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Infinite “Mystery Curves” from Complex Numbers & Parametric Iteration

Infinite “Mystery Curves” from Complex Numbers & Parametric Iteration
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2021-04-04 19:06
Complex NumberParametric EquationSlider AnimationLocusGraphing
This example visualizes an “infinite mystery curves” effect by iteratively generating and accumulating many parameterized curves, revealing how intricate patterns emerge through stepwise refinement. It is built around complex numbers (tracking real/imaginary parts), parametric curve plotting (Cartesian rendering via CurveCartesian), and geometric interpretation of curve families (related to conic/trajectory-style shape composition). The construction key points include using GeoGebra commands Sum and CurveCartesian, and orchestrating iteration with interactive controls (button1/button2/button3) and parameters such as n, speed, p, f, val, and Cols. It maintains and updates complex-related quantities split into real and imaginary components (MistRe/MistIm and NextMistRe/NextMistIm), along with progression variables (Lt, Lexp, Lr and their next-step counterparts). Suitable for secondary-to-university learners exploring complex-number geometry, parametric plotting, and iterative/accumulative construction in math visualization.

GeoGebra Commands & Parameters

  1. Sum(Zip[real(((r * exp((ί * t))) * exp(((ί * x) * m)))), r, Lr, t, Lt, m, Lexp]) → MistRe
  2. Sum(Zip[imaginary(((r * exp((ί * t))) * exp(((ί * x) * m)))), r, Lr, t, Lt, m, Lexp]) → MistIm
  3. Sum(Zip[real(((r * exp((ί * t))) * exp(((ί * x) * m)))), r, NextLr, t, NextLt, m, NextLexp]) → NextMistRe
  4. Sum(Zip[imaginary(((r * exp((ί * t))) * exp(((ί * x) * m)))), r, NextLr, t, NextLt, m, NextLexp]) → NextMistIm
  5. CurveCartesian((MistRe(t) * val) + (NextMistRe(t) * (1 - val)), (MistIm(t) * val) + (NextMistIm(t) * (1 - val)), t, 0, (2 * π)) → c

AI Drawing Prompt Reference

Draw a family of infinite mystery curves generated by complex-number iteration. Using parameter t from 0 to 2π, accumulate the real and imaginary parts of multiple complex terms (each combining radius, angle, and exponential factor) into two coordinate components. Then interpolate between two such accumulated results using a blending parameter to form a smooth closed curve. The curve shows intricate intertwined paths that evolve with iteration count and blending, revealing layered nested geometric patterns reminiscent of trajectory families or conic overlays.
The prompt above was compiled from the original drawing commands and is for reference only.