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Parol 2022: Parametric Curves & 3D Shapes Demo

Parol 2022: Parametric Curves & 3D Shapes Demo
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2022-12-06 15:22
PolygonSolid GeometryParametric EquationGeometric Construction
This example demonstrates the creation of complex parametric curves and their derived polygons and solids. Key math concepts include parametric equations, polygon construction, solid geometry, surface generation, and geometric transformations. Construction highlights GeoGebra commands like Intersect, Point, Line, Rotate, Segment, Polygon, Prism, Pyramid, Surface, Sphere, and CurveCartesian. Ideal for high school and advanced learners exploring parametric modeling and 3D geometry applications.

GeoGebra Commands & Parameters

  1. Intersect(yAxis, zAxis) → A
  2. Point(zAxis) → B
  3. Point(zAxis) → C
  4. Line(C, yAxis) → f
  5. Rotate(B, -72°, f) → B'
  6. Rotate(B', -72°, f) → B''
  7. Rotate(B'', -72°, f) → B'''
  8. Rotate(B''', -72°, f) → D
  9. Segment(B, B''') → g
  10. Segment(D, B'') → h
  11. Segment(B', B''') → i
  12. Segment(B, B'') → j
  13. Segment(D, B') → k
  14. Intersect(k, g) → E
  15. Intersect(k, j) → F
  16. Intersect(i, j) → G
  17. Intersect(h, g) → H
  18. Polygon(E, H, A, G, F) → poly1, e, h_1, a_1, g_1, f_1
  19. Prism(poly1, 0.4415996098181485) → c, N, faceEFRN, faceNOPQR, edgeEN, edgeHO, edgeAP, edgeGQ, edgeFR, edgeNO, edgeOP_{1}, edgePQ_{1}, O, edgeQR_{1}, edgeNR_{1}, P, Q, R, faceEHON, faceAHOP, faceAGQP, faceFGQR
  20. Prism(poly1, -0.5) → d, S, faceEFWS, faceSTUVW, edgeES, edgeHT, edgeAU, edgeGV, edgeFW, edgeST, edgeTU_{1}, edgeUV_{1}, T, edgeVW_{1}, edgeSW_{1}, U, V, W, faceEHTS, faceAHTU, faceAGVU, faceFGVW
  21. Pyramid(R, W, S, N, B) → p, faceNRWS, edgeBR, edgeBW, edgeBS, edgeBN, faceBRW, faceBSW, faceBNS, faceBNR, edgeRW_{1}, edgeSW, edgeNS, edgeNR
  22. Rotate(p, 72°, f) → q
  23. Surface(((3 + (0.3 * cos(u))) * cos(t)), ((3 + (0.3 * cos(u))) * sin(t)) + 1, (0.3 * sin(u)), u, 0, (2 * pi), t, 0, (2 * pi)) → a
  24. Rotate(a, 90°, xAxis) → a'
  25. Sphere(B, 0.5) → b
  26. Rotate(b, 72°, f) → b'
  27. Rotate(b, 144°, f) → b'_{1}
  28. Rotate(b, -72°, f) → b'_{2}
  29. Rotate(b, -144°, f) → b'_{3}
  30. Intersect(f, zAxis) → I
  31. CurveCartesian((0.4 * cos((10 * t))) - 1.8, (0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → l
  32. CurveCartesian((-0.4 * cos((10 * t))) - 1.8, (0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → m
  33. CurveCartesian((0.4 * cos((10 * t))) - 1.8, (-0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → n
  34. CurveCartesian((-0.4 * cos((10 * t))) - 1.8, (-0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → o
  35. CurveCartesian((0.4 * cos((10 * t))) + 1.8, (0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → v
  36. CurveCartesian((-0.4 * cos((10 * t))) + 1.8, (0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → w
  37. CurveCartesian((0.4 * cos((10 * t))) + 1.8, (-0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → b_1
  38. CurveCartesian((-0.4 * cos((10 * t))) + 1.8, (-0.4 * sin((10 * t))), t, t, (-1.5 * pi), (-(pi / 2.5))) → c_1
  39. Pyramid(U, P, O, T, B''') → r, faceOPUT, edgeB'''U, edgeB'''P, edgeB'''O, edgeB'''T, faceB'''PU, faceB'''OP, faceB'''OT, faceB'''TU, edgePU_{1}, edgeOP, edgeOT, edgeTU
  40. Pyramid(Q, P, U, V, B'') → s, facePQVU, edgeB''Q, edgeB''P, edgeB''U, edgeB''V, faceB''PQ, faceB''PU, faceB''UV, faceB''QV, edgePQ, edgePU, edgeUV, edgeQV_{1}
  41. Pyramid(R, Q, V, W, B') → t, faceQRWV, edgeB'R, edgeB'Q, edgeB'V, edgeB'W, faceB'QR, faceB'QV, faceB'VW, faceB'RW, edgeQR, edgeQV, edgeVW, edgeRW
  42. Intersect(zAxis, b, 2) → J
  43. Point(zAxis) → K
  44. Segment(K, J) → i_1

AI Drawing Prompt Reference

Create a complex 3D geometric shape, including a polygon and prisms generated by rotation, and a sphere rotated around the z-axis. Start by creating a point B on the z-axis, then rotate to generate a series of points and segments to form a polygon. Use this polygon as the base to construct two prisms and a pyramid. Finally, rotate a sphere around the z-axis and create two points J and K on the z-axis, connecting them to form a line segment.
The prompt above was compiled from the original drawing commands and is for reference only.