(0,0)

(1,0)

(0,1)

\mathcal{R}

f

\text{unité sur }(Ox)=

\text{unité sur }(Oy)=

\text{angle entre }(Ox)\text{ et }(Oy)=

v

v=(x,y)

v=(x,y)

\operatorname{Vect}(v_1)

\operatorname{Vect}(v_1,v_2)

ax+by=c

\operatorname{Vect}(

,

)

v_1

v_2

v_1

v_2

=

)

v_1

v_1

v_1

\textcolor{orange}{t_1v_1}

,\quad t_2=

t_1v_1+t_2v_2

t_1v_1

t_2v_2

\sigma(v_1,v_2)

t_1=

t_1v_1

t_2v_2

[v]_{\mathcal{B}}

t_1v_1

t_1v_1+t_2v_2

[v]_{\mathcal{B}}

[v]_{\mathcal{B}}=\textcolor{orange}{t_1},\quad \mathcal{B}=v_1 

t_1v_1

f : \mathbb{R}^2\rightarrow\mathbb{R}^2

f(x,y)=(

,

)

f(v)=f(x,y)

f

x(1,0)

y(0,1)

A

g(x,y)=(

g(v)=g(x,y)

g

\operatorname{det}(A)

\lambda f : \mathbb{R}^2\rightarrow\mathbb{R}^2

f+g:\mathbb{R}^2\rightarrow\mathbb{R}^2

f\circ g : \mathbb{R}^2\rightarrow\mathbb{R}^2

\lambda f(v)=\lambda f(x,y)

\lambda f

(f+g)(v)=f(v)+g(v)

(f+g)(v)=f(v)+g(v)

f+g

(f\circ g)(v)=f(g(v))

f(g(v))=f(g(x,y))

f\circ g

g(v)=g(x,y)

g

f

\text{rebond en}

\cdots

(?\,,\,?)

1\text{er rebond en}

(?\,,\,?)

x^2+y^2= \,\,\cdots

x^2+y^2=

v_{\theta}

v_{\theta}=(\cos\theta,\sin\theta)

