O

\overrightarrow{i}

\overrightarrow{j}

\mathcal{R}

(0,0)

(1,0)

(0,1)

t

M

\overrightarrow{u}

\left(\begin{array}{} \textcolor{white}{i} \ \textcolor{white}{i} \ \textcolor{white}{i} \end{array} \right.

\left.\begin{array}{} \textcolor{white}{i} \ \textcolor{white}{i} \ \textcolor{white}{i} \end{array} \right)

x+

y=

d

||\overrightarrow{i}||=

,\,||\overrightarrow{j}||=

\text{angle de }\overrightarrow{i}\text{ vers }\overrightarrow{j}=

\left\{\begin{array}{} x= \ \textcolor{white}{i}\ \textcolor{white}{i}  \ y= \end{array}\right. 

+

+

t

t

,\,t\in\mathbb{R}

,\,t=

x

\left\{\begin{array}{} x'= \ \vspace{5mm} \ y'= \end{array} \right.

x

y=

x+

x+

x

y=

\left\{\begin{array}{} x= \ \textcolor{white}{i}\ \textcolor{white}{i}  \ y= \end{array}\right. 

+

+

t

t

,\,t\in\mathbb{R}

,\,t=

\left\{\begin{array}{} x'= \ \vspace{5mm} \ y'= \end{array} \right.

