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Phantom Graph - Devil's Curve

Phantom Graph - Devil's Curve
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2018-07-19 06:17
FunctionCoordinate GeometryGraphing
In collections Phantom Graphs
This example demonstrates the Devil's Curve, a special type of curve with the equation (x^2+y^2-1)^3-x^2y^3=0. Using GeoGebra commands such as Plane, ComplexRoot, and CurveCartesian, this curve can be constructed. Suitable for high school mathematics education and learning coordinate geometry.

GeoGebra Commands & Parameters

  1. Plane((0, c, 0), (1, c, 0), (0, c, 1)) → p
  2. ComplexRoot(c^(4) - x^(4) + (a * c^(2)) + (b * x^(2))) → z_1, z_2, z_3, z_4
  3. CurveCartesian(0, sqrt(((-a) + sqrt(a^(2) + ((4 * b) * t^(2)) + (4 * t^(4)))) / 2), t, t, -50, 50) → e
  4. CurveCartesian(0, (-sqrt(((-a) + sqrt(a^(2) + ((4 * b) * t^(2)) + (4 * t^(4)))) / 2)), t, t, -50, 50) → f
  5. CurveCartesian(t, sqrt(((-a) + sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2), sqrt(t^(2) - b / 2), t, -100, 100) → h
  6. CurveCartesian(t, sqrt(((-a) - sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2), sqrt(t^(2) - b / 2), t, -100, 100) → i
  7. CurveCartesian(t, sqrt(((-a) + sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2), (-sqrt(t^(2) - b / 2)), t, -100, 100) → l
  8. CurveCartesian(t, sqrt(((-a) - sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2), (-sqrt(t^(2) - b / 2)), t, -100, 100) → m
  9. CurveCartesian(t, (-sqrt(((-a) + sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2)), sqrt(t^(2) - b / 2), t, -100, 100) → n
  10. CurveCartesian(t, (-sqrt(((-a) + sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2)), (-sqrt(t^(2) - b / 2)), t, -100, 100) → o
  11. CurveCartesian(t, (-sqrt(((-a) - sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2)), sqrt(t^(2) - b / 2), t, -100, 100) → q
  12. CurveCartesian(t, (-sqrt(((-a) - sqrt((-16 * t^(4)) + ((8 * b) * t^(2)) + a^(2) - b^(2))) / 2)), (-sqrt(t^(2) - b / 2)), t, -100, 100) → j

AI Drawing Prompt Reference

Draw the Devil's Curve defined by the equation (x^2+y^2-1)^3-x^2y^3=0. Start by drawing a plane, then use parametric equations to plot the curve, which consists of four main parts representing the four branches of the curve.
The prompt above was compiled from the original drawing commands and is for reference only.