,\,\theta=

[f(v)]_{\mathcal{B}}

[f(v)]_{\mathcal{B}}=[f]_{\mathcal{B}}[v]_{\mathcal{B}}

[f]_{\mathcal{B}}

b_{11} = 5.42 (slider from 0 to 10, step 0.01)
n = 0 (slider from 0 to 1, step 1)
α = 1.5707963267948966 rad (slider from -3.141592653589793 to 3.141592653589793, step 0.017453292519943295)
b = 1 (slider from 0.01 to 10, step 0.01)
a = 1 (slider from 0.01 to 10, step 0.01)
C = (0.19258490812216766, 0.6788860228097988)
B = (0.5426134319450314, 0.01687555384047414)
A: (0, 0)
d = Circle(A, b)
c = Circle(A, a)
D = Point(c)
I = If(n ≟ 0, B, D)
D' = Rotate(D, α, A)
f = Ray(A, D')
E = Intersect(d, f)
J = If(n ≟ 0, C, E)
m1: {{x(I), x(J)}, {y(I), y(J)}}
e = Determinant(m1)
m2 = If(e ≟ 0, {{0, 0}, {0, 0}}, m1^(-1))
eq5: ((m2(1, 1) * x) + (m2(1, 2) * y))^(2) + ((m2(2, 1) * x) + (m2(2, 2) * y))^(2) = b_{11}
g_6 = 0.41 (slider from 0.1 to 1, step 0.01)
f_6 = 0.41 (slider from 0.1 to 1, step 0.01)
e_6 = 1.82 (slider from 0.1 to 5, step 0.01)
R_3 = (1.6823381635070431, -5.683834177159883)
Q_3 = (-0.8751826908745403, -8.070853641249357)
w_6 = Vector(Q_3, R_3)
S_3 = Rotate(R_3, 90°, Q_3)
v_6 = Vector(Q_3, S_3)
V_3: Q_3 + (g_6 * w_6) + (f_6 * v_6)
U_3: R_3 + (f_6 * v_6)
T_3: Q_3 + (e_6 * v_6)
W_3: T_3 + (g_6 * w_6)
poly1, h_6, p_6, q_6, r_6, s_6, t_6 = Polygon(Q_3, R_3, U_3, V_3, W_3, T_3)
b_5(x, y) = 0
a_5(x, y) = 0
n_1 = 0 (slider from 0 to 1, step 1)
N: (2, 1)
T: (m1 * N)
M = (2.3079746825299052, 0.8310723375153904)
I_3 = If(n_1 ≟ 0, M, T)
Z: (m2 * M)
A_3 = If(n_1 ≟ 0, Z, N)
B_3: A + (a_5(A_3) * I) + (b_5(A_3) * J)
D_2: (0, 0)
G_2: (m1 * D_2)
f_2 = If(e ≟ 0 ∨ D_2 ≟ A, y = 0, Line[A, G_2])
k_3 = 1 (slider from -2 to 1, step 1)
j_3 = 1 (slider from 1 to 2, step 1)
o = false
i_3 = 1 (slider from 1 to 3, step 1)
j_5 = 2 (slider from -1 to 2, step 1)
i_5 = 1 (slider from 1 to 3, step 1)
F: (x(A), y(A))
G = (15.834829113315058, 26.150396509959755)
H = (15.937963651396757, 23.262629443673873)
K = (20.269614250825637, 11.505292102366909)
ChampTexte1 = Textfield(a)
ChampTexte2 = Textfield(b)
ChampTexte3 = Textfield(α)
g = true
i = 1 (slider from -1 to 2, step 1)
j = 1 (slider from 1 to 2, step 1)
L = If(j ≟ 1, H, K)
h = If(e ≟ 0, Line[(0, 0), (1, 0)], Line[A, I])
k = If(e ≟ 0, Line[(0, 0), (1, 0)], Line[A, J])
l = false
O: (0.6, -0.8)
P: (-4, 5)
Q: (3, -9)
R: (3, 9)
S: (9, 3)
m = 1 (slider from 1 to 6, step 1)
ChampTexte4 = Textfield(N)
i_1 = 2 (slider from -1 to 2, step 1)
U = If(n_1 ≟ 0, M, T)
V = If(e ≟ 0, A, Intersect[Line[U, h], k])
W = If(e ≟ 0, A, Intersect[Line[U, k], h])
p = Segment(U, V)
q = Segment(U, W)
j_1 = 1 (slider from 0 to 2, step 1)
A_1 = (-22.634353591136602, 23.623600326959593)
ChampTexte5 = Textfield(O)
ChampTexte6 = Textfield(P)
