Torsion of an Ellipsoid

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2016-02-16 08:41

In collections surface 3D surface 3D

This example demonstrates the calculation and visualization of the torsion exponent of an ellipsoid. It involves core mathematical concepts such as surfaces, definite integrals, angles, vector fields, and parametric equations. The construction uses GeoGebra commands like If, Surface, Segment, along with objects like cf, fb, fd, f2, f1, a, h, b, f, nt, s1, s2, s3, s4, d, e, i, texte1, texte2, texte3, texte4, texte5, c, g, j, k, l. It is suitable for university students in mathematics, physics, or engineering to understand the geometric properties and computational methods of spatial surfaces.

GeoGebra Commands & Parameters

If(cf, f2, f1) → f
Surface((a * sin((v * π / 2))), (cos((v * π / 2)) * (((h * cos(u)) * cos((((nt * 2) * π) * f(v)))) - ((b * sin(u)) * sin((((nt * 2) * π) * f(v)))))), (cos((v * π / 2)) * (((h * cos(u)) * sin((((nt * 2) * π) * f(v)))) + ((b * sin(u)) * cos((((nt * 2) * π) * f(v)))))), u, 0, π / 2, v, -1, 1) → s1
Surface((a * sin((v * π / 2))), (cos((v * π / 2)) * (((h * cos(u)) * cos((((nt * 2) * π) * f(v)))) - ((b * sin(u)) * sin((((nt * 2) * π) * f(v)))))), (cos((v * π / 2)) * (((h * cos(u)) * sin((((nt * 2) * π) * f(v)))) + ((b * sin(u)) * cos((((nt * 2) * π) * f(v)))))), u, π / 2, π, v, -1, 1) → s2
Surface((a * sin((v * π / 2))), (cos((v * π / 2)) * (((h * cos(u)) * cos((((nt * 2) * π) * f(v)))) - ((b * sin(u)) * sin((((nt * 2) * π) * f(v)))))), (cos((v * π / 2)) * (((h * cos(u)) * sin((((nt * 2) * π) * f(v)))) + ((b * sin(u)) * cos((((nt * 2) * π) * f(v)))))), u, π, (3 * π / 2), v, -1, 1) → s3
Surface((a * sin((v * π / 2))), (cos((v * π / 2)) * (((h * cos(u)) * cos((((nt * 2) * π) * f(v)))) - ((b * sin(u)) * sin((((nt * 2) * π) * f(v)))))), (cos((v * π / 2)) * (((h * cos(u)) * sin((((nt * 2) * π) * f(v)))) + ((b * sin(u)) * cos((((nt * 2) * π) * f(v)))))), u, (3 * π / 2), (2 * π), v, -1, 1) → s4
Segment((fd, 0), (fd, 1)) → i