Plotting Spirals, Part 2

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• Questions?
• Plotting
  • Finish lesson in class today, or on your own afterwards
  • Equations for spirals—a way to find them from the descriptions you were given
    • Archimedean (radius increases by constant increment per turn) as an example
    • What you know about radius in one turn (turn i+1) based on radius in previous one (turn i)
      
      • r_i+1 = r_i + c
      • r₀ = 0 (or any other initial radius you use)
    • This is an example of something called a “recurrence relation”
    • Can solve it by thinking about how it expands into a definition of r_i+1
      • r_i+1= r_i + c = r_i-1 + 2c = … = r₀ + (i+2)c?
      • Could prove by induction
      • And as evidence for why this is a good idea, my guess that r_i+1 = r₀ + c(i+2) is wrong
      • It should be r_i+1 = r₀ + c(i+1)
      • Proof by induction on i:
        • Base case, i = 0
          • Plugging i = 0 into the original recurrence relation for r_i+1 yields r₁ = r₀ + c
          • And the corrected equation r_i+1 = r₀ + c(i+1) yields r₁ = r₀ + c(1)
          • = r₀ + c, which agrees with the recurrence relation
        • Induction step
          • Assuming r_k = r₀ + kc…
          • r_k+1 = r_k + c (from recurrence relation)
          • = r₀ + kc + c (induction hypothesis)
          • = r₀ + (k+1)c (combine “c” terms)
          • Which agrees with the corrected equation
    • Note that i = number of turns = t / 2π, assuming the basic spiral is generated by parametric equations involving r * cos(t) and r * sin(t)
• Next
  • Matrices and image processing
  • Specifically, “green screen” effect that puts me on Mars…
    
    • …from me in front of a colored screen…
      
    • …and a NASA photo from Mars:
      
  • Read…
    • Section 13.2, images, especially part re true-color images
    • Section 3.7, functions

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