ChampTexte7 = Textfield(Q)
ChampTexte8 = Textfield(R)
ChampTexte9 = Textfield(S)
k_1 = 1 (slider from 0 to 1, step 1)
l_1 = 1 (slider from 0 to 1, step 1)
m_1 = 1 (slider from 0 to 1, step 1)
n_2 = 1 (slider from 0 to 1, step 1)
i_2 = 1 (slider from 0 to 1, step 1)
B_1: (m1 * O)
C_1: (m1 * P)
D_1: (m1 * Q)
E_1: (m1 * R)
F_1: (m1 * S)
j_2 = 1 (slider from 0 to 1, step 1)
texte15: "" + (LaTeX[O]) + ""
k_2 = 1 (slider from 0 to 1, step 1)
texte16: "" + (LaTeX[P]) + ""
l_2 = 1 (slider from 0 to 1, step 1)
m_2 = 1 (slider from 0 to 1, step 1)
n_3 = 1 (slider from 0 to 1, step 1)
texte17: "" + (LaTeX[Q]) + ""
texte18: "" + (LaTeX[R]) + ""
texte19: "" + (LaTeX[S]) + ""
G_1 = (-14.641426889809503, 23.417331250796313)
I_1: (2, 0)
ChampTexte11 = Textfield(I_1)
H_1: (3, 1)
ChampTexte10 = Textfield(H_1)
J_1 = (-12.063063437768504, 23.468898519837133)
K_1 = (-6.493798381359943, 23.468898519837133)
L_1: (0, 0)
M_1: (0, 0)
ChampTexte12 = Textfield(L_1)
ChampTexte13 = Textfield(M_1)
N_1: (m1 * H_1)
O_1: (m1 * I_1)
P_1: (m1 * M_1)
Q_1: (m1 * L_1)
l_3 = 2 (slider from 0 to 3, step 1)
m_3 = 2 (slider from 0 to 3, step 1)
texte28: "" + (LaTeX[H_1]) + ""
texte29: "" + (LaTeX[I_1]) + ""
m3: {{x(H_1), x(I_1)}, {y(H_1), y(I_1)}}
r = MatrixRank(m3)
s = If((¬(r ≟ 1)) ∨ e ≟ 0, Line[(0, 0), (1, 0)], ¬(O_1 ≟ A), Line[A, O_1], Line[A, N_1])
t = If(e ≟ 0 ∨ (¬(r ≟ 2)), Line[(0, 0), (1, 0)], Line[A, O_1])
f_1 = If(e ≟ 0 ∨ (¬(r ≟ 2)), Line[(0, 0), (1, 0)], Line[A, N_1])
R_1 = If(e ≟ 0 ∨ (¬(r ≟ 2)), (0, 0), Intersect[Line[U, f_1], t])
S_1 = If(e ≟ 0 ∨ (¬(r ≟ 2)), (0, 0), Intersect[Line[U, t], f_1])
g_1 = Segment(U, R_1)
h_1 = Segment(U, S_1)
n_4 = 1 (slider from -3 to 1, step 1)
i_4 = 2 (slider from -1 to 2, step 1)
texte32: "" + (LaTeX[L_1]) + ""
texte33: "" + (LaTeX[M_1]) + ""
m4: {{x(L_1), x(M_1)}, {y(L_1), y(M_1)}}
v = MatrixRank(m4)
p_1 = If((¬(v ≟ 1)) ∨ e ≟ 0, Line[(0, 0), (1, 0)], ¬(P_1 ≟ A), Line[A, P_1], Line[A, Q_1])
j_4 = 2 (slider from -1 to 2, step 1)
q_1 = If(e ≟ 0 ∨ (¬(v ≟ 2)), Line[(0, 0), (1, 0)], Line[A, Q_1])
r_1 = If(e ≟ 0 ∨ (¬(v ≟ 2)), Line[(0, 0), (1, 0)], Line[A, P_1])
T_1 = If(e ≟ 0 ∨ (¬(v ≟ 2)), (0, 0), Intersect[Line[U, r_1], q_1])
U_1 = If(e ≟ 0 ∨ (¬(v ≟ 2)), (0, 0), Intersect[Line[U, q_1], r_1])
s_1 = Segment(U, T_1)
t_1 = Segment(U, U_1)
u(x, y) = x + (3 * y)
ChampTexte14 = Textfield(u)
w = 0 (slider from -50 to 50, step 0.01)
ChampTexte15 = Textfield(w)
k_4 = 1 (slider from 1 to 3, step 1)
m5 = Transpose(m2)
a_1: u(1, 0)
b_1: u(0, 1)
c_1: a_1^(2) + b_1^(2)
V_1: (a_1, b_1)
W_1: (m5 * V_1)
eq1 = If(e ≟ 0 ∨ c_1 ≟ 0, y = 0, (x(W_1) * x) + (y(W_1) * y) = w)
l_4 = 1 (slider from 0 to 1, step 1)
d_1(x, y) = x - y
ChampTexte16 = Textfield(d_1)
