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Twisting a Sphere

Twisting a Sphere
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2015-04-19 19:15
FunctionCoordinate GeometrySolid GeometryOptimization
In collections surface 3D surface 3D My 3D Collection
This example demonstrates the twisting transformation of a spherical surface. It involves core mathematical concepts such as surfaces, linear programming or linear optimization, triangles, rhombuses, parabolas, and addition. The construction utilizes GeoGebra commands/objects like Surface, parameters n and a, points V, b, V_0, numerical values h, c_1, d_1, ns, c, d, g, f, p, q, and objects s1 to s12, along with buttons Bouton1 to Bouton7. It is suitable for high school and above students or teachers studying solid geometry, coordinate geometry, and surface constructions.

GeoGebra Commands & Parameters

  1. Surface(((cos(u - (b * v)) + (a * sin((n * u) - (b * v)))) * cos(v)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (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)), ((sin(u - (b * v)) + (a * cos((n * u) - (b * v)))) * cos(v)), (h * q(v)), u, (22 * π / (n + 1)), (24 * π / (n + 1)), v, V_0, V) → s12

AI Drawing Prompt Reference

Draw a twisted sphere composed of twelve surface patches. Each patch is defined by parametric equations involving sine and cosine functions, with a twist parameter and an integer parameter controlling the distortion. The sphere is divided evenly into twelve segments along the latitude direction, each covering an equal angular range. The overall shape is a sphere with a spiral twisted texture, showing visible twisted stripes on the surface. Adjustable parameters change the twist intensity and form.
The prompt above was compiled from the original drawing commands and is for reference only.