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Phantom Graph of Pear-shaped Quartic

Phantom Graph of Pear-shaped Quartic
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2018-07-20 10:18
FunctionCoordinate GeometryComplex Number
In collections Phantom Graphs
This example demonstrates the phantom graph of a pear-shaped quartic. It involves the properties of quartic curves and the concept of complex roots. Using GeoGebra commands such as Plane, ComplexRoot, and CurveCartesian, this special curve can be constructed. Suitable for high school mathematics education and research.

GeoGebra Commands & Parameters

  1. Plane((0, c, 0), (1, c, 0), (0, c, 1)) → p
  2. ComplexRoot((b^(2) * c^(2)) - (x^(3) * (a - x))) → z_1, z_2, z_3, z_4
  3. CurveCartesian(t, sqrt((((-96 * a) * t^(5)) + ((48 * a^(2)) * t^(4)) - ((8 * a^(3)) * t^(3)) + (64 * t^(6))) / (((-8 * a) * t) + a^(2) + (16 * t^(2)))) / abs(b), (sqrt((3 * a^(2)) - ((16 * a) * t) + (16 * t^(2))) * abs(t) / (a - (4 * t))), t, -25, 25) → e
  4. CurveCartesian(t, sqrt((((-96 * a) * t^(5)) + ((48 * a^(2)) * t^(4)) - ((8 * a^(3)) * t^(3)) + (64 * t^(6))) / (((-8 * a) * t) + a^(2) + (16 * t^(2)))) / abs(b), ((-sqrt((3 * a^(2)) - ((16 * a) * t) + (16 * t^(2)))) * abs(t) / (a - (4 * t))), t, -25, 25) → f
  5. CurveCartesian(t, ((-sqrt((((-96 * a) * t^(5)) + ((48 * a^(2)) * t^(4)) - ((8 * a^(3)) * t^(3)) + (64 * t^(6))) / (((-8 * a) * t) + a^(2) + (16 * t^(2)))))) / abs(b), (sqrt((3 * a^(2)) - ((16 * a) * t) + (16 * t^(2))) * abs(t) / (a - (4 * t))), t, -25, 25) → h
  6. CurveCartesian(t, ((-sqrt((((-96 * a) * t^(5)) + ((48 * a^(2)) * t^(4)) - ((8 * a^(3)) * t^(3)) + (64 * t^(6))) / (((-8 * a) * t) + a^(2) + (16 * t^(2)))))) / abs(b), ((-sqrt((3 * a^(2)) - ((16 * a) * t) + (16 * t^(2)))) * abs(t) / (a - (4 * t))), t, -25, 25) → i

AI Drawing Prompt Reference

Draw the phantom graph of a pear-shaped quartic curve. First, define a plane p, then calculate the complex roots z1 to z4, and finally use Cartesian equations to draw curves e, f, h, and i, forming the phantom graph of the pear-shaped quartic curve.
The prompt above was compiled from the original drawing commands and is for reference only.