e_1 = 0 (slider from -50 to 50)
ChampTexte17 = Textfield(e_1)
m_4 = 1 (slider from 0 to 1, step 1)
o_1: d_1(1, 0)
u_1: d_1(0, 1)
v_1: o_1^(2) + u_1^(2)
Z_1: (o_1, u_1)
A_2: (m5 * Z_1)
eq2 = If(e ≟ 0 ∨ v_1 ≟ 0, y = 0, (x(A_2) * x) + (y(A_2) * y) = e_1)
w_1(x, y) = x
ChampTexte18 = Textfield(w_1)
a_2 = 0 (slider from -50 to 50, step 0.01)
ChampTexte19 = Textfield(a_2)
n_5 = 1 (slider from 0 to 1, step 1)
b_2: w_1(1, 0)
c_2: w_1(0, 1)
d_2: b_2^(2) + c_2^(2)
B_2: (b_2, c_2)
C_2: (m5 * B_2)
eq3 = If(e ≟ 0 ∨ d_2 ≟ 0, y = 0, (x(C_2) * x) + (y(C_2) * y) = a_2)
ChampTexte20 = Textfield(D_2)
E_2: (0, 0)
F_2: (0, 0)
ChampTexte21 = Textfield(E_2)
ChampTexte22 = Textfield(F_2)
k_5 = 2 (slider from 0 to 2, step 1)
l_5 = 2 (slider from 0 to 2, step 1)
m_5 = 1 (slider from -1 to 1, step 1)
n_6 = -1 (slider from -1 to 1, step 1)
i_6 = -1 (slider from -1 to 1, step 1)
texte37: "" + (LaTeX[D_2]) + ""
H_2: (m1 * E_2)
texte39: "" + (LaTeX[E_2]) + ""
g_2 = If(e ≟ 0 ∨ E_2 ≟ A, y = 0, Line[A, H_2])
I_2: (m1 * F_2)
texte41: "" + (LaTeX[F_2]) + ""
h_2 = If(e ≟ 0 ∨ F_2 ≟ A, y = 0, Line[A, I_2])
texte42: "t_1 v_1\,,\quad v_1=" + (LaTeX[D_2]) + "\,,\quad t_1="
e_2 = 0 (slider from -5 to 5, step 0.01)
ChampTexte23 = Textfield(e_2)
j_6 = 0 (slider from 0 to 1, step 1)
J_2: (e_2 * G_2)
K_2: (e_2 * D_2)
texte44: "" + (LaTeX[K_2]) + ""
k_6 = 0 (slider from 0 to 2, step 1)
o_2 = 0 (slider from -5 to 5, step 0.01)
p_2 = 0 (slider from -5 to 5, step 0.01)
ChampTexte24 = Textfield(o_2)
ChampTexte25 = Textfield(p_2)
L_2: (o_2 * N_1) + (p_2 * O_1)
M_2: (o_2 * H_1)
N_2 = Textfield(I_1)
O_2: (p_2 * I_1)
P_2: (o_2 * N_1)
Q_2: (p_2 * O_1)
texte49: "" + (LaTeX[M_2]) + ""
texte50: "" + (LaTeX[O_2]) + ""
R_2: M_2 + O_2
texte48: "" + (LaTeX[R_2]) + ""
q_2 = Segment(A, P_2)
r_2 = Segment(A, Q_2)
s_2 = Segment(P_2, L_2)
t_2 = Segment(Q_2, L_2)
f_3: y = y(G)
u_2 = false
l_6 = 1 (slider from 1 to 3, step 1)
g_3 = Segment(A, N_1)
h_3 = Segment(A, O_1)
S_2: O_1 + N_1
p_3 = Segment(N_1, S_2)
q_3 = Segment(O_1, S_2)
q1, a_3, r_3, s_3, t_3 = Polygon(A, N_1, S_2, O_1)
T_2: P_1 + Q_1
b_3 = Segment(A, Q_1)
c_3 = Segment(A, P_1)
d_3 = Segment(Q_1, T_2)
e_3 = Segment(P_1, T_2)
q2, a_4, f_4, g_4, h_4 = Polygon(A, Q_1, T_2, P_1)
texte45: "t_1v_1+t_2v_2,\quad v_1=" + (LaTeX[H_1]) + "\,,\quad v_2=" + (LaTeX[I_1]) + "\,,"
m6 = If(r ≟ 2, m3^(-1), {{0, 0}, {0, 0}})
U_2 = If(n_1 ≟ 0, Z, N)
v_2: x(U_2)
w_2: y(U_2)
V_2: (m6 * U_2)
o_3: x(V_2)
u_3: y(V_2)
v_3: m6(1, 1)
w_3(x, y) = (m6(1, 1) * x) + (m6(1, 2) * y)
b_4(x, y) = (m6(2, 1) * x) + (m6(2, 2) * y)
texte60: "(" + (LaTeX[w_3]) + ")" + (LaTeX[H_1]) + ""
texte61: "(" + (LaTeX[b_4]) + ")" + (LaTeX[I_1]) + ""
texte62: "" + (LaTeX[o_3]) + "" + (LaTeX[H_1]) + ""
