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Surface Area and Critical Points of the Octa-point Model's Functional Surface

Surface Area and Critical Points of the Octa-point Model's Functional Surface
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2019-03-11 00:32
FunctionCalculusOptimizationParametric Equation
In collections ₉Extreme Verteilung der 8 Punkten auf der Kugel
This example demonstrates the variation of the functional surface area of the octa-point model as parameters α and θ change. Core mathematical concepts include functions, derivatives, surface area, critical points, and optimization problems. It is constructed using GeoGebra commands/objects like Function, Derivative, Plane, Polygon, Textfield, and Point, as summarized. Suitable for high school students and math enthusiasts studying calculus, geometric constructions, and optimization problems.

GeoGebra Commands & Parameters

  1. Function(GFläche(u, v), u, -1°, 60°, v, -1°, 90°) → m
  2. Derivative(GFläche(x, y), x) → GFläche_x
  3. Derivative(GFläche(x, y), y) → GFläche_y
  4. Plane(z = z0) → Plane0
  5. Polygon({(-1°, -1°), (60°, -1°), (60°, 90°), (-1°, 90°), (-1°, -1°)}) → Poly_{αθ}
  6. Textfield(Δθ) → InputBox1
  7. Point(xAxis) → Ca4
  8. Plane(z = 1.014575944622939) → Pl_{Lö2}
  9. Plane(z = 1) → Pl_{Lö1}

AI Drawing Prompt Reference

Draw a parametric surface showing how its surface area varies with parameters α and θ, defined over -1° to 60° and -1° to 90°, along with its two partial derivatives, and mark the critical points. Also, represent the parameter ranges of α and θ with a polygon on a plane, include an input box, and place a point on the x-axis.
The prompt above was compiled from the original drawing commands and is for reference only.