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Spiral Line and Geometric Construction

Spiral Line and Geometric Construction
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2023-03-15 20:57
VectorCoordinate GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the construction and application of a spiral line in geometry. The key mathematical concepts involved include vectors, area, inequalities, and geometric construction. The construction involves using GeoGebra commands such as Segment, OrthogonalLine, Circle, Intersect, Ellipse, and Angle with various objects. It is suitable for high school students or math enthusiasts learning geometric construction and graph analysis.

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, xAxis) → zz2
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz2, 1) → ZZ4
  14. Intersect(zz0, zz2, 2) → ZZ3
  15. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  16. Point(zElipse2) → Z
  17. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  18. Angle(ZZ8, C, Z) → zAngInc
  19. OrthogonalLine(C, yAxis) → zz1
  20. Intersect(zz0, zz1, 2) → ZZ2
  21. Intersect(zz0, zz1, 1) → ZZ1
  22. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  23. Point(zElipse1) → XY
  24. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  25. Angle(ZZ6, C, XY) → zAngRot
  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 / 10 * ((((5 * α / 2) * cos(u)) * sin(zAngRot)) + (((5 * α / 2) * sin(u)) * cos(zAngRot)))), (r / 10 * (((((-5 * α / 2) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((((5 * α / 2) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (cos((-3 * u) + (5 * α / 2)) * cos(zAngInc)))), u, u1, u2) → curva2
  32. CurveCartesian(((r / 4 * v) * sin(zAngRot)), (r / 10 * ((((-5 * v / 2) * cos(zAngRot)) * sin(zAngInc)) + (cos((5 * v / 2)) * cos(zAngInc)))), v, v1, (2 * v2 / 5)) → curva1
  33. CurveCartesian((r / 10 * (((v * cos(β)) * sin(zAngRot)) + ((v * sin(β)) * cos(zAngRot)))), (r / 10 * (((((-v) * cos(β)) * cos(zAngRot)) * sin(zAngInc)) + (((v * sin(β)) * sin(zAngRot)) * sin(zAngInc)) + (cos((-3 * β) + v) * cos(zAngInc)))), v, v1, v2) → a1
  34. Sequence(CurveCartesian[(r / 10 * (((v * cos(u)) * sin(zAngRot)) + ((v * sin(u)) * cos(zAngRot)))), (r / 10 * (((((-v) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((v * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (cos((-3 * u) + v) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso / 8) → list2
  35. Sequence(CurveCartesian[(r / 10 * (((v * cos(u)) * sin(zAngRot)) + ((v * sin(u)) * cos(zAngRot)))), (r / 10 * (((((-v) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((v * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (cos((-3 * u) + v) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Draw a geometric construction with multiple segments and points, including two intersecting ellipses oriented horizontally and vertically around a central point. Place points on the ellipses and connect relevant points with segments. Then generate a family of spiral curves defined by parametric equations, rotating around the center to form a dynamic spiral pattern. The figure illustrates the application of spiral lines in geometry, involving vectors, area, and inequalities.
The prompt above was compiled from the original drawing commands and is for reference only.