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True Spirograph

True Spirograph
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2020-12-02 05:43
Coordinate GeometryTransformationParametric Equation
This example demonstrates the true Spirograph patterns. It involves core mathematical concepts such as parametric curves and geometric transformations. Using GeoGebra commands like CurveCartesian, Vector, Sequence, Segment, Midpoint, Circle, complex Spirograph patterns can be constructed. Suitable for high school math education and students exploring parametric equations and geometric transformations.

GeoGebra Commands & Parameters

  1. CurveCartesian(((R + (0.01 * cos((n_1 * θ)))) * cos(θ)), ((R + (0.01 * cos((n_1 * θ)))) * sin(θ)), θ, 0, 6.283185307179586) → E_1
  2. CurveCartesian(x(C_{1}) + ((r + (0.01 * cos((n_2 * θ)))) * cos(θ + ((1 - R / r) * t))), y(C_{1}) + ((r + (0.01 * cos((n_2 * θ)))) * sin(θ + ((1 - R / r) * t))), θ, 0, 6.283185307179586) → E_2
  3. Vector(C_{1}, A) → u
  4. Sequence(Segment[C_{1}, (x(C_{1}) + ((r + (0.01 * cos((n_2 * θ)))) * cos(θ + ((1 - R / r) * t))), y(C_{1}) + ((r + (0.01 * cos((n_2 * θ)))) * sin(θ + ((1 - R / r) * t))))], θ, 0, 6.28319, 0.4487989505128276) → rayons
  5. Segment(A, C_{1}) → f
  6. CurveCartesian(x(O) + ((R - r) * cos(i)) + (r * cos(((1 + w1) * i))), y(O) + ((R - r) * sin(i)) + (r * sin(((w1 + 1) * i))), i, 0, t) → b
  7. Midpoint(C_{1}, A) → C
  8. Midpoint(C, A) → B
  9. CurveCartesian(x(O) + ((R - r) * cos(i)) + ((0.5 * r) * cos(((1 + w1) * i))), y(O) + ((R - r) * sin(i)) + ((0.5 * r) * sin(((w1 + 1) * i))), i, 0, t) → e
  10. CurveCartesian(x(O) + ((R - r) * cos(i)) + ((0.75 * r) * cos(((1 + w1) * i))), y(O) + ((R - r) * sin(i)) + ((0.75 * r) * sin(((w1 + 1) * i))), i, 0, t) → c
  11. Midpoint(C_{1}, C) → D
  12. CurveCartesian(x(O) + ((R - r) * cos(i)) + ((0.25 * r) * cos(((1 + w1) * i))), y(O) + ((R - r) * sin(i)) + ((0.25 * r) * sin(((w1 + 1) * i))), i, 0, t) → d
  13. CurveCartesian(((R + (0.01 * cos((n_1 * θ)))) * cos(θ)), ((R + (0.01 * cos((n_1 * θ)))) * sin(θ)), θ, 0, 6.28319) → E_3
  14. CurveCartesian(x(O) + ((R - r) * cos(i)) + ((k * r) * cos(((1 + w1) * i))), y(O) + ((R - r) * sin(i)) + ((k * r) * sin(((w1 + 1) * i))), i, 0, t) → o
  15. Circle(O, R + 0.125) → g

AI Drawing Prompt Reference

Draw a true spirograph pattern, showcasing parametric curves and geometric transformations. Start by drawing a base circle, then generate complex spiral lines using multiple parametric equations that rotate around the circle's center and overlap each other, creating a rich pattern. Use different radii and angles in the parametric equations to draw multiple curves that radiate from the center, rotate around the circle, and interweave, forming a complex spirograph design.
The prompt above was compiled from the original drawing commands and is for reference only.