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Klein (botella de) Geometric Construction and Function Graphing

Klein (botella de) Geometric Construction and Function Graphing
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2023-03-15 20:39
FunctionCoordinate GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to create function graphs using geometric construction methods and explores the relationship between quadratic functions and similarity transformations. Key mathematical concepts include function graphing, quadratic functions, similarity transformations, and geometric construction. The construction involves GeoGebra commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, etc. It is suitable for high school students or college undergraduates learning analytic geometry and function graph drawing techniques.

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, xAxis) → zz2
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz2, 1) → ZZ4
  12. Intersect(zz0, zz2, 2) → ZZ3
  13. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  14. Point(zElipse2) → Z
  15. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  16. Angle(ZZ8, C, Z) → zAngInc
  17. OrthogonalLine(C, yAxis) → zz1
  18. Intersect(zz0, zz1, 2) → ZZ2
  19. Intersect(zz0, zz1, 1) → ZZ1
  20. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  21. Point(zElipse1) → XY
  22. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  23. Angle(ZZ6, C, XY) → zAngRot
  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 / 10 * (((((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)))), u, u1 + p, u2) → curva1B
  32. CurveCartesian((r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2))) * sin(zAngRot)) + (((16 * sin(u)) + ((4 * (1 - cos(u) / 2)) * sin(u))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + (4 * (1 - cos(u) / 2)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + ((4 * (1 - cos(u) / 2)) * sin(u))) * sin(zAngRot)) * sin(zAngInc)))), u, u1, u2 - p) → curva1A
  33. CurveCartesian((r / 10 * (((((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v + 3.14159))) * sin(zAngRot)) + ((16 * sin(α)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v + 3.14159)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(α) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2) → curva2B
  34. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v))) * sin(zAngRot)) + (((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), u, u1, u2 - p], v, v1, v2, paso) → list2
  35. CurveCartesian((r / 10 * (((((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v))) * sin(zAngRot)) + (((16 * sin(α)) + (((4 * (1 - cos(α) / 2)) * sin(α)) * cos(v))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(α)) * (1 + sin(α))) + ((4 * (1 - cos(α) / 2)) * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(α)) + (((4 * (1 - cos(α) / 2)) * sin(α)) * cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(α) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2) → curva2A
  36. CurveCartesian((r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β))) * sin(zAngRot)) + (((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(β))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(β))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(β)) * cos(zAngInc)))), u, u1, u2 - p) → a1
  37. CurveCartesian((r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(β)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(β)) * cos(zAngInc)))), u, u1 + p, u2) → a2
  38. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2], u, u1 + p, u2, paso) → list3
  39. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v))) * sin(zAngRot)) + (((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v)))) * cos(zAngRot)) * sin(zAngInc)) + ((((16 * sin(u)) + (((4 * (1 - cos(u) / 2)) * sin(u)) * cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), v, v1, v2], u, u1, u2 - p, paso) → list4
  40. Sequence(CurveCartesian[(r / 10 * (((((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159))) * sin(zAngRot)) + ((16 * sin(u)) * cos(zAngRot)))), (r / 10 * ((((-(((6 * cos(u)) * (1 + sin(u))) + ((4 * (1 - cos(u) / 2)) * cos(v + 3.14159)))) * cos(zAngRot)) * sin(zAngInc)) + (((16 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (((4 * (1 - cos(u) / 2)) * sin(v)) * cos(zAngInc)))), u, u1 + p, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Draw a figure featuring two ellipses and two curves. Start by drawing two perpendicular lines, then create a circle centered at their intersection point. Mark the four intersection points of the circle with the lines. Connect these points to form a quadrilateral, then draw two ellipses using pairs of adjacent vertices as foci. Next, plot two curves based on angle parameters, connecting points on the ellipses and quadrilateral. Combine these curves with the ellipses and quadrilateral to illustrate the relationship between function graphs and geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.