texte63: "" + (LaTeX[u_3]) + "" + (LaTeX[I_1]) + ""
texte55: "[v]_{\mathcal{B}}=P^{-1}\left(\begin{array}x \\ y  \end{array}\right)=\left(\begin{array}\textcolor{orange}{t_1} \\ \textcolor{blue}{t_2}  \end{array}\right),\,\quad v=(x,y),\quad\mathcal{B}=" + (LaTeX[texte26]) + "," + (LaTeX[texte27]) + " "
m_6 = 0 (slider from 0 to 2, step 1)
c_4 = true
texte14: "v=" + (LaTeX[Z]) + ""
texte13: "v=" + (LaTeX[N]) + ""
l1 = Sequence(Line[(l * I), k], l, -50, 50)
l2 = Sequence(Line[(l * J), h], l, -50, 50)
i_7 = 5 (slider from -5 to 5)
d_4: i_3 ≟ 1 ∧ i_5 ≟ 1 ∧ o ∧ j_5 ≤ 1 ∧ j_5 ≥ 0 ∧ (i_7 ≟ 1 ∨ i_7 ≟ 2)
e_4 = false
p_4 = Segment(A, J_2)
n_7 = 1 (slider from 1 to 2, step 1)
o_4 = false
q_4: i_3 ≟ 2 ∧ j_3 ≟ 1 ∧ o ∧ k_3 ≤ 0 ∧ k_3 ≥ -1 ∧ (i_7 ≟ 1 ∨ i_7 ≟ 2)
j_7 = 1 (slider from 1 to 2, step 1)
texte56: "[v]_{\mathcal{B}}=" + (LaTeX[m3]) + "^{-1}\left(\begin{array}x \\ y \end{array}\right)=\left(\begin{array}\textcolor{orange}{" + (LaTeX[w_3]) + "} \\\textcolor{blue}{ " + (LaTeX[b_4]) + "} \end{array}\right),\quad v=(x,y),\quad \mathcal{B}=" + (LaTeX[H_1]) + "," + (LaTeX[I_1]) + ""
r_4: r ≟ 2 ∧ k_3 ≟ -2 ∧ i_1 ≤ 1 ∧ m ≟ 1 ∧ (i_7 ≟ 1 ∨ i_7 ≟ 2) ∧ i_3 ≟ 2 ∧ l ∧ o
texte57: "[v]_{\mathcal{B}}=" + (LaTeX[m3]) + "^{-1}\left(\begin{array}" + (LaTeX[v_2]) + " \\ " + (LaTeX[w_2]) + "  \end{array}\right)=\left(\begin{array}\textcolor{orange}{" + (LaTeX[o_3]) + "} \\\textcolor{blue}{ " + (LaTeX[u_3]) + "} \end{array}\right),\quad v=" + (LaTeX[U_2]) + ",\quad \mathcal{B}=" + (LaTeX[H_1]) + "," + (LaTeX[I_1]) + ""
s_4: m ≟ 1 ∧ i_1 ≥ 1 ∧ i_3 ≟ 1 ∧ i_5 ≟ 1 ∧ j_5 ≟ -1 ∧ U ∈ f_2
t_4 = false
k_7 = 0 (slider from 0 to 1, step 1)
W_2 = If(n_1 ≟ 0, Z, N)
u_4: x(D_2)
v_4: y(D_2)
w_4 = If(¬(u_4 ≟ 0), x(W_2) / u_4, ¬(v_4 ≟ 0), y(W_2) / v_4, 0)
texte68: "[v]_{\mathcal{B}}=\textcolor{orange}{" + (LaTeX[w_4]) + "},\quad v=" + (LaTeX[W_2]) + ",\quad \mathcal{B}=" + (LaTeX[D_2]) + ""
Z_2: (w_4 * G_2)
texte70: "" + (LaTeX[w_4]) + "" + (LaTeX[W_2]) + ""
l_7 = 0 (slider from 0 to 2, step 1)
texte71: "[v]_{\mathcal{B}}=\left(\begin{array}\textcolor{orange}{t_1} \\ \textcolor{blue}{t_2}  \end{array}\right),\,\quad v=(x,y),\quad\mathcal{B}= " + (LaTeX[texte26]) + "," + (LaTeX[texte27]) + "  "
texte72: "[v]_{\mathcal{B}}=\left(\begin{array}\textcolor{orange}{" + (LaTeX[o_3]) + " } \\\textcolor{blue}{" + (LaTeX[u_3]) + "  } \end{array}\right),\quad v=" + (LaTeX[U_2]) + " ,\quad \mathcal{B}=" + (LaTeX[H_1]) + " , " + (LaTeX[I_1]) + ""
ChampTexte26 = Textfield(a_5)
ChampTexte27 = Textfield(b_5)
c_5 = false
d_5: i_7 ≥ 3 ∧ y(G) > y(L)
m7: {{a_5(1, 0), a_5(0, 1)}, {b_5(1, 0), b_5(0, 1)}}
m_7 = 1 (slider from -2 to 1, step 1)
C_3: (a_5(A_3), b_5(A_3))
texte77: "f(v)=" + (LaTeX[C_3]) + ""
D_3 = Rotate(B_3, -60°, I_3)
