restart:
kernelopts(version)
;
OneFrame := proc(k)
local Courbe, T, a, b, c, t, P, Q, NormM, F, Ell, sol, N1, N2, dr, tx;
local MySol:
uses geometry:
_EnvHorizontalName := x: _EnvVerticalName := y:
a := 11; b := 7; c := sqrt(a^2 - b^2); t := 1/3*Pi;
Ell := x^2/a^2 + y^2/b^2 = 1;
point(T, (a^2 - b^2)*cos(t)^3/a, -(a^2 - b^2)*sin(t)^3/b);
Courbe := plots:-implicitplot(Ell, x = -a - 10 .. a + 10, y = -b - 10 .. b + 10, scaling = constrained, color = blue, grid=[50, 50]);
NormM := plots:-implicitplot(y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), x = -a - 5 .. a + 10, y = -b - 10 .. b + 10, color = orange); 
line(Per, y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), [x, y]);
point(P, [solve(subs(y = 0, lhs(Equation(Per)))), 0]);
point(Q, [0, solve(subs(x = 0, lhs(Equation(Per))))]);
point(M, [a*cos(t), b*sin(t)]);
point(N1, [a*cos(k), b*sin(k)]);
point(F, [2.329411765, -2.567510609]);
line(L, [N1, F]);
sol := solve({Equation(L), Ell}, {x, y},explicit);
min(map(t -> eval(y, t), [%]));
MySol := select(s -> has(s, y=%), [sol])[];
point(N2, eval([x, y], MySol));
segment(sg, N1, N2);
tx := plots:-textplot([[coordinates(M)[], "M"],[coordinates(N1)[], "N1"],[coordinates(N2)[], "N2"],[coordinates(P)[], "P"],[coordinates(Q)[], "Q"],[coordinates(F)[], "F point de Frégier"],[coordinates(T)[], "T"]], font = [times, bold, 16], align = [above, left]);
dr := draw([sg(color = magenta, linestyle = dash),Per(color = black), P(color = red, symbol = solidcircle, symbolsize = 12),Q(color = red, symbol = solidcircle, symbolsize = 12),M(color = black, symbol = solidcircle, symbolsize = 12),F(color = red, symbol = solidcircle, symbolsize = 12),N1(color = black, symbol = solidcircle, symbolsize = 8)]);
plots:-display(Courbe, tx, dr, scaling = constrained, axes=normal, view=[-15..15, default]); 
end proc:


plots:-animate('OneFrame', [k], k = Pi/3 .. Pi, frames = 50);