球面区域：几何与微积分结合的探索

GeoGebra 命令及参数

1. Surface(((rmax * cos(u)) * sin(v)), ((rmax * sin(u)) * sin(v)), (rmax * cos(v)), u, themin, themax, v, phimin, phimax) → Front
2. Surface(((rmin * cos(u)) * sin(v)), ((rmin * sin(u)) * sin(v)), (rmin * cos(v)), u, themin, themax, v, phimin, phimax) → Back
3. Surface(((r * cos(themax)) * sin(v)), ((r * sin(themax)) * sin(v)), (r * cos(v)), r, rmin, rmax, v, phimin, phimax) → Side1
4. Surface(((r * cos(themin)) * sin(v)), ((r * sin(themin)) * sin(v)), (r * cos(v)), r, rmin, rmax, v, phimin, phimax) → Side2
5. Surface(((r * cos(v)) * sin(phimax)), ((r * sin(v)) * sin(phimax)), (r * cos(phimax)), r, rmin, rmax, v, themin, themax) → Side3
6. Surface(((r * cos(v)) * sin(phimin)), ((r * sin(v)) * sin(phimin)), (r * cos(phimin)), r, rmin, rmax, v, themin, themax) → Side4
7. CurveCartesian((rmax * cos(u)), (rmax * sin(u)), 0, u, 0, π / 2) → hc1
8. CurveCartesian((rmin * cos(u)), (rmin * sin(u)), 0, u, 0, π / 2) → hc2
9. Sphere((0, 0, 0), rmax) → Sph
10. Distance(P1, zAxis) → r1p1
11. Distance(P2, zAxis) → r2p2
12. Distance(P3, zAxis) → r3p3
13. Distance(P4, zAxis) → r4p4
14. CurveCartesian((r1p1 * cos(u)), (r1p1 * sin(u)), z(P1), u, 0, π / 2) → hc3
15. CurveCartesian((r2p2 * cos(u)), (r2p2 * sin(u)), z(P2), u, 0, π / 2) → hc4
16. CurveCartesian((r3p3 * cos(u)), (r3p3 * sin(u)), z(P3), u, 0, π / 2) → hc5
17. CurveCartesian((r4p4 * cos(u)), (r4p4 * sin(u)), z(P4), u, 0, π / 2) → hc6
18. Rotate(hc1, ((-π)) / 2, yAxis) → vc1
19. Rotate(hc2, ((-π)) / 2, yAxis) → vc2
20. Rotate(hc1, π / 2, xAxis) → vc3
21. Rotate(hc2, π / 2, xAxis) → vc4
22. Rotate(vc3, themin, zAxis) → vc5
23. Rotate(vc4, themin, zAxis) → vc6
24. Rotate(vc3, themax, zAxis) → vc7
25. Rotate(vc4, themax, zAxis) → vc8
26. Segment(O, P1) → l1
27. Segment(O, P2) → l2
28. Rotate(O, themax - themin, zAxis) → O'
29. Rotate(P1, themax - themin, zAxis) → P1'
30. Segment(Rotate[O, themax - themin, zAxis], Rotate[P1, themax - themin, zAxis]) → l3
31. Rotate(O, themax - themin, zAxis) → O'_{1}
32. Rotate(P2, themax - themin, zAxis) → P2'
33. Segment(Rotate[O, themax - themin, zAxis], Rotate[P2, themax - themin, zAxis]) → l4