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Surface Fit

Surface Fit
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2020-06-20 15:44
Coordinate GeometryOptimization
This example demonstrates how to fit a surface through a set of data points. It involves core mathematical concepts such as surface fitting, geometric distribution, unit circle, inequalities, and rotation. Key GeoGebra commands used include PointIn, Point, Sequence, FitPoly, Coefficients, Surface, and CurveCartesian for creating and fitting the surface. Suitable for high school and above students to learn about data fitting and 3D geometry.

GeoGebra Commands & Parameters

  1. PointIn(Plane[A, Plane[xAxis, zAxis]]) → B
  2. PointIn(Plane[A, Plane[xAxis, zAxis]]) → C
  3. PointIn(Plane[A, Plane[xAxis, zAxis]]) → D
  4. PointIn(Plane[A, Plane[yAxis, zAxis]]) → E
  5. Point(IntersectPath[Plane[B, Plane[yAxis, zAxis]], Plane[E, Plane[xAxis, zAxis]]]) → F
  6. Point(IntersectPath[Plane[C, Plane[yAxis, zAxis]], Plane[E, Plane[xAxis, zAxis]]]) → G
  7. Point(IntersectPath[Plane[D, Plane[yAxis, zAxis]], Plane[E, Plane[xAxis, zAxis]]]) → H
  8. PointIn(Plane[A, Plane[yAxis, zAxis]]) → I
  9. PointIn(Plane[A, Plane[yAxis, zAxis]]) → M
  10. Point(IntersectPath[Plane[B, Plane[yAxis, zAxis]], Plane[I, Plane[xAxis, zAxis]]]) → J
  11. Point(IntersectPath[Plane[C, Plane[yAxis, zAxis]], Plane[I, Plane[xAxis, zAxis]]]) → K
  12. Point(IntersectPath[Plane[D, Plane[yAxis, zAxis]], Plane[I, Plane[xAxis, zAxis]]]) → L
  13. Point(IntersectPath[Plane[B, Plane[yAxis, zAxis]], Plane[M, Plane[xAxis, zAxis]]]) → N
  14. Point(IntersectPath[Plane[C, Plane[yAxis, zAxis]], Plane[M, Plane[xAxis, zAxis]]]) → O
  15. Point(IntersectPath[Plane[D, Plane[yAxis, zAxis]], Plane[M, Plane[xAxis, zAxis]]]) → P
  16. Sequence((x(Element[la, i]), z(Element[la, i])), i, 1, 4) → lax
  17. Sequence((x(Element[le, i]), z(Element[le, i])), i, 1, 4) → lex
  18. Sequence((x(Element[li, i]), z(Element[li, i])), i, 1, 4) → lix
  19. Sequence((x(Element[lm, i]), z(Element[lm, i])), i, 1, 4) → lmx
  20. FitPoly(lax, 3) → fax
  21. FitPoly(lex, 3) → fex
  22. FitPoly(lix, 3) → fix
  23. FitPoly(lmx, 3) → fmx
  24. Coefficients(fax) → cax
  25. Coefficients(fex) → cex
  26. Coefficients(fix) → cix
  27. Coefficients(fmx) → cmx
  28. FitPoly(lcy3, 3) → fy3
  29. FitPoly(lcy2, 3) → fy2
  30. FitPoly(lcy1, 3) → fy1
  31. FitPoly(lcy0, 3) → fy0
  32. Surface(u, v, f(u, v), u, x(A), x(D), v, y(A), y(M)) → a
  33. CurveCartesian(t, y(E), fex(t), t, x(A), x(D)) → ce
  34. CurveCartesian(t, y(I), fix(t), t, x(A), x(D)) → ci
  35. CurveCartesian(t, y(A), fax(t), t, x(A), x(D)) → ca
  36. CurveCartesian(t, y(M), fmx(t), t, x(A), x(D)) → cm
  37. CurveCartesian(x(A), t, f(x(A), t), t, y(A), y(M)) → caa
  38. CurveCartesian(x(B), t, f(x(B), t), t, y(A), y(M)) → cb
  39. CurveCartesian(x(C), t, f(x(C), t), t, y(A), y(M)) → cc
  40. CurveCartesian(x(D), t, f(x(D), t), t, y(A), y(M)) → cd
  41. Surface(u, v, f(u, v), u, A1, A2, v, y(A), y(M)) → B1
  42. Surface(u, v, f(u, v), u, A2, A3, v, y(A), y(M)) → B2
  43. Surface(u, v, f(u, v), u, A3, A4, v, y(A), y(M)) → B3
  44. Surface(u, v, f(u, v), u, A4, A5, v, y(A), y(M)) → B4
  45. Surface(u, v, f(u, v), u, A5, A6, v, y(A), y(M)) → B5
  46. Surface(u, v, f(u, v), u, A6, A7, v, y(A), y(M)) → B6
