Lesson 9: Parametric Surfaces

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A parametric surface is function from a subset of the plane into space.

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Lesson 9: Parametric Surfaces

  1. 1. Section 10.5 Parametric Surfaces Math 21a February 25, 2008 Announcements Problem Sessions: Monday, 8:30 (Sophie); Thursday, 7:30 (Jeremy); SC 103b Office hours Tuesday, Wednesday 2–4pm SC 323. Mathematica assignment due February 29. Image: Mike Baird
  2. 2. Outline Explicit versus implicit descriptions Easy parametrizations Graphs Planes Other coordinate surfaces Surfaces of revolution Other parametrizations
  3. 3. An implicit description of a surface is an equation satisfied by all points in the surface.
  4. 4. An implicit description of a surface is an equation satisfied by all points in the surface. Example The unit sphere in R3 is the set of all points (x, y , z) such that x2 + y2 + z2 = 1 Image: dharmesh84
  5. 5. An explicit description of a surface is as the image of a function r : D → R3 , where D is a subset of the plane.
  6. 6. An explicit description of a surface is as the image of a function r : D → R3 , where D is a subset of the plane. Example The unit sphere can be described as the image of two maps: r+ : D → R3 , (x, y ) → (x, y , 1 − x 2 − y 2) r− : D → R3 , (x, y ) → (x, y , − 1 − x 2 − y 2 ) Here D is the unit disk in the plane: D = (x, y ) x 2 + y 2 ≤ 1
  7. 7. An explicit description of a surface is as the image of a function r : D → R3 , where D is a subset of the plane. Example The unit sphere can be described as the image of two maps: r+ : D → R3 , (x, y ) → (x, y , 1 − x 2 − y 2) r− : D → R3 , (x, y ) → (x, y , − 1 − x 2 − y 2 ) Here D is the unit disk in the plane: D = (x, y ) x 2 + y 2 ≤ 1 It can also be described as the image of one map r : I → R3 , (θ, ϕ) → (cos θ sin ϕ, sin θ sin ϕ, cos ϕ) Here I = [0, 2π] × [0, π].
  8. 8. Goals Given a surface, find a parametrization r of it Given a function r : D → R3 , find the image surface.
  9. 9. Outline Explicit versus implicit descriptions Easy parametrizations Graphs Planes Other coordinate surfaces Surfaces of revolution Other parametrizations
  10. 10. Parametrizing graphs If S is the graph of a function f : D → R, then the function can be used for a parametrization: r : D → R3 , (x, y ) → (x, y , f (x, y ))
  11. 11. Parametrizing graphs If S is the graph of a function f : D → R, then the function can be used for a parametrization: r : D → R3 , (x, y ) → (x, y , f (x, y )) The grid lines x = constant and y = constant trace out curves on the surface.
  12. 12. Parametrizing graphs If S is the graph of a function f : D → R, then the function can be used for a parametrization: r : D → R3 , (x, y ) → (x, y , f (x, y )) The grid lines x = constant and y = constant trace out curves on the surface. Advantages/Disadvantages Often this is easy bad if f is not differentiable at points in D sometimes you need more than one
  13. 13. Planes An implicit description of a surface is n · (r − r0 ) = 0 A parametric description would be as the image of r : R2 → R3 , (s, t) → r0 + su + tv
  14. 14. Planes An implicit description of a surface is n · (r − r0 ) = 0 A parametric description would be as the image of r : R2 → R3 , (s, t) → r0 + su + tv Example (Worksheet problem 1) Write a parameterization for the plane through the point (2, −1, 3) containing the vectors u = 2i + 3j − k and v = i − 4j + 5k.
  15. 15. Planes An implicit description of a surface is n · (r − r0 ) = 0 A parametric description would be as the image of r : R2 → R3 , (s, t) → r0 + su + tv Example (Worksheet problem 1) Write a parameterization for the plane through the point (2, −1, 3) containing the vectors u = 2i + 3j − k and v = i − 4j + 5k. Answer Take r(s, t) = 2, −1, 3 + s 2, 3, −1 + t 1, −4, 5 = 2 + 2s + t, −1 + 3s − 4t, 3 − s + 5t
  16. 16. Example Find a parametrization for the plane x + y + z = 1.
  17. 17. Example Find a parametrization for the plane x + y + z = 1. Solution The normal vector is n = 1, 1, 1 ; the plane passes through (1, 0, 0). We still need two vectors perpendicular to n: 1, 1, −2 and 1, −1, 0 will work (there are other choices). We get r (s, t) = 1, 0, 0 + s 1, 1, −2 + t 1, −1, 0 = 1 + s + t, s − t, −2s Notice that x(s, t) + y (s, t) + z(s, t) = 1 for all s and t.
  18. 18. Other coordinate surfaces The conversion from other coordinate systems to rectangular coordinates is a kind of parametrization.
  19. 19. Other coordinate surfaces The conversion from other coordinate systems to rectangular coordinates is a kind of parametrization. Example (Worksheet problem 2) Write an equation in x, y , and z for the parametric surface x = 3 sin s y = 3 cos s z = t + 1, where 0 ≤ s ≤ π and 0 ≤ t ≤ 1.
  20. 20. Other coordinate surfaces The conversion from other coordinate systems to rectangular coordinates is a kind of parametrization. Example (Worksheet problem 2) Write an equation in x, y , and z for the parametric surface x = 3 sin s y = 3 cos s z = t + 1, where 0 ≤ s ≤ π and 0 ≤ t ≤ 1. Answer The image is the part of the cylinder x 2 + y 2 = 9 which also has 1 ≤ z ≤ 2 and x ≥ 0.
  21. 21. 0 1 2 3 2.0 1.5 1.0 2 0 2
  22. 22. Surfaces of revolution These can be parametrized by drawing circles whose radius is the function value. Example The graph of y = sin x on 0 ≤ x ≤ π is revolved around the x-axis. Find a parametrization of the the surface.
  23. 23. Surfaces of revolution These can be parametrized by drawing circles whose radius is the function value. Example The graph of y = sin x on 0 ≤ x ≤ π is revolved around the x-axis. Find a parametrization of the the surface. Solution For each x0 , a circles of radius f (x0 ) is traced out in the plane x = x0 . So a parametrization could be r → [0, π] × [0, 2π] → R3 (x, θ) → (x, f (x) cos θ, f (x) sin θ)
  24. 24. Outline Explicit versus implicit descriptions Easy parametrizations Graphs Planes Other coordinate surfaces Surfaces of revolution Other parametrizations
  25. 25. Rest of Worksheet problems Image: Erick Cifuentes

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