M' = Rotate(I_3, -3°, D_3)
E_3 = Rotate(B_3, 3°, D_3)
F_3 = Rotate(B_3, 8°, D_3)
u_5 = Vector(F_3, E_3)
e_5 = CircleArc(D_3, F_3, M')
G_3: A + (a_5(A_3) * I)
H_3: A + (b_5(A_3) * J)
f_5 = Segment(B_3, G_3)
g_5 = Segment(H_3, B_3)
n_8 = 0 (slider from 0 to 1, step 1)
h_5: i_7 ≥ 3
M'_{1} = Rotate(I_3, -30°, D_3)
o_5 = MatrixRank(m7)
m8: {{a_5(1, 0), a_5(0, 1), x(D_2)}, {b_5(1, 0), b_5(0, 1), y(D_2)}}
p_5 = MatrixRank(m8)
i_8 = 0 (slider from 0 to 1, step 1)
q_5 = If(¬(x(D_2) ≟ 0), m7(1, 1) / x(D_2), ¬(y(D_2) ≟ 0), m7(2, 1) / y(D_2), 0)
r_5 = If(¬(x(D_2) ≟ 0), m7(1, 2) / x(D_2), ¬(y(D_2) ≟ 0), m7(2, 2) / y(D_2), 0)
s_5(x, y) = (q_5 * x) + (r_5 * y)
texte80: "\fbox{f(x,y)=(" + (LaTeX[s_5]) + ")" + (LaTeX[D_2]) + "}"
t_5: x(A_3)
v_5: y(A_3)
texte84: "" + (LaTeX[v_5]) + "(0,1)"
texte83: "" + (LaTeX[t_5]) + "(1,0)"
w_5 = false
texte86: "A=" + (LaTeX[m7]) + ""
m9: {{x(D_2)}, {y(D_2)}}
m10: {{q_5, r_5}}
a_6: i_7 ≥ 3 ∧ y(G) > y(L)
j_8 = 0 (slider from 0 to 1, step 1)
texte87: "A=" + (LaTeX[m7]) + "=" + (LaTeX[m9]) + "" + (LaTeX[m10]) + ""
k_8 = 1 (slider from 1 to 2, step 1)
b_6(x, y) = 0
c_6(x, y) = 0
ChampTexte28 = Textfield(b_6)
ChampTexte29 = Textfield(c_6)
l_8 = 1 (slider from -2 to 1, step 1)
J_3: A + (b_6(A_3) * I) + (c_6(A_3) * J)
K_3: (b_6(A_3), c_6(A_3))
texte90: "g(v)=" + (LaTeX[K_3]) + ""
L_3 = Rotate(J_3, -60°, I_3)
M_3 = Rotate(I_3, -3°, L_3)
N_3 = Rotate(J_3, 3°, L_3)
O_3 = Rotate(J_3, 8°, L_3)
P_3 = Rotate(J_3, 30°, L_3)
d_6 = CircleArc(L_3, O_3, M_3)
u_6 = Vector(O_3, N_3)
m11: {{b_6(1, 0), b_6(0, 1)}, {c_6(1, 0), c_6(0, 1)}}
texte92: "B=" + (LaTeX[m11]) + ""
Q3 = NaN
m12: (m7 * m2)
Z_3: (m12 * Q_3)
A_4: (m12 * R_3)
B_4: (m12 * U_3)
C_4: (m12 * V_3)
D_4: (m12 * W_3)
E_4: (m12 * T_3)
F_4: A + (x(Z_3) * I) + (y(Z_3) * J)
G_4: A + (x(A_4) * I) + (y(A_4) * J)
H_4: A + (x(B_4) * I) + (y(B_4) * J)
I_4: A + (x(C_4) * I) + (y(C_4) * J)
J_4: A + (x(D_4) * I) + (y(D_4) * J)
K_4: A + (x(E_4) * I) + (y(E_4) * J)
poly2, f_7, g_7, h_7, p_7, q_7, r_7 = Polygon(G_4, F_4, K_4, J_4, I_4, H_4)
o_6 = false
m_8 = 0 (slider from 0 to 1, step 1)
n_9 = 0 (slider from 0 to 2, step 1)
a_7: i_7 ≟ 4 ∧ y(G) < y(L)
b_7: i_7 ≟ 4
texte94: "f(x,y)=(" + (LaTeX[a_5]) + "," + (LaTeX[b_5]) + ")\,,\quad A=" + (LaTeX[m7]) + ""
c_7 = Determinant(m7)
texte95: "\operatorname{det}(A)=" + (LaTeX[c_7]) + ""
s_7 = Segment(Q_3, R_3)
t_7 = Segment(R_3, U_3)
d_7 = Segment(U_3, V_3)
e_7 = Segment(V_3, W_3)
f_8 = Segment(W_3, T_3)
g_8 = Segment(Q_3, T_3)
h_8 = Segment(G_4, F_4)
p_8 = Segment(F_4, K_4)
q_8 = Segment(K_4, J_4)
r_8 = Segment(J_4, I_4)
s_8 = Segment(I_4, H_4)
t_8 = Segment(H_4, G_4)
o_7: i_7 ≟ 3.5
u_7: i_7 ≟ 3.5 ∧ y(G) < y(L)
v_7 = false
w_7 = false
a_8 = false
