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Intersection of Cone and Cylinder by Patrick Clément

Intersection of Cone and Cylinder by Patrick Clément
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2015-08-22 22:24
Solid GeometryOptimizationGeometric Construction
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 a cone and a cylinder, involving concepts like solid geometry, parametric surfaces, and spatial intersections. Construction uses GeoGebra commands such as Cylinder, InfiniteCone, and Surface to visualize the 3D intersection. Suitable for high school students or college freshmen learning spatial geometry and parametric modeling.

GeoGebra Commands & Parameters

  1. Point(Segment[(0, 0, d), (0, 15, d)]) → A
  2. Cylinder(A, Mirror[A, zAxis], a) → e, f, g, h
  3. InfiniteCone((0, 0, 0), zAxis, atan(k)) → i
  4. If(d < a, 1, -100) → lminb
  5. If(d < a, 100, 1) → lmaxb
  6. Surface(((v * a) * cos(u)), ((-v) * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, t4, t2, v, lminb, lmaxb) → con14
  7. Surface(((v * a) * cos(u)), (v * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, t4, t2, v, lminb, lmaxb) → con13
  8. If(d < (-a), 1, 100) → lmaxh
  9. If(d < (-a), -100, 1) → lminh
  10. Surface(((v * a) * cos(u)), ((-v) * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, t1, t3, v, lminh, lmaxh) → con12
  11. Surface(((v * a) * cos(u)), (v * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, t1, t3, v, lminh, lmaxh) → con11
  12. CurveCartesian((a * cos(t)), sqrt((k^(2) * (d + (a * sin(t)))^(2)) - (a^(2) * cos(t)^(2))), t, -3.141592653589793, 3.141592653589793) → d_1
  13. Min(Sequence[arg(d_1(t)), t, 0, 6.28319, (2 * π / n)]) → ttmin
  14. Surface(((v * a) * cos(u)), ((-v) * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, ttmin + π / 2, (5 * π / 2) - ttmin, v, 1, 15) → con2
  15. Surface(((v * a) * cos(u)), (v * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, π / 2 - ttmin, π / 2 + ttmin, v, -15, 1) → con3
  16. Surface(((v * a) * cos(u)), ((-v) * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, π / 2 - ttmin, π / 2 + ttmin, v, -15, 1) → con4
  17. Surface(((k * v) * cos(u)), ((k * v) * sin(u)), v, u, (-ttmin), ttmin, v, -15, 15) → con5
  18. Surface(((k * v) * cos(u)), ((k * v) * sin(u)), v, u, π - ttmin, π + ttmin, v, -15, 15) → con6
  19. Surface((a * cos(u)), v, d + (a * sin(u)), u, t2, t1, v, -10, 10) → cyl3
  20. Surface((a * cos(u)), v, d + (a * sin(u)), u, t3, (2 * π) + t4, v, -10, 10) → cyl4
  21. CurveCartesian((a * cos(t)), sqrt((k^(2) * (d + (a * sin(t)))^(2)) - (a^(2) * cos(t)^(2))), d + (a * sin(t)), t, -3.141592653589793, 3.141592653589793) → b
  22. CurveCartesian((a * cos(t)), (-sqrt((k^(2) * (d + (a * sin(t)))^(2)) - (a^(2) * cos(t)^(2)))), d + (a * sin(t)), t, -3.141592653589793, 3.141592653589793) → c
  23. Surface((a * cos(u)), v + sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2))), d + (a * sin(u)), u, -3.141592653589793, 3.141592653589793, v, 0, 10) → cyl1
  24. Surface((a * cos(u)), v - sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2))), d + (a * sin(u)), u, -3.14159, 3.141592653589793, v, -10, 0) → cyl2
  25. Surface(((v * a) * cos(u)), (v * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))), u, ttmin + π / 2, (5 * π / 2) - ttmin, v, 1, 15) → con1
  26. Surface(p + (a * cos(u)), (v * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (d * (1 - p / 5)) + (a * sin(u)), u, -3.14159, 3.141592653589793, v, -1, 1) → icyl
  27. Surface(q + (a * cos(u)), ((-sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))) * (1 - v)) - (v * sqrt((k^(2) * (d + (a * sin((-u))))^(2)) - (a^(2) * cos((-u))^(2)))), ((d + (a * sin(u))) * (1 - v)) + (v * (d + (a * sin((-u))))) - (d * p / 5), u, 0, 3.141592653589793, v, 0, 1) → icon3
  28. Surface(q + (a * cos(u)), (sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2))) * (1 - v)) + (v * sqrt((k^(2) * (d + (a * sin((-u))))^(2)) - (a^(2) * cos((-u))^(2)))), ((d + (a * sin(u))) * (1 - v)) + (v * (d + (a * sin((-u))))) - (d * p / 5), u, 0, 3.141592653589793, v, 0, 1) → icon6
  29. Surface(q + ((v * a) * cos(u)), (v * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))) - (d * p / 5), u, -3.14159, 3.141592653589793, v, 0, 1) → icon1
  30. Surface(q + ((v * a) * cos(u)), ((-v) * sqrt((k^(2) * (d + (a * sin(u)))^(2)) - (a^(2) * cos(u)^(2)))), (v * (d + (a * sin(u)))) - (d * p / 5), u, -3.14159, 3.141592653589793, v, 0, 1) → icon2
  31. Surface(q + ((a * cos((t1 * (1 - u)) + (u * t3))) * (1 - v)) + ((v * a) * cos((t2 * (1 - u)) + (u * t4))), (sqrt((k^(2) * (d + (a * sin((t1 * (1 - u)) + (u * t3))))^(2)) - (a^(2) * cos((t1 * (1 - u)) + (u * t3))^(2))) * (1 - v)) + (v * sqrt((k^(2) * (d + (a * sin((t2 * (1 - u)) + (u * t4))))^(2)) - (a^(2) * cos((t2 * (1 - u)) + (u * t4))^(2)))), ((d + (a * sin((t1 * (1 - u)) + (u * t3)))) * (1 - v)) + (v * (d + (a * sin((t2 * (1 - u)) + (u * t4))))) - (d * p / 5), u, 0, 1, v, 0, 1) → icon4
  32. Surface(q + ((a * cos((t1 * (1 - u)) + (u * t3))) * (1 - v)) + ((v * a) * cos((t2 * (1 - u)) + (u * t4))), ((-sqrt((k^(2) * (d + (a * sin((t1 * (1 - u)) + (u * t3))))^(2)) - (a^(2) * cos((t1 * (1 - u)) + (u * t3))^(2)))) * (1 - v)) - (v * sqrt((k^(2) * (d + (a * sin((t2 * (1 - u)) + (u * t4))))^(2)) - (a^(2) * cos((t2 * (1 - u)) + (u * t4))^(2)))), ((d + (a * sin((t1 * (1 - u)) + (u * t3)))) * (1 - v)) + (v * (d + (a * sin((t2 * (1 - u)) + (u * t4))))) - (d * p / 5), u, 0, 1, v, 0, 1) → icon5

AI Drawing Prompt Reference

Create a 3D geometric figure showing the intersection between a cylinder and an infinite cone in space. The cylinder has its base center positioned at an offset along the z-axis with radius a, while the cone is positioned with its vertex at the origin and axis along the z-axis, with its slant angle determined by parameter k. The visualization should clearly display the closed intersection curves where the two surfaces meet, along with multiple smooth transitional surfaces filling the intersecting regions, illustrating the complete spatial intersection between the cylinder and cone.
The prompt above was compiled from the original drawing commands and is for reference only.