圆柱曲面最短连接线：展开法与对称反射

GeoGebra 命令及参数

1. Circle(O, B, xOyPlane) → c
2. Circle(M, B, xOyPlane) → d
3. Radius(d) → R
4. Rotate(B, (r / R * 360°), M) → B'
5. CircleArc(M, B, B', xOyPlane) → u_1
6. Segment(B, B_1) → u_2
7. OrthogonalLine(O, xOyPlane) → f
8. Point(f) → A
9. Cylinder(O, A, r) → zyl, e, Deckfl, mantel
10. Surface((R * cos(u)), y(M) + (R * sin(u)), v, u, -90°, α - 90°, v, 0, H) → fl
11. Polygon((0, 0, 0), (0, (-H), 0), (((2 * r) * pi), (-H), 0), (((2 * r) * pi), 0, 0)) → v1, i, j, k, l
12. Rotate(v1, 270° + ((t - 1) * 90°), xAxis) → v1'
13. Rotate(C, ((t - 1) * 90°), xAxis) → C'
14. Line(C', C' + (1, 0, 0)) → g
15. Rotate(Translate[Deckfl, Vector[C' - C]], ((2 * (t - 1)) * 90°), g) → Deckfl'
16. Translate(u_1, Vector[(0, 0, H)]) → u_1'
17. Translate(B', Vector[(0, 0, H)]) → B''
18. Segment(B', B'') → m
19. OrthogonalLine(M, xOyPlane) → dreh
20. PointIn(mantel) → D
21. PointIn(mantel) → E
22. If(y(D) < r, (-1 * Angle[(1, r, z(D)), D_O, D]), (1 * Angle[(1, r, z(D)), D_O, D])) → α_D
23. CircleArc(D_O, D', D) → bog_D
24. If(x(D) < 0, ((-bog_D), 0, z(D)), (bog_D, 0, z(D))) → D_1
25. If(x(D) < 0, Rotate[D', ((-360°) / (((2 * r) * pi)) * bog_D / R), dreh], Rotate[D', ((360°) / (((2 * r) * pi)) * bog_D / R), dreh]) → D''
26. CircleArc(E_O, E', E) → bog_E
27. If(x(E) < 0, ((-bog_E), 0, z(E)), (bog_E, 0, z(E))) → E_1
28. If(x(E) < 0, Rotate[E', ((-360°) / (((2 * r) * pi)) * bog_E / R), dreh], Rotate[E', ((360°) / (((2 * r) * pi)) * bog_E / R), dreh]) → E''
29. If(y(E) < r, (-1 * Angle[(1, r, z(E)), E_O, E]), (1 * Angle[(1, r, z(E)), E_O, E])) → α_E
30. Rotate(D_1, ((t - 1) * 90°), xAxis) → D_2
31. Rotate(E_1, ((t - 1) * 90°), xAxis) → E_2
32. Segment(D_2, E_2) → str1
33. CurveCartesian((r * cos(α_D + (t * (α_E - α_D)))), r + (r * sin(α_D + (t * (α_E - α_D)))), z(D) + (t * (z(E) - z(D))), t, 0, 1) → spiral_1
34. If(y(D'') < y(M), (-1 * Angle[(1, R, z(D)), D_M, D'']), (1 * Angle[(1, R, z(D)), D_M, D''])) → α_{DM}
35. If(y(E'') < y(M), (-1 * Angle[(1, R, z(E)), E_M, E'']), (1 * Angle[(1, R, z(E)), E_M, E''])) → α_{EM}
36. CurveCartesian((R * cos(α_{DM} + (t * (α_{EM} - α_{DM})))), R + (R * sin(α_{DM} + (t * (α_{EM} - α_{DM})))), z(D'') + (t * (z(E'') - z(D''))), t, 0, 1) → spiral_2
37. CurveCartesian((r * cos(α_D + (t * (α_E - α_D)))), r + (r * sin(α_D + (t * (α_E - α_D)))), z(D) + (t * (z(E) - z(D))), t, (-pi), pi) → spiral_3
38. Line(D_2, E_2) → h