Parametric Curves

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• Misc
  • Colloquium
    • “Controllability of Graphs”
    • About graphs as networks of related things
      
    • Dr. Cesar Aguilar, California State University Bakersfield
    • Monday (Feb. 22), 4:00 PM, Newton 201
    • Extra credit for short write-ups on how you connect the talk to other courses, interests, etc.
• Questions?
• Parametric Equations
  • “Parametric Curves — Basic Graphing” video
    • How to sketch curve from parametric equation
    • Eliminating parameter
  • Example: Lissajous curves
    • Parametric equations used in this example (technically not completely general Lissajous curves)
      • x(t) = cos (at)
      • y(t) = cos (bt)
    • Read a Matlab function that draws Lissajous curves
    • Result of lissajous( 3, 5 )
      
  • Some parametric “tricks”
    • Put y = f(x) into clearly parametric form?
      • x(t) = t
      • y(t) = f(t)
      • i.e., you can do everything in parametric form that you can do in the usual explicit form, plus more
    • Line between p₁ = (x₁,y₁) and p₂ = (x₂,y₂)
      • (1-t) p₁ + t p₂
      • i.e., x(t) = (1-t)x₁ + tx₂; y(t) = (1-t)y₁ + ty₂
      • Weighted sum of p₁ and p₂
    • Sine and cosine can produce circle-like figures
  • Example: cubic Bezier curves
    • Parametric curves that get some coefficients from “control points” that govern the exact shape of the curve
    • Parametric equations:
      • x(t) = p_1,x(1-t)³ + 3p_2,xt(1-t)² + 3p_3,xt²(1-t) + p_4,xt³
      • y(t) = p_1,y(1-t)³ + 3p_2,yt(1-t)² + 3p_3,yt²(1-t) + p_4,yt³
    • A Matlab function to draw a cubic Bezier curve
    • Result of bezier( [0,0], [3,13], [1,5], [-8,-3] )
      
    • Result of bezier( [0,0], [2,2], [-1,2], [1,0] )
      
    • Bezier curves aren’t limited to using degree-3 polynomials and 4 control points
      • You can handle n control points with a degree n-1 polynomial following the same basic form as above
      • Coefficient for t^k(1-t)^n-k term is n choose k
      • In fact, the parametric form for a line given earlier is a Bezier curve with 2 control points
• Next
  • Lab day for plotting lab

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