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Nautilus Logarithmic Spiral Parametric Construction

Nautilus Logarithmic Spiral Parametric Construction
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2023-03-15 19:27
FunctionSequenceParametric EquationGeometric ConstructionGraphing
In collections geometriadinamica.es
This GeoGebra example demonstrates how to construct a nautilus (sea shell) logarithmic spiral using parametric equations. The core mathematical concept is the equiangular (logarithmic) spiral, described by x=a*e^(b*θ)*cos(θ), y=a*e^(b*θ)*sin(θ) in Cartesian coordinates. Key construction commands include: CurveCartesian for plotting the parametric curve, Sequence for generating θ values, Ellipse and Circle for auxiliary geometric structures, Point for placing key points, and Angle for measuring spiral angles. This example is suitable for high school mathematics (coordinate geometry, parametric equations) and college calculus (curve theory, polar coordinates), serving as an excellent case study for mathematical modeling of natural phenomena.

GeoGebra Commands & Parameters

  1. Segment(Q3, Q4) → p2
  2. Point(p2) → P3
  3. Segment(P3, Q4) → p4
  4. Point(p4) → P4
  5. Segment(Q1, Q2) → p1
  6. Point(p1) → P1
  7. Segment(P1, Q2) → p3
  8. Point(p3) → P2
  9. OrthogonalLine(C, yAxis) → zz1
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz1, 1) → ZZ1
  12. Intersect(zz0, zz1, 2) → ZZ2
  13. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  14. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  15. Point(zElipse1) → XY
  16. Angle(ZZ6, C, XY) → zAngRot
  17. OrthogonalLine(C, xAxis) → zz2
  18. Intersect(zz0, zz2, 2) → ZZ3
  19. Intersect(zz0, zz2, 1) → ZZ4
  20. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  21. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  22. Point(zElipse2) → Z
  23. Angle(ZZ8, C, Z) → zAngInc
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  26. Segment(ZZ5, ZZ6) → zz4
  27. Segment(ZZ7, ZZ8) → zz3
  28. Intersect(zElipse2, zElipse1, 1) → zPunto1
  29. Intersect(zElipse2, zElipse1, 4) → zPunto2
  30. Segment(zPunto1, zPunto2) → zSeg2
  31. CurveCartesian((r * ((((exp((f * u)) * (d + (a * cos(α)))) * cos((c * u))) * sin(zAngRot)) + (((exp((f * u)) * (d + (a * cos(α)))) * sin((c * u))) * cos(zAngRot)))), (r * ((((((-exp((f * u))) * (d + (a * cos(α)))) * cos((c * u))) * cos(zAngRot)) * sin(zAngInc)) + ((((exp((f * u)) * (d + (a * cos(α)))) * sin((c * u))) * sin(zAngRot)) * sin(zAngInc)) + ((exp((f * u)) * ((-1.4 * e) + (b * sin(α)))) * cos(zAngInc)))), u, u1, u2) → curva2
  32. CurveCartesian((r * ((((exp((f * β)) * (d + (a * cos(v)))) * cos((c * β))) * sin(zAngRot)) + (((exp((f * β)) * (d + (a * cos(v)))) * sin((c * β))) * cos(zAngRot)))), (r * ((((((-exp((f * β))) * (d + (a * cos(v)))) * cos((c * β))) * cos(zAngRot)) * sin(zAngInc)) + ((((exp((f * β)) * (d + (a * cos(v)))) * sin((c * β))) * sin(zAngRot)) * sin(zAngInc)) + ((exp((f * β)) * ((-1.4 * e) + (b * sin(v)))) * cos(zAngInc)))), v, v1, v2) → a1
  33. CurveCartesian((r * ((((exp((f * max)) * (d + (a * cos(v)))) * cos((c * max))) * sin(zAngRot)) + (((exp((f * max)) * (d + (a * cos(v)))) * sin((c * max))) * cos(zAngRot)))), (r * ((((((-exp((f * max))) * (d + (a * cos(v)))) * cos((c * max))) * cos(zAngRot)) * sin(zAngInc)) + ((((exp((f * max)) * (d + (a * cos(v)))) * sin((c * max))) * sin(zAngRot)) * sin(zAngInc)) + ((exp((f * max)) * ((-1.4 * e) + (b * sin(v)))) * cos(zAngInc)))), v, v1, v2) → curva1
  34. Sequence(CurveCartesian[(r * ((((exp((f * u)) * (d + (a * cos(v)))) * cos((c * u))) * sin(zAngRot)) + (((exp((f * u)) * (d + (a * cos(v)))) * sin((c * u))) * cos(zAngRot)))), (r * ((((((-exp((f * u))) * (d + (a * cos(v)))) * cos((c * u))) * cos(zAngRot)) * sin(zAngInc)) + ((((exp((f * u)) * (d + (a * cos(v)))) * sin((c * u))) * sin(zAngRot)) * sin(zAngInc)) + ((exp((f * u)) * ((-1.4 * e) + (b * sin(v)))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r * ((((exp((f * u)) * (d + (a * cos(v)))) * cos((c * u))) * sin(zAngRot)) + (((exp((f * u)) * (d + (a * cos(v)))) * sin((c * u))) * cos(zAngRot)))), (r * ((((((-exp((f * u))) * (d + (a * cos(v)))) * cos((c * u))) * cos(zAngRot)) * sin(zAngInc)) + ((((exp((f * u)) * (d + (a * cos(v)))) * sin((c * u))) * sin(zAngRot)) * sin(zAngInc)) + ((exp((f * u)) * ((-1.4 * e) + (b * sin(v)))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Draw a nautilus-shaped logarithmic spiral, including a central circle, two ellipses, two lines connecting the ellipse centers, two intersection points, two angle measurement points, and a spiral curve. The spiral curve is defined by two parametric equations, describing the shape and size of the spiral.
The prompt above was compiled from the original drawing commands and is for reference only.