  47. Surface(u, v, f(u, v), u, A7, A8, v, y(A), y(M)) → B7
  48. Surface(u, v, f(u, v), u, A8, A9, v, y(A), y(M)) → B8
  49. Surface(u, v, f(u, v), u, A9, A10, v, y(A), y(M)) → B9
  50. Surface(u, v, f(u, v), u, A10, A11, v, y(A), y(M)) → B10
  51. Surface(u, v, f(u, v), u, A11, A12, v, y(A), y(M)) → B11
  52. Surface(u, v, f(u, v), u, A12, A13, v, y(A), y(M)) → B12
  53. Surface(u, v, f(u, v), u, A13, A14, v, y(A), y(M)) → B13
  54. Surface(u, v, f(u, v), u, A14, A15, v, y(A), y(M)) → B14
  55. Surface(u, v, f(u, v), u, A15, A16, v, y(A), y(M)) → B15
  56. Surface(u, v, f(u, v), u, A16, A17, v, y(A), y(M)) → B16
  57. Surface(u, v, f(u, v), u, A17, A18, v, y(A), y(M)) → B17
  58. Surface(u, v, f(u, v), u, A18, A19, v, y(A), y(M)) → B18
  59. Surface(u, v, f(u, v), u, A19, A20, v, y(A), y(M)) → B19
  60. Surface(u, v, f(u, v), u, A20, A21, v, y(A), y(M)) → B20
  61. Surface(u, v, f(u, v), u, A21, A22, v, y(A), y(M)) → B21
  62. Surface(u, v, f(u, v), u, A22, A23, v, y(A), y(M)) → B22
  63. Surface(u, v, f(u, v), u, A23, A24, v, y(A), y(M)) → B23
  64. Surface(u, v, f(u, v), u, A24, A25, v, y(A), y(M)) → B24
  65. Surface(u, v, f(u, v), u, A25, A26, v, y(A), y(M)) → B25
  66. Surface(u, v, f(u, v), u, A26, A27, v, y(A), y(M)) → B26
  67. Surface(u, v, f(u, v), u, A27, A28, v, y(A), y(M)) → B27
  68. Surface(u, v, f(u, v), u, A28, A29, v, y(A), y(M)) → B28
  69. Surface(u, v, f(u, v), u, A29, A30, v, y(A), y(M)) → B29
  70. Surface(u, v, f(u, v), u, A30, A31, v, y(A), y(M)) → B30
  71. Surface(u, v, f(u, v), u, A31, A32, v, y(A), y(M)) → B31
  72. Surface(u, v, f(u, v), u, A32, A33, v, y(A), y(M)) → B32
  73. Surface(u, v, f(u, v), u, A33, A34, v, y(A), y(M)) → B33
  74. Surface(u, v, f(u, v), u, A34, A35, v, y(A), y(M)) → B34
  75. Surface(u, v, f(u, v), u, A35, A36, v, y(A), y(M)) → B35
  76. Surface(u, v, f(u, v), u, A36, A37, v, y(A), y(M)) → B36
  77. Surface(u, v, f(u, v), u, A37, A38, v, y(A), y(M)) → B37
  78. Surface(u, v, f(u, v), u, A38, A39, v, y(A), y(M)) → B38
  79. Surface(u, v, f(u, v), u, A39, A40, v, y(A), y(M)) → B39
  80. Surface(u, v, f(u, v), u, A40, A41, v, y(A), y(M)) → B40
  81. Surface(u, v, f(u, v), u, A41, A42, v, y(A), y(M)) → B41
  82. Surface(u, v, f(u, v), u, A42, A43, v, y(A), y(M)) → B42
  83. Surface(u, v, f(u, v), u, A43, A44, v, y(A), y(M)) → B43
  84. Surface(u, v, f(u, v), u, A44, A45, v, y(A), y(M)) → B44
  85. Surface(u, v, f(u, v), u, A45, A46, v, y(A), y(M)) → B45
  86. Surface(u, v, f(u, v), u, A46, A47, v, y(A), y(M)) → B46
  87. Surface(u, v, f(u, v), u, A47, A48, v, y(A), y(M)) → B47
  88. Surface(u, v, f(u, v), u, A48, A49, v, y(A), y(M)) → B48
  89. Surface(u, v, f(u, v), u, A49, A50, v, y(A), y(M)) → B49
  90. Surface(u, v, f(u, v), u, A50, A51, v, y(A), y(M)) → B50
  91. Surface(u, v, f(u, v), u, A51, A52, v, y(A), y(M)) → B51

AI Drawing Prompt Reference

Create a surface that fits a set of data points. Start by creating points A, B, C, D, E, F, G, H, I, and M on specified planes, then generate intersection points of paths, fit polynomial functions to these intersection points, and finally draw the surface using the fitted polynomial functions, including paths from point A to point M and from point A to point D.
The prompt above was compiled from the original drawing commands and is for reference only.