f

g

g\circ f

q_1 = 0 (slider from -10 to 10, step 0.1)
p_1 = 0 (slider from -10 to 10, step 0.1)
A = (-37.72247273253754, -3.077880584482521)
B = (-26.905305434140047, 5.870437788004608)
C: (x(A), y(B))
D: (x(B), y(A))
q1, a, d, b, c = Polygon(A, D, B, C)
E = PointIn(q1)
F = PointIn(q1)
G = PointIn(q1)
e = 1 (slider from 0.1 to 5, step 0.1)
g = Circle(E, e)
H = Point(g)
f = 1 (slider from 0.1 to 5, step 0.1)
h: f / e
α = 1.5707963267948966 rad (slider from -3.141592653589793 to 3.141592653589793, step 0.017453292519943295)
H' = Rotate(H, α, E)
L: E + (h * Vector[E, H'])
n = 0 (slider from 0 to 1, step 1)
J = If(n ≟ 0, G, L)
v = Vector(E, J)
I = If(n ≟ 0, F, H)
u = Vector(E, I)
V = (-27.164123056281326, 5.263743275423243)
i = 0 (slider from 0 to 1, step 1)
K: E + (0.5 * u)
M: E + (0.5 * v)
j = true
k = 1 (slider from 0 to 2, step 1)
N: (0, 0)
ChampTexte1 = Textfield(N)
m1: {{x(u), x(v)}, {y(u), y(v)}}
O: E + (x(N) * u) + (y(N) * v)
l = Line(E, u)
m = Line(E, v)
w: (2, 1)
Q = (9.850382614414434, 4.532544654623457)
o = 0 (slider from -50 to 50, step 0.1)
p = 0 (slider from -10 to 10, step 0.1)
q = 0 (slider from -10 to 10, step 0.1)
ChampTexte5 = Textfield(o)
ChampTexte6 = Textfield(p)
ChampTexte7 = Textfield(q)
r: o^(2) + p^(2)
s = If(r ≠ 0, ((o * q)) / r)
a_1 = If(r ≠ 0, ((p * q)) / r)
f_1 = If(r ≠ 0, Line[E + (s * u) + (a_1 * v), Vector[((-p) * u) + (o * v)]])
b_1 = 0 (slider from -10 to 10, step 0.1)
c_1 = 0 (slider from -10 to 10, step 0.1)
d_1 = 0 (slider from -10 to 10, step 0.1)
e_1 = 0 (slider from -10 to 10, step 0.1)
ChampTexte8 = Textfield(b_1)
ChampTexte9 = Textfield(c_1)
ChampTexte10 = Textfield(d_1)
ChampTexte11 = Textfield(e_1)
g_1 = 1.85 (slider from -5 to 5, step 0.01)
h_1 = If(c_1^(2) + e_1^(2) ≠ 0, Line[E + (b_1 * u) + (d_1 * v), Vector[(c_1 * u) + (e_1 * v)]])
R: E + ((b_1 + (g_1 * c_1)) * u) + ((d_1 + (e_1 * g_1)) * v)
S: (0, 0)
T: (0, 0)
U: (0, 0)
W: (0, 0)
Z: (0, 0)
ChampTexte12 = Textfield(S)
ChampTexte13 = Textfield(T)
ChampTexte14 = Textfield(U)
ChampTexte15 = Textfield(W)
ChampTexte16 = Textfield(Z)
texte14: "" + (LaTeX[U]) + ""
texte15: "" + (LaTeX[W]) + ""
texte16: "" + (LaTeX[Z]) + ""
A_1: E + (x(S) * u) + (y(S) * v)
B_1: E + (x(T) * u) + (y(T) * v)
C_1: E + (x(U) * u) + (y(U) * v)
D_1: E + (x(W) * u) + (y(W) * v)
E_1: E + (x(Z) * u) + (y(Z) * v)
n_1 = 0 (slider from 0 to 1, step 1)
i_1 = 0 (slider from 0 to 1, step 1)
j_1 = 0 (slider from 0 to 1, step 1)
k_1 = 0 (slider from 0 to 1, step 1)
l_1 = 0 (slider from 0 to 1, step 1)
m_1 = 0 (slider from 0 to 1, step 1)
F_1: (b_1 + (g_1 * c_1), d_1 + (e_1 * g_1))
texte17: "" + (LaTeX[F_1]) + ""
ChampTexte2 = Textfield(g_1)
