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Chain Torsion

Chain Torsion
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2016-10-23 21:30
CalculusTransformationGeometric Construction
This example demonstrates the torsion of a chain. Key mathematical concepts include geometric construction and transformation. It uses Surface, objects r, R, k, d, torsion, rotation, etc., in GeoGebra. Suitable for high school or university geometry and calculus courses.

GeoGebra Commands & Parameters

  1. Surface((((R + (r * cos(v))) * sin(u)) * cos(rotation + (π / 2 * (1 - torsion + (torsion * sin(u / 2)))))) - ((sin(rotation + (π / 2 * (1 - torsion + (torsion * sin(u / 2))))) * r) * sin(v)), d + (((k * R) + (r * cos(v))) * cos(u)), (((R + (r * cos(v))) * sin(u)) * sin(rotation + (π / 2 * (1 - torsion + (torsion * sin(u / 2)))))) + ((cos(rotation + (π / 2 * (1 - torsion + (torsion * sin(u / 2))))) * r) * sin(v)), u, 0, (2 * π), v, (-π), π) → a
  2. Surface((((R + (r * cos(v))) * sin(u)) * cos(rotation + ((π / 2 * torsion) * (2 + sin(u / 2))))) - ((sin(rotation + ((π / 2 * torsion) * (2 + sin(u / 2)))) * r) * sin(v)), (-2 * d) + (((k * R) + (r * cos(v))) * cos(u)), (((R + (r * cos(v))) * sin(u)) * sin(rotation + ((π / 2 * torsion) * (2 + sin(u / 2))))) + ((cos(rotation + ((π / 2 * torsion) * (2 + sin(u / 2)))) * r) * sin(v)), u, 0, (2 * π), v, (-π), π) → aa
  3. Surface((((R + (r * cos(v))) * sin(u)) * cos(rotation + ((π / 2 * torsion) * (-2 + sin(u / 2))))) - ((sin(rotation + ((π / 2 * torsion) * (-2 + sin(u / 2)))) * r) * sin(v)), (2 * d) + (((k * R) + (r * cos(v))) * cos(u)), (((R + (r * cos(v))) * sin(u)) * sin(rotation + ((π / 2 * torsion) * (-2 + sin(u / 2))))) + ((cos(rotation + ((π / 2 * torsion) * (-2 + sin(u / 2)))) * r) * sin(v)), u, 0, (2 * π), v, (-π), π) → bb
  4. Surface((((R + (r * cos(v))) * sin(u)) * cos(rotation + ((π / 2 * torsion) * sin(u / 2)))) - ((sin(rotation + ((π / 2 * torsion) * sin(u / 2))) * r) * sin(v)), (((k * R) + (r * cos(v))) * cos(u)), (((R + (r * cos(v))) * sin(u)) * sin(rotation + ((π / 2 * torsion) * sin(u / 2)))) + ((cos(rotation + ((π / 2 * torsion) * sin(u / 2))) * r) * sin(v)), u, 0, (2 * π), v, (-π), π) → c
  5. Surface((((R + (r * cos(v))) * sin(u)) * cos((-rotation) + (π / 2 * (1 - torsion + (torsion * sin(((-u)) / 2)))))) - ((sin((-rotation) + (π / 2 * (1 - torsion + (torsion * sin(((-u)) / 2))))) * r) * sin(v)), (-d) + (((k * R) + (r * cos(v))) * cos(u)), (-((((R + (r * cos(v))) * sin(u)) * sin((-rotation) + (π / 2 * (1 - torsion + (torsion * sin(((-u)) / 2)))))) + ((cos((-rotation) + (π / 2 * (1 - torsion + (torsion * sin(((-u)) / 2))))) * r) * sin(v)))), u, 0, (2 * π), v, (-π), π) → b

AI Drawing Prompt Reference

Draw a chain-like twisted shape with four surfaces. First, draw a twisted chain surface using parameters R, r, k, d, torsion, and rotation. Then, draw three symmetric twisted surfaces at positions d, -2d, and 2d, using the same parameters. Finally, draw a rotated twisted surface using parameters torsion and sine functions of u/v.
The prompt above was compiled from the original drawing commands and is for reference only.