射影几何：仿射变换、交叉比、调和共轭、极性

GeoGebra 命令及参数

1. Point(xAxis) → P
2. If(x(P) ≟ x(P), x(P), Infinity) → p
3. Point(xAxis) → N
4. If(x(N) ≟ x(N), x(N), Infinity) → n
5. Point(xAxis) → I
6. If(x(I) ≟ x(I), x(I), Infinity) → d
7. Point(xAxis) → E
8. If(x(E) ≟ x(E), x(E), Infinity) → e
9. If(abs(n) ≠ Infinity, Line[V, N], Line[V, xAxis]) → lVN
10. Point(lVN) → N'
11. If(abs(p) ≠ Infinity, Line[V, P], Line[V, xAxis]) → lVP
12. Point(lVP) → P'
13. Line(N', P') → l
14. If(abs(d) ≠ Infinity, Line[V, I], Line[V, xAxis]) → lVI
15. Intersect(l, lVI) → I'
16. If(abs(e) ≠ Infinity, Line[V, E], Line[V, xAxis]) → lVE
17. Intersect(l, lVE) → E'
18. Angle(N, V, P) → View
19. If(n ≠ n ∨ abs(n) ≟ Infinity, (2 * p) - e, If[p ≠ p ∨ abs(p) ≟ Infinity, (2 * n) - e, If[e ≠ e ∨ abs(e) ≟ Infinity, (n + p) / 2, (((2 * n) * p) - ((n + p) * e)) / (n + p - (2 * e))]]) → dh
20. If(n ≠ n ∨ abs(n) ≟ Infinity, (2 * p) - d, If[p ≠ p ∨ abs(p) ≟ Infinity, (2 * n) - d, If[d ≠ d ∨ abs(d) ≟ Infinity, (n + p) / 2, (((2 * n) * p) - ((n + p) * d)) / (n + p - (2 * d))]]) → eh
21. If(e ≠ e ∨ abs(e) ≟ Infinity, (2 * d) - p, If[d ≠ d ∨ abs(d) ≟ Infinity, (2 * e) - p, If[p ≠ p ∨ abs(p) ≟ Infinity, (d + e) / 2, (((2 * d) * e) - ((d + e) * p)) / (e + d - (2 * p))]]) → nh
22. If(e ≠ e ∨ abs(e) ≟ Infinity, (2 * d) - n, If[d ≠ d ∨ abs(d) ≟ Infinity, (2 * e) - n, If[n ≠ n ∨ abs(n) ≟ Infinity, (d + e) / 2, (((2 * d) * e) - ((d + e) * n)) / (d + e - (2 * n))]]) → ph
23. If(e ≠ e ∨ abs(e) ≟ Infinity, (d - p) / (d - n), If[d ≠ d ∨ abs(d) ≟ Infinity, (e - n) / (e - p), If[p ≠ p ∨ abs(p) ≟ Infinity, (e - n) / (d - n), If[n ≠ n ∨ abs(n) ≟ Infinity, (d - p) / (e - p), (((e - n) * (d - p))) / (((d - n) * (e - p)))]]]) → cr
24. If(x(N') ≟ x(N') ∧ x(E') ≟ x(E'), (Vector[N', E'] * vU'), Infinity) → mN'E'
25. If(x(N') ≟ x(N') ∧ x(I') ≟ x(I'), (Vector[N', I'] * vU'), Infinity) → mN'I'
26. If(x(P') ≟ x(P') ∧ x(E') ≟ x(E'), (Vector[P', E'] * vU'), Infinity) → mP'E'
27. If(x(P') ≟ x(P') ∧ x(I') ≟ x(I'), (Vector[P', I'] * vU'), Infinity) → mP'I'
28. If(x(E') ≠ x(E') ∨ abs(x(E')) ≟ Infinity, mP'I' / mN'I', If[x(I') ≠ x(I') ∨ abs(x(I')) ≟ Infinity, mN'E' / mP'E', If[x(N') ≠ x(N') ∨ abs(x(N')) ≟ Infinity, mP'I' / mP'E', If[x(P') ≠ x(P') ∨ abs(x(P')) ≟ Infinity, mN'E' / mN'I', ((mN'E' * mP'I')) / ((mN'I' * mP'E'))]]]) → cr'
29. Line(N', lVE) → lEParallel
30. Line(P', lVI) → lIParallel
31. OrthogonalLine(I, lVE) → lIPerpendicular
32. Intersect(lEParallel, lVP) → P'p
33. Intersect(lIPerpendicular, lVE) → Vp
34. Intersect(lIParallel, lVN) → N'p
35. Segment(N', E') → seN'E'
36. Segment(N', I') → seN'I'
37. Segment(N, E) → seNE
38. Segment(N, I) → seNI
39. Segment(P', E') → seP'E'
40. Segment(P', I') → seP'I'
41. Segment(P, E) → sePE
42. Segment(P, I) → sePI
43. Segment(N', E') → ssN'E'
44. Segment(N', I') → ssN'I'
45. Segment(P', E') → ssP'E'
46. Segment(P', I') → ssP'I'