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Intersection of Two Orthogonal Cylinders

Intersection of Two Orthogonal Cylinders
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2015-06-01 23:35
Solid GeometryGeometric ConstructionGraphing
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
This example demonstrates the intersection of two orthogonal cylinders. Key math concepts include solid geometry, intersection, and cylinders. It is constructed using GeoGebra commands like Ray, Point, CurveCartesian, Cylinder, Surface, If. Suitable for high school solid geometry learning or introductory mathematical modeling.

GeoGebra Commands & Parameters

  1. Ray((0, b, c), (0, b + 1, c)) → dda'
  2. Point(dda') → A'
  3. Ray((0, (-b), c), (0, (-b) - 1, c)) → dda
  4. Point(dda) → A
  5. Ray(((-a), 0, (-c)), ((-a) - 1, 0, (-c))) → ddb
  6. Point(ddb) → B
  7. Ray((a, 0, (-c)), (a + 1, 0, (-c))) → ddb'
  8. Point(ddb') → B'
  9. CurveCartesian((a * cos(t)), sqrt(b^(2) - ((2 * c) + (a * sin(t)))^(2)), c + (a * sin(t)), t, 0, 6.283185307179586) → d
  10. CurveCartesian((a * cos(t)), (-sqrt(b^(2) - ((2 * c) + (a * sin(t)))^(2))), c + (a * sin(t)), t, 0, 6.28319) → e
  11. Cylinder(B', B, b) → cylb, f, g, j
  12. Cylinder(A, A', a) → cyla, k, h, i
  13. Surface((a * cos(u)), (y(A') * (1 - v)) + (v * sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, 0, 6.28319, v, 0, 1) → cpa1
  14. Surface((a * cos(u)), (y(A) * (1 - v)) - (v * sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, 0, 6.28319, v, 0, 1) → cpa2
  15. Surface(x(A) + (a * cos(u)), (y(A') * v) + ((1 - v) * y(A)), z(A) + (a * sin(u)), u, π - asin(((-b) - (2 * c)) / a), (2 * π) + asin(((-b) - (2 * c)) / a), v, 0, 1) → cpa4
  16. Surface((x(B') * (1 - v)) + ((v * a) * cos(u)), sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), c + (a * sin(u)), u, -1.5707963267948966, 1.5708, v, 0, 1) → cpb1
  17. Surface((x(B) * (1 - v)) + ((v * a) * cos(u)), sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), c + (a * sin(u)), u, 1.5708, 4.71238898038469, v, 0, 1) → cpb2
  18. Surface((x(B') * (1 - v)) + ((v * a) * cos(u)), (-sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, -1.5707963267948966, 1.5707963267948966, v, 0, 1) → cpb3
  19. Surface((x(B) * (1 - v)) + ((v * a) * cos(u)), (-sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), c + (a * sin(u)), u, 1.5707963267948966, 4.71238898038469, v, 0, 1) → cpb4
  20. Surface((x(B') * (1 - v)) + (v * x(B)), (b * cos(u)), z(B') + (b * sin(u)), u, asin((a + (2 * c)) / b), π - asin((a + (2 * c)) / b), v, 0, 1) → cpb5
  21. If(-1 ≤ (b - (2 * c)) / a ≤ 1, asin((b - (2 * c)) / a), If[-1 ≤ (b + (2 * c)) / a ≤ 1, π + asin((b + (2 * c)) / a), 0]) → uad
  22. If(-1 ≤ (b - (2 * c)) / a ≤ 1, π - asin((b - (2 * c)) / a), If[-1 ≤ (b + (2 * c)) / a ≤ 1, (2 * π) - asin((b + (2 * c)) / a), 0]) → uaf
  23. Surface(x(A) + (a * cos(u)), (y(A') * v) + ((1 - v) * y(A)), z(A) + (a * sin(u)), u, uad, uaf, v, 0, 1) → cpa3
  24. If(-1 ≤ (a - (2 * c)) / b ≤ 1, π + asin((a - (2 * c)) / b), If[-1 ≤ (a + (2 * c)) / b ≤ 1, asin((a + (2 * c)) / b), 0]) → ubd
  25. If(-1 ≤ (a - (2 * c)) / b ≤ 1, (2 * π) - asin((a - (2 * c)) / b), If[-1 ≤ (a + (2 * c)) / b ≤ 1, π - asin((a + (2 * c)) / b), 0]) → ubf
  26. Surface((x(B') * (1 - v)) + (v * x(B)), (b * cos(u)), z(B') + (b * sin(u)), u, ubd, ubf, v, 0, 1) → cpb
  27. Surface((10 * p) + ((v * a) * cos(u)), (-10 * p) - sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), (10 * p) + c + (a * sin(u)), u, 0, 6.28319, v, -1, 1) → sb'
  28. Surface((10 * p) + (a * cos(u)), (20 * q) - (10 * p) + (v * sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2))), (10 * p) + c + (a * sin(u)), u, 0, 6.28319, v, -1, 1) → sa
  29. Surface((10 * p) + ((v * a) * cos(u)), (-10 * p) + sqrt(b^(2) - ((2 * c) + (a * sin(u)))^(2)), (10 * p) + c + (a * sin(u)), u, 0, 6.283185307179586, v, -1, 1) → sb

AI Drawing Prompt Reference

Draw two orthogonal cylinders and show their intersection. Start by drawing two perpendicular rays, one from (0, b, c) and the other from (0, -b, c), each of length 1. Mark points A' and A on these rays. Then draw two more perpendicular rays from (-a, 0, -c) and (a, 0, -c), each of length 1, and mark points B and B' on them. Next, create two cylinders, one with bases A and B and radius a, and the other with bases A' and B' and radius b. Finally, use surfaces to visualize the intersection of the two cylinders.
The prompt above was compiled from the original drawing commands and is for reference only.