Le surfeur
Author: Guy Marion
Date: août 2006

Cette feuille de travail est une première approche de la notion de tangente à une courbe en un point. Nous donnerons une définition plus mathématique de la tangente  lors de la prochaine activité 
Déplace le point M et observe bien les variations du coefficient directeur de la tangente .
Quels liens peut-on faire entre ce coefficient et la position du surfeur sur la vague ?

Para modificar el valor de cada parámetro, 

desliza los puntos verdes o bien pulsa + o - tras 

seleccionar el parámetro en la ventana de la izquierda 

Déplacez le point P

∆ x = 1

La planche affleure à la surface de la vague : 
La droite Δ est tangente à la courbe C_f au point M

C_f

Le coefficient directeur de la tangente au point M vaut

Y = (-2.3, -0.7666666666666666)
a = 3
f(x) = -(x - 3.7333333333333334)^(3) / 150 - (x - 3.7333333333333334)^(2) / 10 + 10.586666666666666
M = Point(f)
d = Circle(M, a)
b = Tangent(M, f)
B, C_3 = Intersect(d, b)
Δ_1 = Segment(B, C_3)
i = OrthogonalLine(M, Δ_1)
c = OrthogonalLine(M, yAxis)
g = Segment(M, (x(M), 0))
A_4 = (8.8, 7)
j = Segment(A_4, M)
Q = Midpoint(B, M)
R = Midpoint(Q, M)
S = Midpoint(M, C_3)
T = Midpoint(M, S)
b_1 = Segment(R, T)
o = Integral(f, -30, 20)
h = OrthogonalLine(A_4, xAxis)
k = OrthogonalLine(M, xAxis)
l = Segment(A_4, M)
n = OrthogonalLine(M, Δ_1)
K = Midpoint(M, T)
L = Mirror(K, M)
m = OrthogonalLine(A_4, h)
D = Point(h)
E = Point(m)
F = Mirror(E, A_4)
G = Mirror(A_4, D)
p = Segment(E, D)
q = Segment(D, F)
H = Mirror(D, G)
r = Circle(G, H)
s = OrthogonalLine(G, h)
I, J = Intersect(r, s)
t = Segment(J, I)
M_2 = Midpoint(H, G)
N = Midpoint(H, M_2)
c_1 = Circle(N, H)
a_1 = Segment(D, M_2)
d_1 = Line(A_4, Δ_1)
O = Point(m)
e_1 = OrthogonalLine(O, m)
U = Intersect(d_1, e_1)
f_1 = AngularBisector(E, A_4, U)
V = Mirror(G, f_1)
W = Mirror(D, f_1)
Z = Mirror(M_2, f_1)
A_1 = Mirror(N, f_1)
B_1 = Mirror(F, f_1)
g_1 = OrthogonalLine(M, Δ_1)
h_1 = Segment(A_4, M)
i_1 = Line(W, h_1)
j_1 = Line(V, h_1)
k_1 = Line(Z, h_1)
l_1 = Line(A_1, h_1)
C_1 = Intersect(i_1, g_1)
D_1 = Intersect(j_1, g_1)
E_1 = Intersect(k_1, g_1)
F_1 = Intersect(l_1, g_1)
m_1 = Line(B_1, h_1)
G_1 = Intersect(m_1, Δ_1)
H_1 = Mirror(G_1, M)
n_1 = Segment(G_1, C_1)
p_1 = Segment(H_1, C_1)
q_1 = Segment(C_1, E_1)
r_1 = Circle(F_1, E_1)
s_1 = Circle(D_1, C_1)
t_1 = OrthogonalLine(D_1, g_1)
I_1, J_1 = Intersect(s_1, t_1)
a_2 = Segment(I_1, J_1)
K_1: (x(M) + 1, y(M))
b_2 = Segment(M, K_1)
c_2 = OrthogonalLine(K_1, b_2)
L_1 = Intersect(b, c_2)
P_1, l_2, p_2, k_2 = Polygon(M, K_1, L_1)
dy: y(L_1) - y(K_1)
M_1 = (4.566666666666666, 13.7)
N_1: (x(M_1) + 2, y(M_1))
O_1: (x(N_1), y(N_1) + 2 dy)
Q_1, o_1, m_2, n_2 = Polygon(M_1, N_1, O_1)
u = Vector(M_1, N_1)
v = Vector(N_1, O_1)
T8: "∆ y = " + dy
R_1 = Midpoint(N_1, O_1)
T5: "" + dy
S_1 = (0.5, 15.5)
T_1 = (9.2, 15.5)
U_1 = (9.2, 8.3)
V_1 = (0.5, 8.3)
W_1, s_2, t_2, u_1, v_1 = Polygon(S_1, T_1, U_1, V_1)
Y_1 = (8.9, 8.57)
Z_1 = (7.77, 8.57)
A_2: (x(M) + 2, y(M))
e_2 = Circle(M, A_2)
B_2, C_2 = Intersect(e_2, Δ_1)
A = Point(Δ_1)
P = Mirror(A, M)
Δ_2 = Segment(A, P)
C = Point(Δ_2)
D_2 = Mirror(C, M)
Δ = Segment(C, D_2)