Et, oui !!!

En fait, tous les cas n'ont donc pas été étudiés   ...

Alors où est le défaut ? (pourquoi ne voit-on pas le plan ici ?)

Et comment y remédier ? 

Dans quel cas précis se trouve-t-on ici :

I est sur l'arête supérieure droite et ... ?

Comment obtiendrait-on le plan ?

Quel(s) autre(s) cas va (vont) se présenter ?
(en gardant I sur une arête)

Les points I, J et K appartiennent aux faces de dessus, de droite et de devant.

Sauf oubli involontaire, on a regardé tous les cas possibles et on obtient toujours la section du cube par le plan (IJK).

Prenez le temps de vérifier ceci.

Si du texte apparait, l'oubli était volontaire ... !

Remarque : Le cas dans lequel le point est choisi sur une arête de sa face n'a été fait que pour le seul point I.

Dans quel cas précis se trouve-t-on ici :

I est sur l'arête supérieure droite et ... ?

Comment obtiendrait-on le plan ?

Quel(s) autre(s) cas va (vont) se présenter ?
(en gardant I sur une arête)

A = (6, 0)
B = (-8, 0)
C = (-8, -14)
D = (6, -14)
poly1, a, b, c, d = Polygon(A, B, C, D)
K = PointIn(poly1)
F = (0, 6)
E = (14, 6)
poly2, a_1, b_1, f, g = Polygon(A, B, F, E)
H = (14, -8)
poly3, a_2, d_1, h, g_1 = Polygon(A, D, H, E)
I = PointIn(poly2)
J = PointIn(poly3)
G = (0, -8)
e = Segment(G, C)
i = Segment(G, H)
j = Segment(G, F)
k = Line(K, d)
l = Line(J, d)
m = Line(J, g)
n = Line(I, g)
p = Line(I, a)
q = Line(K, a)
Kx = Intersect(k, a)
Kz = Intersect(q, d)
Jz = Intersect(m, d)
Ix = Intersect(n, a)
Iy = Intersect(p, g)
Jy = Intersect(l, g)
x_I = Distance(A, Ix)
y_I = Distance(A, Iy)
y_J = Distance(A, Jy)
z_J = Distance(A, Jz)
x_K = Distance(A, Kx)
z_K = Distance(A, Kz)
r_x: (x_I * y_J * z_K + x_K * y_I * z_J) / (y_J * z_K - y_I * z_K + y_I * z_J)
r = Circle(A, r_x)
L = Intersect(r, a, 1)
r_y: (x_K * y_I * z_J + x_I * y_J * z_K) / (x_I * z_K - x_I * z_J + x_K * z_J)
c_1 = Circle(A, r_y)
M = Intersect(c_1, g)
r_z: (x_K * y_I * z_J + x_I * y_J * z_K) / (x_I * y_J - x_K * y_J + x_K * y_I)
s = Circle(A, r_z)
N = Intersect(s, d)
h_1 = Segment(I, Ix)
i_1 = Segment(I, Iy)
j_1 = Segment(Kx, K)
k_1 = Segment(K, Kz)
l_1 = Segment(Jy, J)
m_1 = Segment(J, Jz)
t = Ray(L, I)
O = Intersect(b_1, t)
P = Intersect(t, f)
e_1 = Ray(L, K)
Q = Intersect(e_1, c)
R = Intersect(e_1, b)
f_1 = Ray(N, K)
S = Intersect(f_1, b)
T = Intersect(f_1, c)
n_1 = Ray(N, J)
U = Intersect(n_1, h)
V = Intersect(n_1, d_1)
p_1 = Ray(M, J)
W = Intersect(p_1, d_1)
Z = Intersect(p_1, h)
q_1 = Segment(L, M)
r_1 = Segment(M, Z)
s_1 = Segment(L, R)
t_1 = Segment(W, M)
b_2 = Segment(N, M)
c_2 = Segment(N, S)
d_2 = Ray(M, I)
A_1 = Intersect(d_2, b_1)
B_1 = Intersect(f, d_2)
e_2 = Segment(M, B_1)
f_2 = Segment(N, T)
g_2 = Segment(M, A_1)
h_2 = Segment(L, P)
i_2 = Segment(L, Q)
j_2 = Segment(N, L)
k_2 = Segment(N, U)