b_8 = 0 (slider from -5 to 5, step 0.1)
m13: (b_8 * m7)
c_8(x, y) = (m13(1, 1) * x) + (m13(1, 2) * y)
d_8(x, y) = (m13(2, 1) * x) + (m13(2, 2) * y)
ChampTexte30 = Textfield(b_8)
texte99: "" + (LaTeX[b_8]) + "f(x,y)=(" + (LaTeX[c_8]) + "," + (LaTeX[d_8]) + ")"
L_4: (b_8 * B_3)
M_4: (b_8 * C_3)
texte101: "" + (LaTeX[b_8]) + " f(v)=" + (LaTeX[M_4]) + ""
i_9 = 1 (slider from 0 to 2, step 1)
j_9 = 0 (slider from 0 to 3, step 1)
M'_{2} = Rotate(M, 60°, L_4)
M'_{3} = Rotate(M, -3°, M'_{2})
N_4 = Rotate(L_4, 3°, M'_{2})
O_4 = Rotate(L_4, 8°, M'_{2})
e_8 = CircleArc(M'_{2}, O_4, M'_{3})
u_8 = Vector(O_4, N_4)
P_4 = Rotate(L_4, 30°, M'_{2})
m14: m7 + m11
o_8(x, y) = (m14(1, 1) * x) + (m14(1, 2) * y)
v_8(x, y) = (m14(2, 1) * x) + (m14(2, 2) * y)
texte104: "(f+g)(x,y)=(" + (LaTeX[o_8]) + "," + (LaTeX[v_8]) + ")"
k_9 = 0 (slider from 0 to 2, step 1)
l_9 = 0 (slider from 0 to 3, step 1)
Q_4: B_3 + J_3
R_4: C_3 + K_3
texte107: "(f+g)(v)=" + (LaTeX[R_4]) + ""
texte102: "" + (LaTeX[b_8]) + "f(v)"
m_9 = 0 (slider from 0 to 1, step 1)
M'_{4} = Rotate(M, 60°, Q_4)
M'_{5} = Rotate(M, -3°, M'_{4})
S_4 = Rotate(Q_4, 3°, M'_{4})
T_4 = Rotate(Q_4, 8°, M'_{4})
M'_{6} = Rotate(M, -30°, M'_{4})
c_9 = CircleArc(M'_{4}, T_4, M'_{5})
w_8 = Vector(T_4, S_4)
n_{10} = 1 (slider from 0 to 1, step 1)
i_{10} = 1 (slider from 0 to 1, step 1)
f_9 = If(A ≟ B_3, y = 0, Line[A, B_3])
g_9 = Segment(A, J_3)
h_9 = Segment(J_3, Q_4)
p_9 = Segment(Q_4, B_3)
q_9 = Segment(B_3, A)
j_{10} = 1 (slider from 0 to 2, step 1)
m15: (m7 * m11)
a_9(x, y) = (m15(1, 1) * x) + (m15(1, 2) * y)
b_9(x, y) = (m15(2, 1) * x) + (m15(2, 2) * y)
texte110: "f(g(x,y))=(" + (LaTeX[a_9]) + "," + (LaTeX[b_9]) + ")"
k_{10} = 3 (slider from 0 to 3, step 1)
l_{10} = 3 (slider from -1 to 3, step 1)
U_4: A + (a_9(A_3) * I) + (b_9(A_3) * J)
V_4 = Rotate(U_4, -60°, M)
M'_{7} = Rotate(M, -3°, V_4)
W_4 = Rotate(U_4, 3°, V_4)
Z_4 = Rotate(U_4, 8°, V_4)
A_5 = Rotate(U_4, 30°, V_4)
d_9 = CircleArc(V_4, Z_4, M'_{7})
u_9 = Vector(Z_4, W_4)
B_5: (a_9(A_3), b_9(A_3))
texte112: "f(g(v))=" + (LaTeX[B_5]) + ""
m_{10} = 1 (slider from 0 to 1, step 1)
C_5: (x(J_3), y(J_3))
texte115: "g(v)=" + (LaTeX[K_3]) + ""
D_5 = Rotate(C_5, 60°, M)
M'_{8} = Rotate(M, 3°, D_5)
E_5 = Rotate(C_5, -3°, D_5)
F_5 = Rotate(C_5, -8°, D_5)
G_5 = Rotate(C_5, -30°, D_5)
e_9 = CircleArc(D_5, M'_{8}, F_5)
v_9 = Vector(F_5, E_5)
H_5 = Rotate(C_5, -60°, U_4)
I_5 = Rotate(U_4, -3°, H_5)
J_5 = Rotate(C_5, 3°, H_5)
K_5 = Rotate(U_4, -8°, H_5)
r_9 = CircleArc(H_5, J_5, K_5)
w_9 = Vector(K_5, I_5)
L_5 = Rotate(U_4, -30°, H_5)
texte118: "\fbox{f(x,y)=(" + (LaTeX[a_5]) + "," + (LaTeX[b_5]) + ")\,\,,\quad g(x,y)=(" + (LaTeX[b_6]) + "," + (LaTeX[c_6]) + ")}"
