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Morin and Apéry Surface: Interactive Exploration of Linear Programming and Geome

Morin and Apéry Surface: Interactive Exploration of Linear Programming and Geome
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2016-09-05 01:42
Solid GeometryOptimizationParametric EquationSlider AnimationGraphing
In collections surface 3D surface 3D My 3D Collection
① This example demonstrates a complex 3D surface constructed in GeoGebra, related to the mathematical work of Morin and Apéry, likely representing a parameterized surface or a geometric visualization of an optimization problem. ② Core mathematical concepts include surface parameterization, linear programming, median value, boxplot, symmetry, and the Pythagorean theorem, all integrated into the construction. ③ Construction highlights: Based on the summary, the Surface command is used to generate the surface, and Element extracts list items; objects include parameters k, n, K and multiple points/curves (c, d, e, f, g, h, i, b, j, l, m, o, p, q, r, s), with list liste1 and numeric nc for dynamic control, likely via sliders to adjust parameters and observe surface changes. ④ Suitable for high school to university-level mathematics education, particularly for exploring intersections of geometry, algebra, and optimization, visualizing feasible regions in linear programming, or as an example in mathematical modeling and computer-aided geometric design.

GeoGebra Commands & Parameters

  1. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, π / n, (2 * π / n)) → c
  2. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (2 * π / n), (3 * π / n)) → d
  3. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (3 * π / n), (4 * π / n)) → e
  4. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (4 * π / n), (5 * π / n)) → f
  5. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (5 * π / n), (6 * π / n)) → g
  6. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (6 * π / n), (7 * π / n)) → h
  7. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (7 * π / n), (8 * π / n)) → i
  8. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, 0, π / n) → b
  9. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (8 * π / n), (9 * π / n)) → j
  10. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (9 * π / n), (10 * π / n)) → l
  11. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (10 * π / n), (11 * π / n)) → m
  12. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (11 * π / n), (12 * π / n)) → o
  13. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (12 * π / n), (13 * π / n)) → p
  14. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (13 * π / n), (14 * π / n)) → q
  15. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (14 * π / n), (15 * π / n)) → r
  16. Surface((K(u, v) * (((2 / (n - 1) * cos(u)) * cos(((n - 1) * v))) + ((sqrt(2) * sin(u)) * cos(v)))), (K(u, v) * (((2 / (n - 1) * cos(u)) * sin(((n - 1) * v))) - ((sqrt(2) * sin(u)) * sin(v)))), (K(u, v) * cos(u)), u, 0, π, v, (15 * π / n), (16 * π / n)) → s
  17. Element(liste1, n) → nc