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Lemniscate of Bernoulli Rational Surfaces of Revolution

Lemniscate of Bernoulli Rational Surfaces of Revolution
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2017-01-28 07:47
VectorSolid GeometryLimitParametric Equation
This example demonstrates a rational surface of revolution generated by rotating the Lemniscate of Bernoulli. It involves core mathematical concepts such as parametric equations, 3D surfaces, and limits. The construction utilizes the GeoGebra Surface command with objects c1, c2, b2, b1. Suitable for high school or university mathematics education to visualize rotational solids and parametric surfaces.

GeoGebra Commands & Parameters

  1. Surface(((((-((2 * u^(3)) + (4 * u)))) / (u^(4) + 4) * 2) * v / (1 + v^(2))), ((2 * u^(3)) - (4 * u)) / (u^(4) + 4), (((-((2 * u^(3)) + (4 * u)))) / (u^(4) + 4) * (1 - v^(2)) / (1 + v^(2))), u, (-sqrt(2)), sqrt(2), v, -1, 1) → c1
  2. Surface(((((2 * u^(3)) + (4 * u)) / (u^(4) + 4) * 2) * v / (1 + v^(2))), ((2 * u^(3)) - (4 * u)) / (u^(4) + 4), (((2 * u^(3)) + (4 * u)) / (u^(4) + 4) * (1 - v^(2)) / (1 + v^(2))), u, (-sqrt(2)), sqrt(2), v, -1, 1) → c2
  3. Surface(((2 * u^(3)) + (4 * u)) / (u^(4) + 4), ((((-((2 * u^(3)) - (4 * u)))) / (u^(4) + 4) * 2) * v / (1 + v^(2))), (((-((2 * u^(3)) - (4 * u)))) / (u^(4) + 4) * (1 - v^(2)) / (1 + v^(2))), u, (-sqrt(2)), sqrt(2), v, -1, 1) → b2
  4. Surface(((2 * u^(3)) + (4 * u)) / (u^(4) + 4), ((((2 * u^(3)) - (4 * u)) / (u^(4) + 4) * 2) * v / (1 + v^(2))), (((2 * u^(3)) - (4 * u)) / (u^(4) + 4) * (1 - v^(2)) / (1 + v^(2))), u, (-sqrt(2)), sqrt(2), v, -1, 1) → b1