MathPlotAI Logo MathPlotAI

Whitney Umbrella Geometric Construction Example

Whitney Umbrella Geometric Construction Example
Loading the math board and drawing, please wait…
2023-03-15 21:16
Coordinate GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the construction of a Whitney umbrella using geometric principles, focusing on symmetry and similarity. Key mathematical concepts include symmetry, similarity transformations, and geometric construction. The construction involves GeoGebra commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, etc. It is suitable for middle or high school mathematics courses to teach geometric transformations and graphical 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. CurveCartesian(((r / 4 * u) * cos(zAngRot)), (((r / 4 * u) * sin(zAngRot)) * sin(zAngInc)), u, u1, u2) → 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 / 4 * (((β * v) * sin(zAngRot)) + (β * cos(zAngRot)))), (r / 4 * (((((-β) * v) * cos(zAngRot)) * sin(zAngInc)) + ((β * sin(zAngRot)) * sin(zAngInc)) + (v^(2) * cos(zAngInc)))), v, v1, v2) → curva2
  33. CurveCartesian((r / 4 * (((u * α) * sin(zAngRot)) + (u * cos(zAngRot)))), (r / 4 * (((((-u) * α) * cos(zAngRot)) * sin(zAngInc)) + ((u * sin(zAngRot)) * sin(zAngInc)) + (α^(2) * cos(zAngInc)))), u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r / 4 * (((u * v) * sin(zAngRot)) + (u * cos(zAngRot)))), (r / 4 * (((((-u) * v) * cos(zAngRot)) * sin(zAngInc)) + ((u * sin(zAngRot)) * sin(zAngInc)) + (v^(2) * cos(zAngInc)))), v, v1, v2], u, u1, u2, (2 * paso)) → list2
  35. Sequence(CurveCartesian[(r / 4 * (((u * v) * sin(zAngRot)) + (u * cos(zAngRot)))), (r / 4 * (((((-u) * v) * cos(zAngRot)) * sin(zAngInc)) + ((u * sin(zAngRot)) * sin(zAngInc)) + (v^(2) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1
  36. CurveCartesian(0, ((r / 4 * u) * cos(zAngInc)), u, -4, 4) → a

AI Drawing Prompt Reference

Construct a Whitney umbrella–style geometric shape in a coordinate plane using analytic geometry. Start from a circle with center C and an x-axis reference, draw the line orthogonal to the x-axis through C, and find intersection points between the circle and these orthogonal lines. Use those points to define two ellipses, then take their intersections with the relevant orthogonal lines to determine an angle at C and another angle coming from the perpendicular setup. With these angle parameters, generate parameterized Cartesian curves (two different parameter families over specified ranges) that share the same geometric symmetry behavior, producing the twisted umbrella-like surface implied by the construction.
The prompt above was compiled from the original drawing commands and is for reference only.