从球体到立方体再到球体：三维变形与对称探索

GeoGebra 命令及参数

1. Angle(A) → α
2. Distance((0, 0), A) → R
3. Rotate(A, π / 2, (0, 0)) → A'
4. CurveCartesian((R * cos(t)), (R * sin(t)), 0, t, α, α + π / 2) → a
5. Polygon(A, A', 4) → pol1, f, g, h_1, i, B, C
6. Midpoint(f) → F
7. Distance(E, F) → h
8. Point(Circle[A', Distance[A, A'], Segment[A, A']]) → G
9. Cube(A, A', G) → cub, H, caraGHLK, caraIJKL, arestaAH, arestaAA', arestaA'G, arestaGH, arestaAI, arestaHL, arestaIL, arestaIJ, I, arestaA'J, arestaJK, arestaGK, arestaKL, J, K, L, caraAA'GH, caraAHLI, caraAA'JI, caraA'GKJ
10. Midpoint(A', L) → M
11. Distance(A, M) → h'
12. CurveCartesian(x(M) + (h' * cos(t)), y(M), z(M) + (h' * sin(t)), t, 0, (2 * π)) → e
13. Angle(N, M, O) → β
14. CurveCartesian(x(M) + (h' * cos(t)), y(M), z(M) + (h' * sin(t)), t, (2 * π) - β, (3 * π / 2)) → k_0
15. Rotate(k_0, α - π, zAxis) → k'
16. Sphere(M, A) → l
17. Surface(k', π / 2, zAxis) → s_2
18. CurveCartesian(AA + (k * (AA' - AA)), k, 0, 1) → n
19. Surface((k * n(t)), k, 0, 1, t, 0, 1) → s_1
20. Surface((s * s_1(k, t)) + ((1 - s) * s_2((c1 * k) + c2, (c3 * t) + c4)), k, 0, 1, t, 0, 1) → c
21. Rotate(c, π / 2, zAxis) → c_1
22. Rotate(c, π, zAxis) → c_2
23. Rotate(c, (3 * π / 2), zAxis) → c_3
24. Mirror(c, pp) → c_4
25. Mirror(c_1, pp) → c_5
26. Mirror(c_2, pp) → c_6
27. Mirror(c_3, pp) → c_7
28. Midpoint(i) → P
29. Midpoint(f) → Q
30. Midpoint(h_1) → S
31. Centroid(caraAHLI) → U
32. Centroid(caraA'GKJ) → V
33. Intersect(zAxis, l) → W, Z
34. Mirror(a, pp) → a'
35. Rotate(e, α - π, zAxis) → e'
36. Angle(Z, M, I) → γ
37. CurveCartesian(e'(t), t, π / 2 + γ, (3 * π / 2) - γ) → o
38. Surface(o, π / 2, zAxis) → s_3
39. CurveCartesian(I + (k * (A - I)), k, 0, 1) → p
40. CurveCartesian(J + (k * (A' - J)), k, 0, 1) → q
41. Surface((k * q(t)) + ((1 - k) * p(t)), k, 0, 1, t, 0, 1) → s_4
42. Surface((s * s_4(k, t)) + ((1 - s) * s_3((c5 * t) + c6, (c7 * k) + c8)), k, 0, 1, t, 0, 1) → d
43. Rotate(d, π / 2, zAxis) → d_1
44. Rotate(d, (2 * π / 2), zAxis) → d_2
45. Rotate(d, (3 * π / 2), zAxis) → d_3