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Geometry of the Conch Shell

Geometry of the Conch Shell
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2023-03-15 19:31
CircleParametric EquationGeometric Construction
In collections geometriadinamica.es
This GeoGebra example demonstrates the geometry of the conch shell. It involves core mathematical concepts such as geometric shapes, circles, and curves. The construction points include using Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, and CurveCartesian commands/objects. This example is suitable for middle school or high school students learning geometric shapes and curve concepts.

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 the geometry of a conch shell, including multiple circles and curves, showing the construction process. Start by drawing two line segments and four points, then use orthogonal lines to construct two circles and find their intersection points. Next, draw two ellipses with the intersection points as foci, and find the intersections of the ellipses with the orthogonal lines. Finally, use the points and angle information to draw complex parametric curves, forming the shape of the conch shell.
The prompt above was compiled from the original drawing commands and is for reference only.