o_9: i_7 ≟ 3.6 ∧ l_4 ≟ 0 ∧ l_1 ≟ 0
s_9 = false
N_5: (3, 1)
ChampTexte31 = Textfield(N_5)
n_{11} = 1 (slider from 0 to 1, step 1)
O_5: (m1 * N_5)
texte124: "" + (LaTeX[N_5]) + ""
M_5 = (8.089578600325023, 1.3696582198844105)
P_5 = If(n_{11} ≟ 0, M_5, O_5)
t_9: P_5 ∈ eq1
f_{10} = Segment(C_1, P_5)
i_{11} = 0 (slider from 0 to 3, step 1)
u_{10} = If(C_1 ≟ P_5, Vector[(0, 0), (1, 0)], Vector[C_1, P_5])
v_{10} = Mirror(u_{10}, eq1)
Q_5 = Translate(P_5, v_{10})
g_{10} = Ray(P_5, Q_5)
R_5 = Mirror(P_5, eq1)
S_5 = Mirror(Q_5, eq1)
j_{11} = 0 (slider from 0 to 1, step 1)
a_{10} = 8.75 (slider from 0 to 10, step 0.05)
T_5 = If(a_{10} ≤ 1, ((1 - a_{10}) * C_1) + (a_{10} * P_5), P_5 + ((a_{10} - 1) * v_{10}))
U_5 = Mirror(T_5, eq1)
V_5 = (-0.9866891873517978, -3.90578611047022)
b_{10}: (x(A_2) * x(v_{10})) + (y(A_2) * y(v_{10}))
c_{10} = If(b_{10} ≟ 0, -12, (e_1 - (x(A_2) * x(P_5)) - (y(A_2) * y(P_5))) / b_{10})
W_5: P_5 + (c_{10} * v_{10})
w_{10} = Mirror(v_{10}, eq2)
Z_5 = Translate(W_5, w_{10})
q_{10} = Ray(W_5, Z_5)
A_6 = Point(q_{10})
r_{10} = Segment(W_5, A_6)
p_{10} = Segment(P_5, W_5)
B_6 = Point(g_{10})
s_{10} = Segment(P_5, B_6)
C_6 = Mirror(W_5, eq2)
D_6 = Mirror(A_6, eq2)
t_{10} = Segment(C_6, D_6)
E_6 = Point(f_{10})
d_{10} = Segment(C_1, E_6)
F_6 = Mirror(P_5, eq1)
G_6 = Mirror(B_6, eq1)
h_{10} = Segment(F_6, G_6)
H_6 = Point(p_{10})
e_{10} = Segment(P_5, H_6)
o_{10}: i_7 ≟ 3.7
a_{11} = false
eq4: ((m2(1, 1) * x) + (m2(1, 2) * y))^(2) + ((m2(2, 1) * x) + (m2(2, 2) * y))^(2) = 1
ChampTexte32 = Textfield(b_{11})
k_{11} = 1 (slider from 0 to 1, step 1)
m16: (Transpose[m7] * m7)
l_{11} = 1 (slider from 0 to 1, step 1)
I_6 = If(n_1 ≟ 0, M, T)
f_{11} = Segment(A, I_6)
g_{11} = Segment(A, B_3)
c_{11} = Determinant(m7)
β = Angle(I_6, A, B_3)
d_{11} = false
J_6 = Point(eq4)
γ = 1.0471975511965976 rad (slider from -3.141592653589793 to 3.141592653589793, step 0.017453292519943295)
K_6: (cos(γ), sin(γ))
L_6: (m1 * K_6)
texte129: "v_{" + (LaTeX[γ]) + "}"
ChampTexte33 = Textfield(γ)
m_{11} = 1 (slider from 0 to 1, step 1)
M_6 = If(m_{11} ≟ 0, J_6, L_6)
h_{11} = Line(A, M_6)
δ = Angle(I, A, M_6)
n_{12} = 0 (slider from 0 to 3, step 1)
i_{12} = 1 (slider from 0 to 1, step 1)
j_{12} = 0 (slider from 0 to 1, step 1)
e_{11}: i_7 ≟ 5
o_{11} = true
texte134: "[f]_{\mathcal{B}}=" + (LaTeX[m3]) + "^{-1}" + (LaTeX[m7]) + "" + (LaTeX[m3]) + ""
m17: ((m6 * m7) * m3)
texte135: "[f]_{\mathcal{B}}=" + (LaTeX[m17]) + ""
p_{11} = true
k_{12} = 2 (slider from 0 to 2, step 1)
l_{12} = 0 (slider from 0 to 1, step 1)
m_{12} = 0 (slider from 0 to 1, step 1)
N_6: ((m6 * m2) * B_3)
O_6: A + (x(N_6) * N_1)
P_6: A + (y(N_6) * O_1)
q_{11} = Segment(B_3, P_6)
r_{11} = Segment(B_3, O_6)