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Surfaces

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Transcript

  • 1. S U R F A C E S
        • By Abdul Ghaffar
  • 2. Chapter 6: Polygonal Meshes
    • 6.4.5 Discretely Swept Surfaces of Revolution
      • Place all spline points at origin, and use rotation for affine transformation.
      • Base polygon called profile
      • Operation equivalent to circularly sweeping shape about axis
      • Resulting shape called surface of revolution.
  • 3. Chapter 6: Polygonal Meshes
    • 6.5.1 Representation of Surfaces
      • Similar to planar patch P(u,v) = C + a u + b v
      • Generalize: P(u,v) = (X(u,v), Y(u,v), Z(u,v)) (point form).
      • If v constant, u varies: v-contour
      • If u constant, v varies: u-contour
    • Implicit Form of Surface
      • F(x,y,z)=0 iff (x,y,z) is on surface.
      • F(x,y,z)<0 iff (x,y,z) is inside surface
      • F(x,y,z)>0 iff (x,y,z) is outside surface
  • 4. Polygonal Meshes
    • 6.5.6 Rules Surfaces
      • Surface is ruled if, through every one of tis points, there passes at least one line that lies entirely on the surface.
      • Rules surfaces are swept out by moving a straight line along a particular trajectory.
      • Parametric form: P(u,v) = (1-v)P 0 (u) +vP 1 (u).
      • P 0 (u) and P 1 (u) define curves in 3D space, defined by components P 0 (u)=(X 0 (u),Y 0 (u),Z 0 (u)).
      • P 0 (u) and P 1 (u) defined on same interval in u.
      • Ruled surface consists of one straight line joining each pair of points P 0 (u’) and P 1 (u’).
  • 5. Chapter 6: Polygonal Meshes
    • Cones
      • Ruled surface for which P 0 (u) is a single point
      • P(u,v) = (1-v)P 0 +vP 1 (u).
  • 6. Chapter 6: Polygonal Meshes
    • Cylinders
      • Ruled surface for which P 1 (u) is a translated version of P 0 (u): P 1 (u) = P 0 (u) + d
      • =>P(u,v)= P 0 (u) + d v
  • 7. Chapter 6: Polygonal Meshes
    • 6.5.8 The Quadric Surfaces
      • 3D analogs of conic sections
  • 8. Chapter 6: Polygonal Meshes
    • 6.5.8 The Quadric Surfaces
  • 9. Chapter 6: Polygonal Meshes
    • Properties of Quadric Surfaces
      • Trace is curve formed when surface is cut by plane
        • All traces of quadric surfaces are conic sections.
      • Principal traces are curves generated when cutting planes aligned with axes.