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棱柱展开与折叠网图探索

棱柱展开与折叠网图探索
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2022-11-11 22:08
多边形角立体几何变换几何作图
该范例演示如何为棱柱“展开成平面网图”，再将网图折叠回立体，并通过改变棱柱尺寸观察其几何结构变化。核心数学概念包括棱柱的面与棱的对应关系、平面展开（net）的组成、长方体/矩形棱柱与一般棱柱的网图规律，以及用几何约束保持边长与相邻面连接。作图要点：利用 GeoGebra 的 Point、Segment、Polygon、Rotate、Intersect 等命令生成并定位各个面与边；用 Prism、Net、Surface 构建/标识棱柱及其展开网；通过 Circle、Ray、Sequence 与 OrthogonalLine 辅助定义几何参考与可视化结构；借助 Cartesian 坐标系组织图元；同时用 Cylinder、Cone 等对象用于拓展同类立体（如圆柱、圆锥）的网图与对比。适合中学 6–8 年级进行立体几何入门与“找展开图”技能训练，也可用于课堂探究活动或几何作图练习。

GeoGebra 命令及参数

  1. Point(yAxis) → CA
  2. Point(yAxis) → CB
  3. Circle(CA, CB) → cCil
  4. Point(cCil) → C
  5. Ray(A, (x(A), y(A), 40)) → h
  6. Point(yAxis) → B
  7. Sequence(Rotate[C, ((t * 2) * pi / n), A], t, 1, n) → vert
  8. Polygon(vert) → poly
  9. OrthogonalLine(C, xOyPlane) → f
  10. Segment(HC, (x(C), y(C), 4)) → g
  11. Point(g) → E
  12. Prism(poly, Segment[C, E]) → a, F, face3, face4, face5, face6, face2, face13, face15, faceFGHIJK, edge1, edge2, G, edge3, edge4, edge5, edge6, edge13, edge16, edgeFG, edgeGH, edgeHI, edgeIJ, H, edgeJK, edgeFK, edge14, edge17, I, J, K, L, M, face1
  13. Point(Segment[(0, 0, 0.1), (0, 0, 4)]) → CD
  14. Pyramid(poly, CD) → b, face7, edge8, edge9, edge10, edge11, edge12, edge15, edge18, face9, face10, face11, face12, face8, face14, face16, edge7
  15. Net(a, (-open)) → prismNet, N, W, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1, O, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, P, S_1, face17, face18, face19, face20, face21, face22, face23, face24, face34, Q, face35, edge19, edge20, edge21, edge22, edge23, edge24, edge25, edge26, edge27, R, edge28, edge29, edge30, edge31, edge32, edge33, edge34, edge35, edge36, edge37, S, edge38, edge39, edge40, edge41, edge42, edge43, edge44, edge45, edge70, edge71, T, edge72, edge73, edge74, edge75, edge76, edge77, edge78, edge79, edge80, edge81, U, V
  16. Net(b, (-open)) → pyrNet, T_1, E_2, F_2, G_2, H_2, I_2, J_2, K_2, face25, face26, face27, U_1, face28, face29, face30, face31, face32, face33, edge46, edge47, edge48, edge49, V_1, edge50, edge51, edge52, edge53, edge54, edge55, edge56, edge57, edge58, edge59, W_1, edge60, edge61, edge62, edge63, edge64, edge65, edge66, edge67, edge68, edge69, Z_1, A_2, B_2, C_2, D_2
  17. Random(3, 8) → nRand
  18. Circle(CA, CB) → d
  19. Cylinder(cCil, 3) → bCil, eCil, fCil
  20. Line(CC, xAxis) → aCil
  21. OrthogonalLine(CD, xOyPlane) → gCil
  22. Circle(gCil, CC) → kCil
  23. Rotate(Rotate[eCil, ϕCil, aCil], ϕCil, xAxis) → eeCil
  24. Rotate(Rotate[eCil, ϕCil, aCil], ϕCil, xAxis) → dCil
  25. Circle(CA, CB) → hCil
  26. If(abs(tCil) < 1, ((rCil * sin((-θCil))), ((rCil * (1 - cos((-θCil)))) * cos(ϕCil)) - (3 * sin(ϕCil)), (3 * cos(ϕCil)) + ((rCil * (1 - cos((-θCil)))) * sin(ϕCil))), ((-pi), (-3 * sin(ϕCil)), (3 * cos(ϕCil)))) → CE