l_2 = Segment(N, V)
m_2 = Segment(L, O)
n_2 = Segment(S, A_1)
p_2 = Line(W, g_2)
q_2 = Line(A_1, t_1)
C_1 = Intersect(q_2, b)
D_1 = Intersect(p_2, c)
r_2 = Segment(A_1, C_1)
s_2 = Segment(C_1, D_1)
t_2 = Segment(D_1, W)
a_3 = Segment(Q, W)
b_3 = Line(Q, h_2)
c_3 = Line(P, i_2)
E_1 = Intersect(c_3, h)
F_1 = Intersect(b_3, d_1)
d_3 = Segment(Q, F_1)
e_3 = Segment(P, E_1)
f_3 = Segment(E_1, F_1)
g_3 = Line(T, g_2)
G_1 = Intersect(g_3, e)
h_3 = Segment(A_1, G_1)
i_3 = Segment(G_1, T)
j_3 = Line(Q, q_1)
H_1 = Intersect(j_3, i)
k_3 = Segment(Z, H_1)
l_3 = Segment(H_1, Q)
m_3 = Segment(U, P)
n_3 = Line(U, j_2)
I_1 = Intersect(n_3, j)
p_3 = Segment(U, I_1)
q_3 = Segment(I_1, O)
r_3 = Line(V, m_2)
s_3 = Line(O, l_2)
J_1 = Intersect(s_3, j)
K_1 = Intersect(r_3, i)
t_3 = Segment(J_1, O)
a_4 = Segment(K_1, J_1)
b_4 = Segment(V, K_1)
L_1 = Intersect(s_3, r_3)
M_1 = Intersect(q_2, p_2)
e_4 = Segment(M_1, O)
N_1 = Intersect(c_3, b_3)
i_4 = Line(U, c_2)
j_4 = Line(S, k_2)
O_1 = Intersect(j_4, b_1)
P_1 = Intersect(i_4, f)
k_4 = Segment(U, P_1)
l_4 = Segment(P_1, O_1)
m_4 = Segment(O_1, S)
n_4 = Line(P, j_2)
Q_1 = Intersect(n_4, i)
p_4 = Segment(P, Q_1)
q_4 = Segment(Q_1, V)
r_4 = Line(S, b_2)
R_1 = Intersect(r_4, j)
s_4 = Segment(B_1, R_1)
t_4 = Segment(R_1, S)
a_5 = Line(T, e_2)
b_5 = Line(B_1, f_2)
S_1 = Intersect(j, b_5)
d_5 = Segment(S_1, B_1)
T_1 = Intersect(e, a_5)
c_5 = Segment(T, T_1)
e_5 = Segment(T_1, S_1)
f_5 = Segment(T, T_1)
U_1 = Intersect(b_5, a_5)
g_5 = Segment(T, U_1)
h_5 = Segment(U_1, B_1)
i_5 = Line(Z, s_1)
V_1 = Intersect(i_5, i)
j_5 = Line(R, r_1)
W_1 = Intersect(j_5, e)
k_5 = Segment(W_1, V_1)
l_5 = Segment(W_1, R)
m_5 = Segment(V_1, Z)
n_5 = Line(W, q_1)
Z_1 = Intersect(e, n_5)
p_5 = Segment(Z_1, R)
q_5 = Segment(Z_1, W)
A_2 = Intersect(b_1, r_4)
r_5 = Segment(S, A_2)
s_5 = Segment(A_2, I)
t_5 = Line(R_1, c_2)
B_2 = Intersect(t_5, f)
a_6 = Segment(R_1, B_2)
b_6 = Segment(I, B_2)
c_4 = Line(U, j_2)
C_2 = Intersect(c_4, f)
d_4 = Segment(I, C_2)
f_4 = Segment(C_2, U)

Tool: suite
Input: f, u_0, n
Output: b, L1, L2, L3, L4, liste1, liste2, liste3, liste4, liste5, c

f(x) = 0.05 * x^(2)
b = OrthogonalLine(u_0, xAxis)
L1 = Sequence(Iteration[f, x(u_0), i], i, 0, n)
L3 = Sequence((Element[L1, i], f(Element[L1, i])), i, 1, n)
L4 = Sequence((Element[L1, i + 2], Element[L1, i + 2]), i, 0, n - 1)
L2 = Sequence((Element[L1, i + 1], 0), i, 1, n)
liste1 = Sequence(Segment[Element[L3, i], Element[L4, i]], i, 1, n)
liste2 = Sequence(Segment[Element[L2, i], Element[L3, i + 1]], i, 1, n - 1)
liste3 = Sequence(Segment[Element[L2, i], Element[L4, i]], i, 1, n)
liste4 = Sequence(Segment[Element[L3, i + 1], Element[L4, i]], i, 1, n - 1)
c = Segment(Element[L3, 1], u_0)
liste5 = Sequence(Text["u_" + (i), Element[L2, i]], i, 1, n)