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Catenoide Transformation Demo

Catenoide Transformation Demo
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2023-03-15 18:58
FunctionTrigonometryCoordinate GeometryParametric EquationGeometric Construction
In collections geometriadinamica.es
This example demonstrates the transformation process and geometric form of the catenoide curve. It involves core mathematical concepts such as analytical geometry, ellipses, parametric equations, and geometric transformations. The construction utilizes GeoGebra commands and objects like segments, points, orthogonal lines, circles, intersections, ellipses, angles, Cartesian curves, and sequences. Suitable for high school and above mathematics learners to understand the generation and transformation properties of complex curves.

GeoGebra Commands & Parameters

  1. Segment(Q5, Q6) → p5
  2. Point(p5) → P5
  3. Segment(Q3, Q4) → p2
  4. Point(p2) → P3
  5. Segment(P3, Q4) → p4
  6. Point(p4) → P4
  7. Segment(Q1, Q2) → p1
  8. Point(p1) → P1
  9. Segment(P1, Q2) → p3
  10. Point(p3) → P2
  11. OrthogonalLine(C, yAxis) → zz1
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz1, 1) → ZZ1
  14. Intersect(zz0, zz1, 2) → ZZ2
  15. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  16. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  17. Point(zElipse1) → XY
  18. Angle(ZZ6, C, XY) → zAngRot
  19. OrthogonalLine(C, xAxis) → zz2
  20. Intersect(zz0, zz2, 2) → ZZ3
  21. Intersect(zz0, zz2, 1) → ZZ4
  22. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  23. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  24. Point(zElipse2) → Z
  25. Angle(ZZ8, C, Z) → zAngInc
  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 / 3 * (((((cos(a) * sinh(α)) * sin(u)) + ((sin(a) * cosh(α)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(α)) * cos(u)) + ((sin(a) * cosh(α)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(α)) * sin(u)) + ((sin(a) * cosh(α)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(α)) * cos(u)) + ((sin(a) * cosh(α)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (α * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), u, u1, u2) → curva2
  32. CurveCartesian((r / 3 * (((((cos(a) * sinh(v)) * sin(β)) + ((sin(a) * cosh(v)) * cos(β))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(β)) + ((sin(a) * cosh(v)) * sin(β))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(β)) + ((sin(a) * cosh(v)) * cos(β)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(β)) + ((sin(a) * cosh(v)) * sin(β))) * sin(zAngRot)) * sin(zAngInc)) + (((β * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2) → a1
  33. CurveCartesian((r / 3 * ((((-sin(a)) * cosh(v)) * sin(zAngRot)) - ((cos(a) * sinh(v)) * cos(zAngRot)))), (r / 3 * ((((sin(a) * cosh(v)) * cos(zAngRot)) * sin(zAngInc)) - (((cos(a) * sinh(v)) * sin(zAngRot)) * sin(zAngInc)) + (((3.14159 * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2) → curva1A
  34. Sequence(CurveCartesian[(r / 3 * (((((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), u, u1, u2], v, v1, v2, paso) → list2
  35. Sequence(CurveCartesian[(r / 3 * (((((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u))) * sin(zAngRot)) + ((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * cos(zAngRot)))), (r / 3 * ((((-(((cos(a) * sinh(v)) * sin(u)) + ((sin(a) * cosh(v)) * cos(u)))) * cos(zAngRot)) * sin(zAngInc)) + (((((cos(a) * sinh(v)) * cos(u)) + ((sin(a) * cosh(v)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)) + (((u * cos(a)) + (v * sin(a))) * cos(zAngInc)))) - 4 + (2 * a), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Draw a catenoid curve. First, draw some basic line segments and points. Then, draw vertical lines for the y-axis and x-axis, and a circle with radius 0.62. Next, find the intersection points of the circle with the vertical lines and use them to draw an ellipse. Again, find the intersection points of the ellipse with the vertical lines and calculate the corresponding angles. Then, connect these intersection points with line segments. Finally, generate a Cartesian curve using parametric equations and draw a sequence of parametric curves to show the transformation process and geometric shape of the catenoid curve. This process involves concepts like analytical geometry, ellipses, parametric equations, and geometric transformations, suitable for high school and above mathematics learners to understand the generation and transformation properties of complex curves.
The prompt above was compiled from the original drawing commands and is for reference only.