MathPlotAI Logo MathPlotAI

Monkey Saddle Surface Construction with Parametric Curves

Monkey Saddle Surface Construction with Parametric Curves
Loading the math board and drawing, please wait…
2023-03-15 21:31
FunctionParametric EquationGeometric ConstructionLocus
In collections geometriadinamica.es
This example demonstrates how to construct a monkey saddle surface (Silla de mono) using multiple parametric curves in GeoGebra. The core mathematical concepts involve parametric equations, surface construction, and curve transformations. Construction points: Define parametric curves using the CurveCartesian command, generate curve sequences with Sequence, construct ellipse and rotated angles using Ellipse and Angle commands, then apply OrthogonalLine and Intersect to complete the curve network. This example is suitable for advanced mathematics, analytic geometry, or differential geometry courses, helping students understand parametric surface construction methods and surface characteristics.

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, yAxis) → zz1
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz1, 1) → ZZ1
  12. Intersect(zz0, zz1, 2) → ZZ2
  13. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  14. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  15. Point(zElipse1) → XY
  16. Angle(ZZ6, C, XY) → zAngRot
  17. OrthogonalLine(C, xAxis) → zz2
  18. Intersect(zz0, zz2, 2) → ZZ3
  19. Intersect(zz0, zz2, 1) → ZZ4
  20. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  21. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  22. Point(zElipse2) → Z
  23. Angle(ZZ8, C, Z) → zAngInc
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  26. CurveCartesian(((r / 1.5 * u) * sin(zAngRot)), (r / 1.5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + (u^(3) * cos(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 / 1.5 * ((α * sin(zAngRot)) + (v * cos(zAngRot)))), (r / 1.5 * ((((-α) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + ((α^(3) - ((3 * α) * v^(2))) * cos(zAngInc)))), v, v1, v2) → curva2
  33. CurveCartesian((r / 1.5 * ((u * sin(zAngRot)) + (β * cos(zAngRot)))), (r / 1.5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((β * sin(zAngRot)) * sin(zAngInc)) + ((u^(3) - ((3 * u) * β^(2))) * cos(zAngInc)))), u, u1, u2) → a
  34. Sequence(CurveCartesian[(r / 1.5 * ((u * sin(zAngRot)) + (v * cos(zAngRot)))), (r / 1.5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + ((u^(3) - ((3 * u) * v^(2))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list2
  35. Sequence(CurveCartesian[(r / 1.5 * ((u * sin(zAngRot)) + (v * cos(zAngRot)))), (r / 1.5 * ((((-u) * cos(zAngRot)) * sin(zAngInc)) + ((v * sin(zAngRot)) * sin(zAngInc)) + ((u^(3) - ((3 * u) * v^(2))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, (4 * paso)) → list1

AI Drawing Prompt Reference

To draw a monkey saddle surface, first construct two ellipses and a rotation angle. Steps include defining ellipses, calculating their intersection points, then determining the rotation angle via orthogonal lines. Next, use these ellipses and angles to generate parametric curve sequences, combining them with sequence functions to form the surface. Key objects are ellipses, rotation angles, and parametric curves, connected through intersections and orthogonal relationships, resulting in a surface with specific curvature.
The prompt above was compiled from the original drawing commands and is for reference only.