t = Vector(V, V + (p_1 * u) + (q_1 * v))
P = Midpoint(V, V + (p_1 * u) + (q_1 * v))
ChampTexte3 = Textfield(p_1)
ChampTexte4 = Textfield(q_1)
n_2 = 0 (slider from 0 to 1, step 1)
r_1 = false
i_2 = 1 (slider from -2 to 6, step 1)
s_1 = false
j_2 = 0 (slider from -2 to 5, step 1)
t_1 = false
k_2 = 0 (slider from 0 to 5, step 1)
l_2 = 0 (slider from 0 to 1, step 1)
u_1: abs(p)
texte25: "" + (LaTeX[o]) + " \,x" + (LaTeX[p]) + "\,y=" + (LaTeX[q]) + ""
texte26: "" + (LaTeX[o]) + "\,x+" + (LaTeX[u_1]) + "\,y=" + (LaTeX[q]) + ""
G_1 = Point(f_1)
m_2 = 0 (slider from 0 to 2, step 1)
n_3 = 0 (slider from 0 to 1, step 1)
texte37: "" + (LaTeX[o]) + "\,x+" + (LaTeX[u_1]) + "\,y=" + (LaTeX[q]) + ""
texte38: "" + (LaTeX[o]) + "\,x" + (LaTeX[p]) + "\,y=" + (LaTeX[q]) + ""
l2 = Sequence(Line[E + (k * u), v], k, -20, 20)
l1 = Sequence(Line[E + (k * v), u], k, -20, 20)
o_1 = 0 (slider from -10 to 10, step 0.1)
v_1 = 0 (slider from -10 to 10, step 0.1)
ChampTexte17 = Textfield(o_1)
ChampTexte18 = Textfield(v_1)
i_3 = 0 (slider from 0 to 1, step 1)
H_1 = (-12.30112710919569, 2.3487930901974177)
w_1 = Vector(H_1, H_1 + (o_1 * u) + (v_1 * v))
texte40: "\left(\begin{array}{} " + (LaTeX[o_1]) + " \\ " + (LaTeX[v_1]) + " \end{array} \right)"
I_1 = Midpoint(H_1, H_1 + (o_1 * u) + (v_1 * v))
a_2(x, y) = y
b_2(x, y) = x
J_1 = (-25.36022829101104, -1.074102489714996)
ChampTexte19 = Textfield(a_2)
ChampTexte20 = Textfield(b_2)
K_1: (m1^(-1) * (J_1 - E))
L_1: E + (a_2(x(K_1), y(K_1)) * u) + (b_2(x(K_1), y(K_1)) * v)
M_1 = (13.4, 3.18)
N_1 = (16.76, 3.8)
m2: {{a_2(1, 0) - a_2(0, 0), a_2(0, 1) - a_2(0, 0)}, {b_2(1, 0) - b_2(0, 0), b_2(0, 1) - b_2(0, 0)}}
m3: {{a_2(0, 0)}, {b_2(0, 0)}}
texte43: "\left(\begin{array}{} x' \\ y'  \end{array} \right)=" + (LaTeX[m2]) + "\left(\begin{array}{} x \\ y  \end{array}\right) +" + (LaTeX[m3]) + ""
j_3 = 1 (slider from 1 to 3, step 1)
k_3 = 3 (slider from 0 to 3, step 1)
O_1 = (11.56, 3.6)
ChampTexte21 = Textfield(e)
ChampTexte22 = Textfield(f)
ChampTexte23 = Textfield(α)
l_3 = 0 (slider from 0 to 1, step 1)
d_2 = 0 (slider from -10 to 10, step 0.1)
e_2 = 0 (slider from -10 to 10, step 0.1)
ChampTexte24 = Textfield(d_2)
ChampTexte25 = Textfield(e_2)
P_1 = (-7.845977312355397, 0.4452322281565948)
u_2 = Vector(P_1, P_1 + (d_2 * u) + (e_2 * v))
Q_1 = Midpoint(P_1, P_1 + (d_2 * u) + (e_2 * v))
texte46: "\left(\begin{array}{}" + (LaTeX[d_2]) + "   \\ " + (LaTeX[e_2]) + "  \end{array} \right)"
texte10: "\left(\begin{array}{} " + (LaTeX[p_1]) + "  \\  " + (LaTeX[q_1]) + " \end{array} \right)"
m_3 = 0 (slider from 0 to 1, step 1)
f_2 = 0 (slider from -10 to 10, step 0.1)
