MathPlotAI Logo MathPlotAI

Phantom Graph Tschirnhaus's Cubic

Phantom Graph Tschirnhaus's Cubic
Loading the math board and drawing, please wait…
2018-07-19 06:19
FunctionTrigonometryCoordinate GeometryComplex NumberGeometric Construction
In collections Phantom Graphs
This example demonstrates the geometric graph and properties of Phantom Graph Tschirnhaus's Cubic. It involves core mathematical concepts such as symmetry, real numbers, plane figures, arithmetic operations, and equilateral triangles. The construction uses GeoGebra commands like Plane, ComplexRoot, CurveCartesian, and Surface. Suitable for high school and above students to understand the geometric construction and properties of complex curves.

GeoGebra Commands & Parameters

  1. Plane((0, c, 0), (1, c, 0), (0, c, 1)) → p
  2. ComplexRoot(((3 * a) * c^(2)) - (x * (x - a)^(2))) → z_1, z_2, z_3
  3. CurveCartesian(t, sqrt(((-8 * t^(3)) + ((16 * a) * t^(2)) - ((10 * a^(2)) * t) + (2 * a^(3))) / ((3 * a))), sqrt((3 * t^(2)) - ((4 * a) * t) + a^(2)), t, -25, 25) → b
  4. CurveCartesian(t, (-sqrt(((-8 * t^(3)) + ((16 * a) * t^(2)) - ((10 * a^(2)) * t) + (2 * a^(3))) / ((3 * a)))), sqrt((3 * t^(2)) - ((4 * a) * t) + a^(2)), t, -25, 25) → e
  5. CurveCartesian(t, (-sqrt(((-8 * t^(3)) + ((16 * a) * t^(2)) - ((10 * a^(2)) * t) + (2 * a^(3))) / ((3 * a)))), (-sqrt((3 * t^(2)) - ((4 * a) * t) + a^(2))), t, -25, 25) → f
  6. CurveCartesian(t, sqrt(((-8 * t^(3)) + ((16 * a) * t^(2)) - ((10 * a^(2)) * t) + (2 * a^(3))) / ((3 * a))), (-sqrt((3 * t^(2)) - ((4 * a) * t) + a^(2))), t, -25, 25) → h
  7. CurveCartesian((2 * a / 3) - (a * cosh(t) / 3), (a * sinh(t) / sqrt(3)), t, -25, 25) → i
  8. Surface((2 * a / 3) - (a * cosh(t) / 3), w, (a * sinh(t) / sqrt(3)), t, -25, 25, w, -25, 25) → j