MathPlotAI Logo MathPlotAI

Spherical Curve Families: Tennis and Basketball via Translation and Similarity

Spherical Curve Families: Tennis and Basketball via Translation and Similarity
Loading the math board and drawing, please wait…
2017-06-11 19:34
Solid GeometryTransformationParametric EquationSlider AnimationLocus
This example demonstrates generating parametric curves on a sphere through translation and similarity transformations, and observing how their geometric shapes change. Core mathematical concepts include sphere equations, parametric curves, translation, similarity (scaling) transformations, and intersections of curves with surfaces. Construction highlights: using GeoGebra's Sphere command to create the sphere, CurveCartesian for parametric curves, Plane and IntersectPath for plane-sphere intersections, and Intersect for computing points; objects include the sphere kugel, tennis curve tennis, basketball curve basket, and auxiliary planes and sliders. Suitable for high school or early college students learning solid geometry, parametric equations, and spatial transformations, and for dynamic demonstrations of curve motion on a sphere.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Sphere(A, 1.025) → tennis
  3. If(g, 0.6, 0.8) → a
  4. CurveCartesian((a * cos(t)) + (b * cos((3 * t))), (a * sin(t)) - (b * sin((3 * t))), (c * sin((2 * t))), t, 0, (2 * π)) → k_1
  5. Plane(A, (0, 0, 1), (1, 1, 0)) → e_1
  6. CurveCartesian((a * cos(t)) + (b * cos((3 * t))), (a * sin(t)) - (b * sin((3 * t))), (c * sin((2 * t))), t, 0, (2 * π)) → k_2
  7. Plane(A, (0, 0, 1), (1, -1, 0)) → e_2
  8. Sphere(A, 1.01) → basket
  9. Sphere(A, 1) → kugel
  10. IntersectPath(e_1, kugel) → e
  11. IntersectPath(e_2, kugel) → f
  12. CurveCartesian((s * cos(t)) + ((1 - s) * cos((3 * t))), (s * sin(t)) - ((1 - s) * sin((3 * t))), ((2 * sqrt((s * (1 - s)))) * sin((2 * t))), t, 0, (2 * π)) → k

AI Drawing Prompt Reference

Draw a unit sphere centered at the origin in 3D space. Generate two parametric curves on the sphere: one resembling a tennis ball seam, composed of cosine and sine terms with different frequencies, and another resembling a basketball seam, obtained by scaling and translating the first curve. Add two perpendicular planes passing through the sphere's center, intersecting the sphere to form two great circles. Include a slider to dynamically morph a third curve on the sphere, transitioning between the tennis and basketball shapes. All curves and intersections lie on or inside the sphere, creating a symmetric geometric composition.
The prompt above was compiled from the original drawing commands and is for reference only.