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Geometric Construction of a Cylinder Net

Geometric Construction of a Cylinder Net
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2024-03-13 21:01
CircleSolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates how to construct a cylinder’s net, unfolding the lateral surface and arranging the circular bases into a consistent planar pattern. It helps visualize how 3D cylinder geometry translates into a 2D layout and highlights the geometric constraints that must remain valid after unfolding. Key math ideas include cylinder geometry, circle and circumference relations, intersection-based construction, and using rotations to position the net’s boundary components. Construction highlights: GeoGebra tools Point and Circle create key points and circles; Cylinder and Intersect generate cylinder-related intersection points/curves; Line and OrthogonalLine define boundary lines and perpendicular constraints; Rotate is used to place components correctly for unfolding; If selects construction branches under conditions, and Segment forms the final net outline. The model uses parameters such as θ, r, ϕ and includes multiple computed objects (A, B, C, D, …) plus interactive controls (button1, button2). Suitable for high-school advanced geometry, introductory solid-geometry instruction, and classroom demonstrations of nets/unfolding concepts.

GeoGebra Commands & Parameters

  1. Point(yAxis) → A
  2. Point(yAxis) → B
  3. Circle(A, B) → c
  4. Cylinder(c, 3) → b, e, f
  5. Intersect(zAxis, e) → C,
  6. Line(C, xAxis) → a
  7. OrthogonalLine(D, xOyPlane) → g
  8. Circle(g, C) → k
  9. Rotate(Rotate[e, ϕ, a], ϕ, xAxis) → e'
  10. Rotate(Rotate[e, ϕ, a], ϕ, xAxis) → d
  11. Circle(A, B) → h
  12. If(t < 1, ((r * sin((-θ))), ((r * (1 - cos((-θ)))) * cos(ϕ)) - (3 * sin(ϕ)), (3 * cos(ϕ)) + ((r * (1 - cos((-θ)))) * sin(ϕ))), ((-pi), (-3 * sin(ϕ)), (3 * cos(ϕ)))) → E
  13. If(t < 1, ((r * sin(θ)), ((r * (1 - cos(θ))) * cos(ϕ)) - (3 * sin(ϕ)), (3 * cos(ϕ)) + ((r * (1 - cos(θ))) * sin(ϕ))), (pi, (-3 * sin(ϕ)), (3 * cos(ϕ)))) → F
  14. If(t < 1, ((r * sin((-θ))), ((r * (1 - cos((-θ)))) * cos(ϕ)), ((r * (1 - cos((-θ)))) * sin(ϕ))), (-3.141592653589793, 0, 0)) → G
  15. If(t < 1, ((r * sin(θ)), ((r * (1 - cos(θ))) * cos(ϕ)), ((r * (1 - cos(θ))) * sin(ϕ))), (3.141592653589793, 0, 0)) → H
  16. Segment(E, G) → m
  17. Segment(F, H) → n
  18. Line((0, 1, 0), zAxis) → p
  19. Circle(p, B) → q
  20. Circle(p, B) → s
  21. If(t < 1, Surface[(r * sin((u * θ))), ((r * (1 - cos((u * θ)))) * cos(ϕ)) - (v * sin(ϕ)), (v * cos(ϕ)) + ((r * (1 - cos((u * θ)))) * sin(ϕ)), u, -1, 1, v, 0, 3], Surface[(pi * u), ((-v) * sin(ϕ)), (v * cos(ϕ)), u, -1, 1, v, 0, 3]) → i
  22. If(t < 1, CurveCartesian[(r * sin((u * θ))), ((r * (1 - cos((u * θ)))) * cos(ϕ)), ((r * (1 - cos((u * θ)))) * sin(ϕ)), u, -1, 1], CurveCartesian[(pi * u), 0, 0, u, -1, 1]) → l
  23. If(t < 1, CurveCartesian[(r * sin((u * θ))), ((r * (1 - cos((u * θ)))) * cos(ϕ)) - (3 * sin(ϕ)), (3 * cos(ϕ)) + ((r * (1 - cos((u * θ)))) * sin(ϕ)), u, -1, 1], CurveCartesian[(pi * u), (-3 * sin(ϕ)), (3 * cos(ϕ)), u, -1, 1]) → j

AI Drawing Prompt Reference

Draw a cylinder net with two circular bases and a side unfolding. Start by creating two points A and B, then draw a circle c with them as centers. Next, draw a cylinder b, e, f with circle c as the base. Use rotation and intersection to generate the boundary lines and circle k for the side unfolding. Finally, use conditional statements to draw line segments m and n, and add circles q and s, and surfaces i and j to complete the net.
The prompt above was compiled from the original drawing commands and is for reference only.