g_2 = 0 (slider from -10 to 10, step 0.1)
R_1 = (-8.42422450358058, 3.240093652411581)
v_2 = Vector(R_1, R_1 + (f_2 * u) + (g_2 * v))
ChampTexte26 = Textfield(f_2)
ChampTexte27 = Textfield(g_2)
S_1 = Midpoint(R_1, R_1 + (f_2 * u) + (g_2 * v))
texte47: "\left(\begin{array}{} " + (LaTeX[f_2]) + "   \\  " + (LaTeX[g_2]) + "  \end{array} \right)"
n_4 = 0 (slider from 0 to 1, step 1)
h_2 = 0 (slider from -10 to 10, step 0.1)
o_2 = 0 (slider from -10 to 10, step 0.1)
T_1 = (-6.850107149689851, 1.489289656757595)
w_2 = Vector(T_1, T_1 + (h_2 * u) + (o_2 * v))
ChampTexte28 = Textfield(h_2)
ChampTexte29 = Textfield(o_2)
U_1 = Midpoint(T_1, T_1 + (h_2 * u) + (o_2 * v))
texte48: "\left(\begin{array}{}  " + (LaTeX[h_2]) + "   \\   " + (LaTeX[o_2]) + "  \end{array} \right)"
V_1 = (-28.16794912099816, 2.7470560363029626)
m4: m1^(-1)
W_1: (m4 * Vector[E, V_1])
texte49: "" + (LaTeX[W_1]) + ""
Z_1 = (-27.866885232083387, 3.0127769233633246)
A_2 = (-22.951784106669454, 4.6350815431895)
p_2 = Vector(Z_1, A_2)
B_2 = Midpoint(Z_1, A_2)
C_2: (m4 * p_2)
q_2: x(C_2)
r_2: y(C_2)
texte50: "\left(\begin{array}{}  " + (LaTeX[q_2]) + "  \\  " + (LaTeX[r_2]) + "  \end{array} \right)"
texte13: "C" + (LaTeX[T]) + ""
texte12: "" + (LaTeX[S]) + ""
texte6: "" + (LaTeX[N]) + ""
s_2 = 0 (slider from -10 to 10, step 0.1)
t_2 = 0 (slider from -10 to 10, step 0.1)
a_3 = 0 (slider from -10 to 10, step 0.1)
i_4 = 0 (slider from 0 to 1, step 1)
ChampTexte30 = Textfield(s_2)
ChampTexte31 = Textfield(t_2)
ChampTexte32 = Textfield(a_3)
b_3: s_2^(2) + t_2^(2)
c_3 = If(b_3 ≠ 0, ((s_2 * a_3)) / b_3)
d_3 = If(b_3 ≠ 0, ((t_2 * a_3)) / b_3)
f_3 = If(b_3 ≠ 0, Line[E + (c_3 * u) + (d_3 * v), Vector[((-t_2) * u) + (s_2 * v)]])
e_3 = 0 (slider from -10 to 10, step 0.1)
g_3 = 0 (slider from -10 to 10, step 0.1)
h_3 = 0 (slider from -10 to 10, step 0.1)
j_4 = 0 (slider from 0 to 1, step 1)
ChampTexte33 = Textfield(e_3)
ChampTexte34 = Textfield(g_3)
ChampTexte35 = Textfield(h_3)
o_3: e_3^(2) + g_3^(2)
p_3 = If(o_3 ≠ 0, ((e_3 * h_3)) / o_3)
q_3 = If(o_3 ≠ 0, ((g_3 * h_3)) / o_3)
r_3 = If(o_3 ≠ 0, Line[E + (p_3 * u) + (q_3 * v), Vector[((-g_3) * u) + (e_3 * v)]])
s_3 = 0 (slider from -10 to 10, step 0.1)
t_3 = 0 (slider from -10 to 10, step 0.1)
u_3 = 0 (slider from -10 to 10, step 0.1)
v_3 = 0 (slider from -10 to 10, step 0.1)
ChampTexte36 = Textfield(s_3)
ChampTexte37 = Textfield(t_3)
ChampTexte38 = Textfield(u_3)
ChampTexte39 = Textfield(v_3)
f_4 = If(t_3^(2) + v_3^(2) ≠ 0, Line[E + (s_3 * u) + (u_3 * v), Vector[(t_3 * u) + (v_3 * v)]])
k_4 = 0 (slider from 0 to 2, step 1)
w_3 = 0.75 (slider from -5 to 5, step 0.05)
ChampTexte40 = Textfield(w_3)