  27. If(abs(tCil) < 1, ((rCil * sin(θCil)), ((rCil * (1 - cos(θCil))) * cos(ϕCil)) - (3 * sin(ϕCil)), (3 * cos(ϕCil)) + ((rCil * (1 - cos(θCil))) * sin(ϕCil))), (pi, (-3 * sin(ϕCil)), (3 * cos(ϕCil)))) → CF
  28. If(abs(tCil) < 1, ((rCil * sin((-θCil))), ((rCil * (1 - cos((-θCil)))) * cos(ϕCil)), ((rCil * (1 - cos((-θCil)))) * sin(ϕCil))), ((-pi), 0, 0)) → CG
  29. If(abs(tCil) < 1, ((rCil * sin(θCil)), ((rCil * (1 - cos(θCil))) * cos(ϕCil)), ((rCil * (1 - cos(θCil))) * sin(ϕCil))), (pi, 0, 0)) → CH
  30. Segment(CE, CG) → mCil
  31. Segment(CF, CH) → nCil
  32. Line((0, 1, 0), zAxis) → pCil
  33. Circle(pCil, CB) → qCil
  34. Circle(pCil, CB) → sCil
  35. OrthogonalLine((1, 0, 0), xOyPlane) → aCon
  36. Circle(aCon, (0, 0, 0)) → cCon
  37. Point(aCon) → KA
  38. Intersect(xAxis, aCon) → KB
  39. Cone(KB, KA, 1) → bCon, dCon, eCon
  40. Intersect(yAxis, xAxis) → KC
  41. Segment(KC, KA) → lCon
  42. Distance(KB, KC) → rCon
  43. Angle(KA, KC, KB) → α
  44. Rotate(KA, (abs(tCil) * α), yAxis) → KAA
  45. OrthogonalLine(KAA, xOyPlane) → fCon
  46. Intersect(fCon, xOyPlane) → KBB
  47. Rotate(KC, bet / 2, fCon) → KCC
  48. Rotate(KC, ((-bet)) / 2, fCon) → KCR
  49. CircumcircleArc(KCC, KC, KCR) → gCon
  50. Distance(KAA, KBB) → hCon
  51. Distance(KBB, KCC) → rcCon
  52. Surface(((kCon - ((v * rcCon) * cos((u * bet / 2)))) * cos(ϕCon)) - (((1 - v) * hCon) * sin(ϕCon)), ((v * rcCon) * sin((u * bet / 2))), (((1 - v) * hCon) * cos(ϕCon)) + ((kCon - ((v * rcCon) * cos((u * bet / 2)))) * sin(ϕCon)), u, -1, 1, v, 0, 1) → jCon
  53. CurveCartesian(((kCon - (rcCon * cos((u * bet / 2)))) * cos(ϕCon)), (rcCon * sin((u * bet / 2))), ((kCon - (rcCon * cos((u * bet / 2)))) * sin(ϕCon)), u, -1, 1) → mCon
  54. Segment(KE, KF) → nCon
  55. Segment(KD, KF) → pCon
  56. If(abs(tCil) < 1, CurveCartesian[(rCil * sin((u * θCil))), ((rCil * (1 - cos((u * θCil)))) * cos(ϕCil)) - (3 * sin(ϕCil)), (3 * cos(ϕCil)) + ((rCil * (1 - cos((u * θCil)))) * sin(ϕCil)), u, -1, 1], CurveCartesian[(pi * u), (-3 * sin(ϕCil)), (3 * cos(ϕCil)), u, -1, 1]) → jCil
  57. If(abs(tCil) < 1, CurveCartesian[(rCil * sin((u * θCil))), ((rCil * (1 - cos((u * θCil)))) * cos(ϕCil)), ((rCil * (1 - cos((u * θCil)))) * sin(ϕCil)), u, -1, 1], CurveCartesian[(pi * u), 0, 0, u, -1, 1]) → lCil
  58. If(abs(tCil) < 1, Surface[(rCil * sin((u * θCil))), ((rCil * (1 - cos((u * θCil)))) * cos(ϕCil)) - (v * sin(ϕCil)), (v * cos(ϕCil)) + ((rCil * (1 - cos((u * θCil)))) * sin(ϕCil)), u, -1, 1, v, 0, 3], Surface[(pi * u), ((-v) * sin(ϕCil)), (v * cos(ϕCil)), u, -1, 1, v, 0, 3]) → iCil
  59. Segment(R_1, V) → missingSide

AI 绘图提示语参考

画一个可折叠的棱柱几何图：先在空间中给定一圆以确定底面边界，再以底面多边形为核心构造棱柱的侧面与底面；同时放置一条参考平面并生成棱柱的三维实体可视化。随后把棱柱沿相邻面关系展开成二维网格：各个面以多边形块表示，并在展开图中保留对应的公共边，用旋转与连接使相邻面正确贴合。再展示等价的金字塔网作为对比，并保留可调的棱柱参数，使学生能观察底边尺寸变化后网与实体如何同步调整。
以上提示语为根据原始绘图命令整理而来，仅供参考。