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Clelies Curve on Sphere

Clelies Curve on Sphere
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2020-03-14 02:33
TrigonometryCoordinate GeometrySolid Geometry
This example demonstrates the Clelies curve on a sphere. Clelies curves are a special class of curves on a sphere, involving spherical geometry and trigonometry. Using GeoGebra commands such as Sphere, CurveCartesian, and Rotation, dynamic construction and observation of these curves can be achieved. Suitable for high school mathematics education, particularly in coordinate geometry and trigonometry.

GeoGebra Commands & Parameters

  1. Sphere(O, R) → b
  2. Sphere(A, R) → j
  3. Sphere(B, R) → k
  4. Sphere(C, R) → l
  5. Sphere(D, R) → m
  6. Sphere(E, R) → n
  7. Sphere(F, R) → o
  8. CurveCartesian(x(A) + ((R * cos((3 * i))) * cos(i + Rotation)), y(A) + ((R * cos((3 * i))) * sin(i + Rotation)), z(A) + (R * sin((3 * i))), i, 0, 6.28319) → d
  9. CurveCartesian(x(D) + ((R * cos((9 * i))) * cos(i - Rotation)), y(D) + ((R * cos((9 * i))) * sin(i - Rotation)), z(D) + (R * sin((9 * i))), i, 0, 6.28319) → g
  10. CurveCartesian(x(E) + ((R * cos((5 * i))) * cos(i + Rotation)), y(E) + ((R * cos((5 * i))) * sin(i + Rotation)), z(E) + (R * sin((5 * i))), i, 0, 6.28319) → h
  11. CurveCartesian(x(B) + ((R * cos((3 / 2 * i))) * cos(i - Rotation)), y(B) + ((R * cos((3 / 2 * i))) * sin(i - Rotation)), z(B) + (R * sin((3 / 2 * i))), i, 0, (4 * 6.28319)) → e
  12. CurveCartesian(x(O) + ((R * cos((p / q * i))) * cos(i + Rotation)), y(O) + ((R * cos((p / q * i))) * sin(i + Rotation)), z(O) + (R * sin((p / q * i))), i, 0, (5 * 6.28319)) → a
  13. CurveCartesian(x(C) + ((R * cos((7 / 4 * i))) * cos(i + Rotation)), y(C) + ((R * cos((7 / 4 * i))) * sin(i + Rotation)), z(C) + (R * sin((7 / 4 * i))), i, 0, (10 * π)) → f
  14. CurveCartesian(x(F) + ((R * cos((2 * i))) * cos(i - Rotation)), y(F) + ((R * cos((2 * i))) * sin(i - Rotation)), z(F) + (R * sin((2 * i))), i, 0, (4 * 6.28319)) → i

AI Drawing Prompt Reference

Draw a sphere in 3D space with several colorful curves lying on its surface. The curves start from different points on the sphere or its center, winding around the surface with up-and-down oscillations, creating petal-like or spiral patterns. Each curve has a distinct frequency parameter, making some tightly coiled and others loosely spread, with rotation directions alternating. The overall composition is symmetric and intricate, showcasing the beauty of spherical geometry combined with trigonometric functions.
The prompt above was compiled from the original drawing commands and is for reference only.