D_2: (s_3 + (t_3 * w_3), u_3 + (v_3 * w_3))
E_2: E + (x(D_2) * u) + (y(D_2) * v)
texte64: "" + (LaTeX[D_2]) + ""
a_4(x, y) = 0
b_4(x, y) = 0
ChampTexte41 = Textfield(a_4)
ChampTexte42 = Textfield(b_4)
c_4 = true
l_4 = 1 (slider from 0 to 3, step 1)
m_4 = 1 (slider from 1 to 3, step 1)
g_4: y = y(M_1)
texte66: "\left\{\begin{array}{} x'=& " + (LaTeX[a_4]) + "\\ y'=& " + (LaTeX[b_4]) + "  \end{array}\right. "
texte42: "\left\{\begin{array}{} x'=&" + (LaTeX[a_2]) + "\\ y'=&" + (LaTeX[b_2]) + "  \end{array}\right. "
m5: {{a_4(1, 0) - a_4(0, 0), a_4(0, 1) - a_4(0, 0)}, {b_4(1, 0) - b_4(0, 0), b_4(0, 1) - b_4(0, 0)}}
m6: {{a_4(0, 0)}, {b_4(0, 0)}}
texte67: "\left(\begin{array}{} x' \\ y'  \end{array} \right)= " + (LaTeX[m5]) + "\left(\begin{array}{} x \\ y  \end{array}\right) +" + (LaTeX[m6]) + " "
F_2 = Rotate(J_1, 60°, L_1)
G_2 = Rotate(L_1, 8°, F_2)
H_2 = Rotate(J_1, -5°, F_2)
c_2 = CircleArc(F_2, G_2, H_2)
I_2 = Rotate(L_1, 5°, F_2)
u_4 = Vector(G_2, I_2)
J_2 = Point(c_2)
n_5 = 0 (slider from 0 to 3, step 1)
K_2 = (-24.34011794967402, 1.4935730815784323)
L_2: (m1^(-1) * (K_2 - E))
M_2: E + (a_4(x(L_2), y(L_2)) * u) + (b_4(x(L_2), y(L_2)) * v)
N_2 = Rotate(K_2, 60°, M_2)
O_2 = Rotate(M_2, 8°, N_2)
P_2 = Rotate(M_2, 5°, N_2)
Q_2 = Rotate(K_2, -5°, N_2)
d_4 = CircleArc(N_2, O_2, Q_2)
v_4 = Vector(O_2, P_2)
R_2 = Point(d_4)
texte68: "\left(\begin{array}{} x' \\ y'  \end{array} \right)=  " + (LaTeX[m5]) + "(" + (LaTeX[m2]) + "\left(\begin{array}{} x \\ y  \end{array}\right)+" + (LaTeX[m3]) + ") + " + (LaTeX[m6]) + " "
m7: (m5 * m2)
m8: (m5 * m3) + m6
i_5 = 2 (slider from 0 to 2, step 1)
texte69: "\left(\begin{array}{} x' \\ y'  \end{array} \right)= " + (LaTeX[m7]) + " \left(\begin{array}{} x \\ y  \end{array}\right) +" + (LaTeX[m8]) + "  "
j_5 = 0 (slider from 0 to 3, step 1)
S_2 = (-26.92685540756099, 5.975406965831794)
T_2: (m1^(-1) * (S_2 - E))
U_2: (m7 * T_2)
e_4: x(U_2) + m8(1, 1)
h_4: y(U_2) + m8(2, 1)
V_2: E + (e_4 * u) + (h_4 * v)
W_2 = Rotate(V_2, 60°, S_2)
Z_2 = Rotate(S_2, 5°, W_2)
A_3 = Rotate(V_2, -8°, W_2)
B_3 = Rotate(V_2, -5°, W_2)
p_4 = CircleArc(W_2, Z_2, A_3)
w_4 = Vector(A_3, B_3)
C_3 = Point(p_4)
D_3 = (-29.14931331734106, -3.3665599286384564)
E_3 = (-27.814108173921696, 0.11248366273834687)
F_3 = (-30.177980931247085, 0.8950070582667371)
t1, q_4, r_4, s_4 = Polygon(D_3, E_3, F_3)
G_3: (m1^(-1) * (D_3 - E))
H_3: E + (a_2(x(G_3), y(G_3)) * u) + (b_2(x(G_3), y(G_3)) * v)
I_3: (m1^(-1) * (E_3 - E))
J_3: E + (a_2(I_3) * u) + (b_2(I_3) * v)
K_3: (m1^(-1) * (F_3 - E))
L_3: E + (a_2(K_3) * u) + (b_2(K_3) * v)
t2, t_4, f_5, g_5 = Polygon(J_3, L_3, H_3)
k_5 = 0 (slider from 0 to 2, step 1)