Lines

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• Questions?
• Lines
  • First part of section 12.5
  • Line determined by point and direction vector
    • Think of this as point + offset vectors that are all possible scalings of direction
  • Described parametrically
    • r = p + tv
    • xi + yj + zk = x₀i + y₀j + z₀k + t(v₁i + v₂j + v₃k )
    • r(t) = p₀ + tv = position of point on line L
      • Vector-valued function
    • Parametrically x = x₀+tv₁, y = y₀+tv₂, z = z₀+tv₃
    • Distance from S to line parallel to v = | PS x v | / |v|
  • Examples
    • Line r(t) = ⟨1,-1,0⟩ + t⟨1,0,2⟩ Where is it when t = 2? How about t = -1?
      • r(2) = ⟨1,-1,0⟩ + 2⟨1,0,2⟩ = ⟨1+2, -1+0, 0+4⟩ = ⟨3,-1,4⟩
      • r(-1) = ⟨0,-1,-2⟩
    • Line through (1,0,2) in direction ⟨0,3,-1⟩
      • r(t) = ⟨1,0,2⟩ + t⟨0,3,-1⟩
      • r₂(t) = ⟨1,0,2⟩ + t⟨0,6,-2⟩
    • Line through (1,2,3) and (-1,0,4)
      • r(t) = ⟨1,2,3⟩ + t⟨-2,-2,1⟩
      • (0,1,3.5) = r(1/2)
      • r(0) = (1,2,3), r(1) = (-1,0,4)
      • 0 ≤ t ≤ 1 means point in segment
      • t < 0 means point “before” start
      • t > 1 means point “after” end pf segment
    • Ray tracing
      • “Eye” at (0,-2,0) looking through (1,0,1)
      • Ray = line through (0,-2,0) and (1,0,1)
        • r(t) = ⟨0,-2,0⟩ + t⟨1,2,1⟩
      • Sphere w/ radius 2 at (1,1,1), “sky” at y = 10
      • Use parametric form, x = t, y = 2t - 2, z = t, to find intersections
        • Sky: y = 10, i.e., 2t - 2 = 10, t = 6
      • Intersections: set ray equation(s) = surface equation(s) and solve
      • Sphere:
        • (x-1)² + (y-1)² + (z-1)² = 4
        • (t-1)² + (2t-3)² + (t-1)² = 4
        • Can use muPad to solve this and evaluate ray at t values:
          
• Next
  • Planes
  • Read rest of section 12.5
    • “An Equation for a Plane in Space” through end
    • i.e., bottom 3/4 of page 723 through first 2/3 page 726

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