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Möbius Geometric Construction Example

Möbius Geometric Construction Example
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2023-03-15 20:48
CircleVectorCoordinate GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the interaction between variables, vectors, and circles through geometric construction. Key mathematical concepts include variables, vectors, circles, and geometric construction. Construction involves GeoGebra commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, and CurveCartesian with various objects. Suitable for middle or high school mathematics to understand geometric transformations and relationships.

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. CurveCartesian((r * ((((1 + (v * cos(3.14159 / 2))) * cos(3.14159)) * sin(zAngRot)) + (((1 + (v * cos(3.14159 / 2))) * sin(3.14159)) * cos(zAngRot)))), (r * (((((-(1 + (v * cos(3.14159 / 2)))) * cos(3.14159)) * cos(zAngRot)) * sin(zAngInc)) + ((((1 + (v * cos(3.14159 / 2))) * sin(3.14159)) * sin(zAngRot)) * sin(zAngInc)) + ((v * sin(3.14159 / 2)) * cos(zAngInc)))), v, v1, v2) → curva1
  27. Segment(ZZ5, ZZ6) → zz4
  28. Segment(ZZ7, ZZ8) → zz3
  29. Intersect(zElipse2, zElipse1, 1) → zPunto1
  30. Intersect(zElipse2, zElipse1, 4) → zPunto2
  31. Segment(zPunto1, zPunto2) → zSeg2
  32. CurveCartesian((r * ((((1 + (β * cos(u / 2))) * cos(u)) * sin(zAngRot)) + (((1 + (β * cos(u / 2))) * sin(u)) * cos(zAngRot)))), (r * (((((-(1 + (β * cos(u / 2)))) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((((1 + (β * cos(u / 2))) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((β * sin(u / 2)) * cos(zAngInc)))), u, u1, (2 * u2)) → curva2
  33. CurveCartesian((r * ((((1 + (v * cos(α / 2))) * cos(α)) * sin(zAngRot)) + (((1 + (v * cos(α / 2))) * sin(α)) * cos(zAngRot)))), (r * (((((-(1 + (v * cos(α / 2)))) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + ((((1 + (v * cos(α / 2))) * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + ((v * sin(α / 2)) * cos(zAngInc)))), v, v1, v2) → a1
  34. Sequence(CurveCartesian[(r * ((((1 + (v * cos(u / 2))) * cos(u)) * sin(zAngRot)) + (((1 + (v * cos(u / 2))) * sin(u)) * cos(zAngRot)))), (r * (((((-(1 + (v * cos(u / 2)))) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((((1 + (v * cos(u / 2))) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((v * sin(u / 2)) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list2
  35. Sequence(CurveCartesian[(r * ((((1 + (v * cos(u / 2))) * cos(u)) * sin(zAngRot)) + (((1 + (v * cos(u / 2))) * sin(u)) * cos(zAngRot)))), (r * (((((-(1 + (v * cos(u / 2)))) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((((1 + (v * cos(u / 2))) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((v * sin(u / 2)) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Construct a Möbius strip geometrically, including two ellipses (zElipse1 and zElipse2) connected by intersection points (ZZ1, ZZ2, ZZ3, ZZ4, ZZ5, ZZ6, ZZ7, ZZ8, zPunto1, zPunto2). Start by drawing two circles (C), then construct perpendicular lines and ellipses, and use intersection points and angles to create curves, finally generate the surface of the Möbius strip using curves and ellipses.
The prompt above was compiled from the original drawing commands and is for reference only.