5.2 Intersection problem classification
  • 5.2.1 Classification by dimension
  • 5.2.2 Classification by type of geometry
  • 5.2.3 Classification by number system
  
  
• 5.3 Point/point intersection
• 5.4 Point/curve intersection
  • 5.4.1 Point/implicit algebraic curve intersection
  • 5.4.2 Point/rational polynomial parametric curve intersection
    • 5.4.2.1 Elementary method
    • 5.4.2.2 Bounding box and subdivision followed by minimization method
    • 5.4.2.3 Distance function method
    • 5.4.2.4 Implicitization
  • 5.4.3 Point/procedural parametric curve intersection
  
  
• 5.5 Point/surface intersection
  • 5.5.1 Point/implicit algebraic surface intersection
  • 5.5.2 Point/rational polynomial parametric surface intersection
    • 5.5.2.1 Implicitization
    • 5.5.2.2 Newton's method
    • 5.5.2.3 Bounding box and subdivision followed by minimization method
    • 5.5.2.4 Distance function method
  • 5.5.3 Point/procedural parametric surface intersection
  
  
• 5.6 Curve/curve intersection
  • 5.6.1 Rational polynomial parametric/implicit algebraic curve intersection (Case D3)
    • 5.6.1.1 2-D planar case
    • 5.6.1.2 3-D space curve
  • 5.6.2 Rational polynomial parametric/rational polynomial parametric curve intersection (Case D1)
  • 5.6.3 Rational polynomial parametric/procedural parametric and procedural parametric/procedural parametric curve intersections (Cases D2 and D5)
  • 5.6.4 Procedural parametric/implicit algebraic curve intersection (Case D6)
  • 5.6.5 Implicit algebraic/implicit algebraic curve intersection (Case D8)
    • 5.6.5.1 Implicitization
    • 5.6.5.2 Newton's method
    • 5.6.5.3 Interval Projected Polyhedron solver
  
  
• 5.7 Curve/surface intersection
  • 5.7.1 Rational polynomial parametric curve/implicit algebraic surface intersection (Case E3)
  • 5.7.2 Rational polynomial parametric curve/rational polynomial parametric surface intersection (Case E1)
    • 5.7.2.1 Implicitization
    • 5.7.2.2 Bounding box and subdivision followed by minimization method
    • 5.7.2.3 Interval Projected Polyhedron solver
  • 5.7.3 Rational polynomial parametric/procedural parametric and procedural parametric/procedural parametric curve/surface intersections (Cases E2/E6)
  • 5.7.4 Procedural parametric curve/implicit algebraic surface intersection (Case E7)
  • 5.7.5 Implicit algebraic curve/implicit algebraic surface intersection (Case E11)
  • 5.7.6 Implicit algebraic curve/rational polynomial parametric surface intersection (Case E9)
  
  
• 5.8 Surface/surface intersections
  • 5.8.1 Rational polynomial parametric/implicit algebraic surface intersection (Case F3)
    • 5.8.1.1 Formulation
    • 5.8.1.2 Tracing method
    • 5.8.1.3 Characteristic points
    • 5.8.1.4 Analysis of singular points
    • 5.8.1.5 Computing starting points for all branches
  • 5.8.2 Rational polynomial parametric/rational polynomial parametric surface intersection (Case F1)
    • 5.8.2.1 Lattice methods
    • 5.8.2.2 Subdivision methods
    • 5.8.2.3 Marching methods
  • 5.8.3 Implicit algebraic/implicit algebraic surface intersection (Case F8)
  
  
• 5.9 Overlapping of curves and surfaces
• 5.10 Self-intersection of curves and surfaces
• 5.11 Summary