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Cuerno: Conic Sections and Trigonometric Functions in 2D Geometry

Cuerno: Conic Sections and Trigonometric Functions in 2D Geometry
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2023-03-15 19:40
VectorTrigonometryCoordinate GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the combination of conic sections (such as ellipses) and trigonometric functions in two-dimensional geometry. Using GeoGebra commands like Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, and CurveCartesian, it constructs geometric figures and explores related phenomena. Key mathematical concepts include conic sections, trigonometric functions, vectors, and coordinate geometry. The construction involves specific commands and objects to illustrate these ideas. Suitable for high school students or college undergraduates learning applications of analytic geometry and trigonometry.

GeoGebra Commands & Parameters

  1. Segment(Q1, Q2) → p1
  2. Point(p1) → P1
  3. Segment(P1, Q2) → p3
  4. Point(p3) → P2
  5. Segment(Q5, Q6) → p5
  6. Point(p5) → P5
  7. OrthogonalLine(C, xAxis) → zz2
  8. Circle(C, 0.62) → zz0
  9. Intersect(zz0, zz2, 1) → ZZ4
  10. Intersect(zz0, zz2, 2) → ZZ3
  11. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  12. Point(zElipse2) → Z
  13. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  14. Angle(ZZ8, C, Z) → zAngInc
  15. OrthogonalLine(C, yAxis) → zz1
  16. Intersect(zz0, zz1, 2) → ZZ2
  17. Intersect(zz0, zz1, 1) → ZZ1
  18. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  19. Point(zElipse1) → XY
  20. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  21. Angle(ZZ6, C, XY) → zAngRot
  22. Segment(Q3, Q4) → p2
  23. Point(p2) → P3
  24. Segment(P3, Q4) → p4
  25. Point(p4) → P4
  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(2 + (r / 1.5 * ((((a + (u * cos(α))) * sin(((b * 3.14159) * u))) * sin(zAngRot)) + ((a + (u * cos(α))) * cos(((b * 3.14159) * u))) + ((c * u) * cos(zAngRot)))), (r / 1.5 * (((((-(a + (u * cos(α)))) * sin(((b * 3.14159) * u))) * cos(zAngRot)) * sin(zAngInc)) + ((a + (u * cos(α))) * cos(((b * 3.14159) * u))) + (((c * u) * sin(zAngRot)) * sin(zAngInc)) + ((u * sin(α)) * cos(zAngInc)))) + 3, u, u1, u2) → curva2
  32. CurveCartesian(2 + (r / 1.5 * ((((a + (β * cos(v))) * sin(((b * 3.14159) * β))) * sin(zAngRot)) + ((a + (β * cos(v))) * cos(((b * 3.14159) * β))) + ((c * β) * cos(zAngRot)))), (r / 1.5 * (((((-(a + (β * cos(v)))) * sin(((b * 3.14159) * β))) * cos(zAngRot)) * sin(zAngInc)) + ((a + (β * cos(v))) * cos(((b * 3.14159) * β))) + (((c * β) * sin(zAngRot)) * sin(zAngInc)) + ((β * sin(v)) * cos(zAngInc)))) + 3, v, v1, v2) → a1
  33. CurveCartesian(2 + (r / 1.5 * ((((a + (0.5 * cos(v))) * sin(((b * 3.14159) * 0.5))) * sin(zAngRot)) + ((a + (0.5 * cos(v))) * cos(((b * 3.14159) * 0.5))) + ((c * 0.5) * cos(zAngRot)))), (r / 1.5 * (((((-(a + (0.5 * cos(v)))) * sin(((b * 3.14159) * 0.5))) * cos(zAngRot)) * sin(zAngInc)) + ((a + (0.5 * cos(v))) * cos(((b * 3.14159) * 0.5))) + (((c * 0.5) * sin(zAngRot)) * sin(zAngInc)) + ((0.5 * sin(v)) * cos(zAngInc)))) + 3, v, v1, v2) → curva1
  34. Sequence(CurveCartesian[2 + (r / 1.5 * ((((a + (u * cos(v))) * sin(((b * 3.14159) * u))) * sin(zAngRot)) + ((a + (u * cos(v))) * cos(((b * 3.14159) * u))) + ((c * u) * cos(zAngRot)))), (r / 1.5 * (((((-(a + (u * cos(v)))) * sin(((b * 3.14159) * u))) * cos(zAngRot)) * sin(zAngInc)) + ((a + (u * cos(v))) * cos(((b * 3.14159) * u))) + (((c * u) * sin(zAngRot)) * sin(zAngInc)) + ((u * sin(v)) * cos(zAngInc)))) + 3, u, u1, u2], v, v1, v2, (4 * paso)) → list1
  35. Sequence(CurveCartesian[2 + (r / 1.5 * ((((a + (u * cos(v))) * sin(((b * 3.14159) * u))) * sin(zAngRot)) + ((a + (u * cos(v))) * cos(((b * 3.14159) * u))) + ((c * u) * cos(zAngRot)))), (r / 1.5 * (((((-(a + (u * cos(v)))) * sin(((b * 3.14159) * u))) * cos(zAngRot)) * sin(zAngInc)) + ((a + (u * cos(v))) * cos(((b * 3.14159) * u))) + (((c * u) * sin(zAngRot)) * sin(zAngInc)) + ((u * sin(v)) * cos(zAngInc)))) + 3, v, v1, v2], u, u1, u2, paso) → list2

AI Drawing Prompt Reference

Construct a 2D diagram in a coordinate system using a fixed center defined by the intersections of perpendicular reference lines tied to the x- and y-axes. Build a circle with that center, and from intersections with two perpendicular directions obtain the key points that determine two ellipses sharing the same center and using the same axis scaling factor. Make the two ellipses intersect, then mark their intersection points and connect two of them with a segment. Measure the angle at the center between one reference direction and an ellipse intersection to define a rotation and an inclination relationship. Finally, draw two parameterized curves (Curve 1 and Curve 2) whose shape is controlled by the measured rotation and inclination angles, reflecting trigonometric modulation in their x- and y-coordinates, to depict the resulting trajectory-like pattern in the plane.
The prompt above was compiled from the original drawing commands and is for reference only.