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Phantom Graph Strophoid

Phantom Graph Strophoid
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2018-07-24 03:37
Coordinate GeometryComplex NumberGraphing
In collections Phantom Graphs
This example demonstrates the construction and properties of the Phantom Graph Strophoid, involving key mathematical concepts such as coordinate geometry, complex roots, parametric equations, and geometric transformations. The construction utilizes GeoGebra commands like Plane, ComplexRoot, CurveCartesian, and Segment, making it suitable for high school and advanced learners exploring complex function graphs and transformations.

GeoGebra Commands & Parameters

  1. Plane((0, c, 0), (1, c, 0), (0, c, 1)) → p
  2. ComplexRoot((c^(2) * (a + x)) - (x^(2) * (a - x))) → z_1, z_2, z_3
  3. CurveCartesian(t, sqrt((((-4 * a) * t^(2)) + (a^(2) * t) + (4 * t^(3))) / (a - t)), sqrt(((-a^(3)) * t) + ((2 * a^(2)) * t^(2)) - t^(4)) / (a - t), t, -25, 25) → d
  4. CurveCartesian(t, sqrt((((-4 * a) * t^(2)) + (a^(2) * t) + (4 * t^(3))) / (a - t)), ((-sqrt(((-a^(3)) * t) + ((2 * a^(2)) * t^(2)) - t^(4)))) / (a - t), t, -25, 25) → e
  5. CurveCartesian(t, (-sqrt((((-4 * a) * t^(2)) + (a^(2) * t) + (4 * t^(3))) / (a - t))), ((-sqrt(((-a^(3)) * t) + ((2 * a^(2)) * t^(2)) - t^(4)))) / (a - t), t, -25, 25) → f
  6. CurveCartesian(t, (-sqrt((((-4 * a) * t^(2)) + (a^(2) * t) + (4 * t^(3))) / (a - t))), sqrt(((-a^(3)) * t) + ((2 * a^(2)) * t^(2)) - t^(4)) / (a - t), t, -25, 25) → h
  7. Segment(z_1, A1) → S1
  8. Segment(z_2, A2) → S2
  9. Segment(z_3, A3) → S3