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Three Tori Morphing

Three Tori Morphing
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2019-04-25 23:59
Solid GeometryCalculusTransformationParametric EquationSlider Animation
In collections GeoGebra 3D & AR: PreCalc & Calculus Resources
① This example demonstrates three tori (torus surfaces) continuously morphing and interpenetrating in 3D space, with adjustable parameters to observe smooth shape changes. ② Core mathematical concepts include parametric surfaces, torus geometry, transformations and deformations in 3D space, and dynamic visualization via slider-controlled parameters. ③ Construction highlights: using GeoGebra's Surface command, building torus equations from objects r, R, k, a, b, c, and enhancing solid appearance with the Filling property; sliders allow real-time adjustment of morphing intensity. ④ Suitable for high school to university-level mathematics teaching, especially for solid geometry, calculus (parametric surfaces), and dynamic geometry demonstrations, as well as artistic form generation.

GeoGebra Commands & Parameters

  1. Surface(((R + (r * cos(v))) * cos(u)), ((R + (r * cos(v))) * sin(u)), (r * sin(v)) + (k * u), u, 0, (2 * π), v, 0, (2 * π)) → a
  2. Surface(((R + (r * cos(v))) * cos(u)), ((R + (r * cos(v))) * sin(u)), (r * sin(v)) + (k * u) + (2 * (π * k)), u, 0, ((2 * π) * k), v, 0, (2 * π)) → b
  3. Surface(((R + (r * cos(v))) * cos(u)), ((R + (r * cos(v))) * sin(u)), (r * sin(v)) + (k * u) - (2 * (π * k)), u, 0, (2 * π), v, 0, (2 * π)) → c

AI Drawing Prompt Reference

Create three torus surfaces that continuously morph and interpenetrate in 3D space. Each torus has a major radius and a minor radius, with a helical twist along its surface, giving a dynamic flowing appearance. The three tori are offset vertically—one central, one shifted upward, and one shifted downward—so they intersect and weave through each other. Adjustable parameters control the twist intensity and vertical offsets, allowing smooth blending and separation. Use solid or semi-transparent filling to enhance the 3D solid look, suitable for demonstrating parametric surface transformations.
The prompt above was compiled from the original drawing commands and is for reference only.