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Nefroide: Linear Function and Ellipse Construction

Nefroide: Linear Function and Ellipse Construction
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2023-03-15 20:51
FunctionGeometric Construction
In collections geometriadinamica.es
This example demonstrates the relationship between linear functions and ellipses through geometric construction. Key mathematical concepts include linear functions, ellipse equations, and geometric construction methods. The construction involves GeoGebra commands such as Segment, Circle, Ellipse, and Intersect. Suitable for middle or high school mathematics education to understand the integration of functions and geometry.

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 / 2.5 * (((cos(α) * ((3 * cos(v)) - cos((3 * v)))) * sin(zAngRot)) + ((sin(α) * ((3 * cos(v)) - cos((3 * v)))) * cos(zAngRot)))), (r / 2.5 * (((((-cos(α)) * ((3 * cos(v)) - cos((3 * v)))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(α) * ((3 * cos(v)) - cos((3 * v)))) * sin(zAngRot)) * sin(zAngInc)) + (((3 * sin(v)) - sin((3 * v))) * cos(zAngInc)))), v, v1, v2) → curva2
  32. CurveCartesian((r / 2.5 * (((cos(u) * 2) * sin(zAngRot)) + ((sin(u) * 2) * cos(zAngRot)))), (r / 2.5 * (((((-cos(u)) * 2) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * 2) * sin(zAngRot)) * sin(zAngInc)))), u, u1, u2) → curva1
  33. CurveCartesian((r / 2.5 * (((cos(u) * ((3 * cos(β - 3.14159 / 2)) - cos((3 * (β - 3.14159 / 2))))) * sin(zAngRot)) + ((sin(u) * ((3 * cos(β - 3.14159 / 2)) - cos((3 * (β - 3.14159 / 2))))) * cos(zAngRot)))), (r / 2.5 * (((((-cos(u)) * ((3 * cos(β - 3.14159 / 2)) - cos((3 * (β - 3.14159 / 2))))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * ((3 * cos(β - 3.14159 / 2)) - cos((3 * (β - 3.14159 / 2))))) * sin(zAngRot)) * sin(zAngInc)) + (((3 * sin(β - 3.14159 / 2)) - sin((3 * (β - 3.14159 / 2)))) * cos(zAngInc)))), u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r / 2.5 * (((cos(u) * ((3 * cos(v)) - cos((3 * v)))) * sin(zAngRot)) + ((sin(u) * ((3 * cos(v)) - cos((3 * v)))) * cos(zAngRot)))), (r / 2.5 * (((((-cos(u)) * ((3 * cos(v)) - cos((3 * v)))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * ((3 * cos(v)) - cos((3 * v)))) * sin(zAngRot)) * sin(zAngInc)) + (((3 * sin(v)) - sin((3 * v))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list3
  35. Sequence(CurveCartesian[(r / 2.5 * (((cos(u) * ((3 * cos(v)) - cos((3 * v)))) * sin(zAngRot)) + ((sin(u) * ((3 * cos(v)) - cos((3 * v)))) * cos(zAngRot)))), (r / 2.5 * (((((-cos(u)) * ((3 * cos(v)) - cos((3 * v)))) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * ((3 * cos(v)) - cos((3 * v)))) * sin(zAngRot)) * sin(zAngInc)) + (((3 * sin(v)) - sin((3 * v))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Draw a geometric illustration showing the relationship between linear functions and ellipses. Start by constructing two line segments and two points. Then, draw a circle with one point as the center and intersect it with a line perpendicular to the x-axis to get two points. Use these points to draw an ellipse. Next, draw another circle with the other point as the center, intersect it with the same perpendicular line to get two more points, and draw another ellipse. Finally, connect the intersection points of the two ellipses and plot two curves to demonstrate the integration of functions and geometry.
The prompt above was compiled from the original drawing commands and is for reference only.