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3D Twisting Surface with Volume-Style Calculus

3D Twisting Surface with Volume-Style Calculus
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2021-01-11 05:37
Solid GeometryCalculusIntegralGeometric ConstructionSlider Animation
This example demonstrates how to generate a 3D “twisting/deformed” surface using a parameterized construction and then relate it to volume-like quantities through integral-calculus ideas. Key concepts include solid/3D geometry (e.g., sphere/rotation-style shapes and surfaces), arithmetic operations such as multiplication and roots, and integral calculus for accumulated quantities such as volume. From the provided construction summary, the surface is created with the GeoGebra command Surface, using parameter objects (e.g., n, a, b, h, V, V_0, c_1, d_1) and functional intermediates (e.g., f, g, d, p, q) to form the defining equations. Interactive elements such as Bouton1/Bouton2/Bouton3 together with points or controls (A, B) and auxiliary samples/segments (s1–s12, ns, image1/image2) support stepwise visualization and exploration. Suitable for advanced high school and early university learners studying 3D geometry and calculus via interactive models.

GeoGebra Commands & Parameters

  1. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, 0, (2 * π / (n + 1)), v, V_0, V) → s1
  2. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (2 * π / (n + 1)), (4 * π / (n + 1)), v, V_0, V) → s2
  3. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (4 * π / (n + 1)), (6 * π / (n + 1)), v, V_0, V) → s3
  4. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (6 * π / (n + 1)), (8 * π / (n + 1)), v, V_0, V) → s4
  5. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (8 * π / (n + 1)), (10 * π / (n + 1)), v, V_0, V) → s5
  6. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (10 * π / (n + 1)), (12 * π / (n + 1)), v, V_0, V) → s6
  7. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (12 * π / (n + 1)), (14 * π / (n + 1)), v, V_0, V) → s7
  8. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (14 * π / (n + 1)), (16 * π / (n + 1)), v, V_0, V) → s8
  9. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (16 * π / (n + 1)), (18 * π / (n + 1)), v, V_0, V) → s9
  10. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (18 * π / (n + 1)), (20 * π / (n + 1)), v, V_0, V) → s10
  11. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (20 * π / (n + 1)), (22 * π / (n + 1)), v, V_0, V) → s11
  12. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v + c)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v + c)), (h * q(v)), u, (22 * π / (n + 1)), (24 * π / (n + 1)), v, V_0, V) → s12

AI Drawing Prompt Reference

Create a 3D twisting surface generated by a parameterized construction, formed through rotation and deformation. Use parameters a, b, n, h, V, V_0, c, and d to define the surface equations. Adjust parameters to observe the twisting effect. Start with u values from 0 to 2π/n+1, incrementally increasing u values, and draw multiple surface segments to form the complete twisting surface.
The prompt above was compiled from the original drawing commands and is for reference only.