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Black Hole: Dynamic 3D Surface and Geometric Intersections

Black Hole: Dynamic 3D Surface and Geometric Intersections
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2020-01-18 01:34
FunctionSequenceSolid GeometryParametric EquationGraphing
① This GeoGebra example constructs a dynamic 3D visualization named "Black Hole," showing a funnel-like curved surface interacting with spheres, curves, and intersection points in real time. ② Core mathematical concepts include parametric surfaces (CurveCartesian), spheres (Sphere), sequences (Sequence), and functions (Function), with implicit use of the Pythagorean theorem for distance calculations and orthocenter (Orthocenter) elements in spatial geometry. ③ Construction highlights: uses CurveCartesian to define parametric curves, Sphere to create spheres, Join to connect objects, Sequence to generate point sets or curve families, and Function to control dynamic variables; objects include t, φ, A, b, c, liste1 through liste6, Vitesse, O, d, and images image1 and image2, driven by sliders or speed parameters to animate intersections and nesting effects. ④ Suitable for high school to undergraduate mathematics education, especially for exploring analytic geometry, solid geometry, and parametric equations, as well as for mathematical modeling or creative math art, helping students understand 3D surface construction and dynamic geometric intersections.

GeoGebra Commands & Parameters

  1. CurveCartesian((r * cos(φ)), (r * sin(φ)), -1 / sqrt(((r * cos(φ)))^(2) + ((r * sin(φ)))^(2)), r, 0.1, 3) → b
  2. Sphere(A, 0.5) → c
  3. Join(Sequence[Sequence[(((t - j) * cos(i)), ((t - j) * sin(i)), -1 / sqrt((((t - j) * cos(i)))^(2) + (((t - j) * sin(i)))^(2))), i, 1, 6.28319, 0.62832], j, 3, 7]) → liste2
  4. Sequence(((t * cos(i)), (t * sin(i)), 1 / sqrt(((t * cos(i)))^(2) + ((t * sin(i)))^(2))), i, 0.1, 6.28319, 1.25664) → liste1
  5. Function(-1 / sqrt(x^(2) + y^(2)), -3, 3, -3, 3) → a
  6. Sequence(CurveCartesian[x(O) + s, y(O) + r, z(O) + a(s, r), s, -3, 3], r, -3, 3) → liste3
  7. Sequence(CurveCartesian[x(O) + r, y(O) + s, z(O) + a(r, s), s, -3, 3], r, -3, 3) → liste4
  8. Sequence(Sphere[Element[liste2, i], 0.1], i, 1, Length[liste2]) → liste6
  9. Sequence(Sphere[Element[liste1, i], 0.1], i, 1, Length[liste1]) → liste5

AI Drawing Prompt Reference

Create a dynamic 3D black hole visualization: a funnel-shaped parametric surface with a central depression. Place multiple small spheres inside and around the surface, arranged along spiral curves or grid lines to create nested layers. Overlay two sets of orthogonal curve grids on the surface, and distribute small spheres along these curves. Animate the scene with sliders or speed parameters to show real-time intersections and motion among the surface, spheres, and curves.
The prompt above was compiled from the original drawing commands and is for reference only.