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Parametric Torus

Parametric Torus
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2015-05-13 03:47
Solid GeometryCalculusParametric Equation
This GeoGebra example demonstrates how to describe a torus and other surfaces of revolution using parametric equations. It involves concepts of parametric curves and surfaces, utilizing commands like Segment, Point, Polygon, Surface, and CurveCartesian. Suitable for high school and above students learning calculus and solid geometry.

GeoGebra Commands & Parameters

  1. Segment(A, ((2 * π), y(A))) → a
  2. Point(a) → B
  3. Segment(A, (x(A), (2 * π))) → b
  4. Point(b) → C
  5. Polygon(A, B, D, C) → poly1, a_1, c, f, e
  6. Surface(((5 + (2 * cos(u))) * cos(t)), ((5 + (2 * cos(u))) * sin(t)), (2 * sin(u)), u, x(A), x(B), t, y(A), y(C)) → d
  7. Segment(A, B) → g
  8. Segment(A, C) → h
  9. CurveCartesian(((5 + (2 * cos(u))) * cos(y(A))), ((5 + (2 * cos(u))) * sin(y(A))), (2 * sin(u)), u, x(A), x(B)) → i
  10. CurveCartesian(((5 + (2 * cos(x(A)))) * cos(t)), ((5 + (2 * cos(x(A)))) * sin(t)), (2 * sin(x(A))), t, y(A), y(C)) → j
  11. Segment(C, D) → k
  12. Segment(D, B) → l
  13. CurveCartesian(((5 + (2 * cos(u))) * cos(y(C))), ((5 + (2 * cos(u))) * sin(y(C))), (2 * sin(u)), u, x(A), x(B)) → m
  14. CurveCartesian(((5 + (2 * cos(x(B)))) * cos(t)), ((5 + (2 * cos(x(B)))) * sin(t)), (2 * sin(x(B))), t, y(A), y(C)) → n
  15. Intersect(p, g) → F
  16. Intersect(p, k) → G
  17. Segment(F, G) → q
  18. Segment(G, F) → r
  19. CurveCartesian(((5 + (2 * cos(x(H)))) * cos(t)), ((5 + (2 * cos(x(H)))) * sin(t)), (2 * sin(x(H))), t, y(A), y(C)) → u
  20. CurveCartesian(((5 + (2 * cos(u))) * cos(y(H))), ((5 + (2 * cos(u))) * sin(y(H))), (2 * sin(u)), u, x(A), x(B)) → v
  21. Intersect(b_1, k) → J
  22. Intersect(b_1, g) → K
  23. Segment(J, K) → c_1
  24. Intersect(d_1, h) → L
  25. Intersect(d_1, l) → M
  26. Segment(L, M) → e_1
  27. CurveCartesian(((5 + (2 * cos(x(F)))) * cos(t)), ((5 + (2 * cos(x(F)))) * sin(t)), (2 * sin(x(F))), t, y(A), y(C)) → o
  28. Sequence(CurveCartesian[((5 + (2 * cos(u))) * cos(t)), ((5 + (2 * cos(u))) * sin(t)), (2 * sin(u)), u, x(A), x(B)], t, y(A), y(C), 0.2617993877991494) → list1
  29. Sequence(CurveCartesian[((5 + (2 * cos(u))) * cos(t)), ((5 + (2 * cos(u))) * sin(t)), (2 * sin(u)), t, y(A), y(C)], u, x(A), x(B), 0.2618) → list2
  30. Sequence(Segment[(u, y(A)), (u, y(C))], u, x(A), x(B), 0.2617993877991494) → list3
  31. Sequence(Segment[(x(A), v), (x(B), v)], v, y(A), y(C), 0.2618) → list4

AI Drawing Prompt Reference

Draw a torus using parametric equations (5 + 2*cos(u)) * cos(t) and (5 + 2*cos(u)) * sin(t) for the cross-section, and 2*sin(u) for the height. First, draw the cross-section, then rotate it around the central axis to form the torus. Next, draw the boundary lines and auxiliary lines, and finally use the parametric equations to draw the surface of the torus.
The prompt above was compiled from the original drawing commands and is for reference only.