JAMET

60 Reputation

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2 years, 122 days

MaplePrimes Activity


These are questions asked by JAMET

with(LinearAlgebra); A := `

expand((x-c)^2+(y-d)^2-R^2) = 0; algsubs(-R^2+c^2+d^2 = f, %); P := proc (x, y) options operator, arrow; -2*x*c-2*y*d+x^2+y^2+f = 0 end proc; 2 2 P := (x, y) -> -2 x c - 2 y d + x + y + f = 0 P(a*cos(theta), b*sin(theta)); G := unapply(%, theta); #usage des formules d'Euler simplify(expand(4*(exp(I*theta))^2*subs(cos(theta) = (exp(I*theta)+exp(-I*theta))*(1/2), sin(theta) = (exp(I*theta)-exp(-I*theta))/(2*I), G(theta)))); poly := sort(subs(exp(I*theta) = X, exp((2*I)*theta) = X^2, exp((3*I)*theta) = X^3, exp((4*I)*theta) = X^4, %)); coeff(lhs(poly), X^4)/tcoeff(lhs(poly)); # exp(I*theta1),exp(I*theta2),exp(I*theta3),exp(I*theta4) sont les racines de ce polynôme unitaire : exp(I*theta1)*exp(I*theta2)*exp(I*theta3)*exp(I*theta4) =1 exp(I*(θ1+θ2+θ3+θ4)=1 d'où θ1+θ2+θ3+θ4 ≡ 2*Pi

; restart; with(plots); _local(O); P := b*x*cos(phi)+a*y*sin(phi)-a . b = 0; P := b x cos(phi) + a y sin(phi) - a . b = 0 Q := a*x*sin(phi)-b*y*cos(phi)-c^2*sin(phi)*cos(phi) = 0; 2 Q := a x sin(phi) - b y cos(phi) - c sin(phi) cos(phi) = 0 M := op(solve([P, Q], [x, y])); M := [subs(M, x), subs(M, y)]; X := `&-+`(P/sqrt(b^2*cos(phi)^2+a^2*sin(phi)^2)); Y := `&-+`(Q/sqrt(b^2*cos(phi)^2+a^2*sin(phi)^2)); #L'équation générale des coniques ayant pour axes MN et MT est, par rapport aux nouveaux axes de coordonnées X^2/A+Y^2/B-1 = (0*et)*par*rapport*aux*anciens; P^2/(A*(b^2*cos(phi)^2+a^2*sin(phi)^2))+Q^2/(B*(b^2*cos(phi)^2+a^2*sin(phi)^2))-1 = 0; 2 /b x cos(phi) + a y sin(phi) - a . b \ &-+|----------------------------------- = 0| | (1/2) | |/ 2 2 2 2\ | \\a sin(phi) + cos(phi) b / / --------------------------------------------- A 2 / 2 \ |a x sin(phi) - b y cos(phi) - c sin(phi) cos(phi) | &-+|-------------------------------------------------- = 0| | (1/2) | | / 2 2 2 2\ | \ \a sin(phi) + cos(phi) b / / + ------------------------------------------------------------ B - 1 = 0 #1.-Ecrivons que la conique (1) est tangente en O à Oy : il faut pour cela annuler le coefficient de y et le terme indépendant. #Nous obtenons 2 équations en A et B, d'où nous tirons : A=a² et B=c²cos(phi)² a := 10; b := 7; c := sqrt(a^2-b^2); phi := 4*Pi*(1/5); Ell := implicitplot(x^2/a^2+y^2/b^2-1 = 0, x = -11 .. 11, y = -8 .. 8, color = grey); O := [0, 0]; M := [a*cos(phi), b*sin(phi)]; vec := plot([O, M], color = black, thickness = 1); P := implicitplot(P, x = -20 .. 20, y = -20 .. 20, color = aquamarine); Q := implicitplot(Q, x = -20 .. 20, y = -20 .. 20); F1 := [(a+b)*cos(phi), (a+b)*sin(phi)]; F2 := [2*M[1]-F1[1], 2*M[2]-F1[2]]; F1F2 := plot([F1, F2], color = green, thickness = 3); ELL := implicitplot((b*x*cos(phi)+a*y*sin(phi)-a . b)^2/(a^2*(b^2*cos(phi)^2+a^2*sin(phi)^2))+(a*x*sin(phi)-b*y*cos(phi)-c^2*sin(phi)*cos(phi))^2/(c^2*cos(phi)^2*(b^2*cos(phi)^2+a^2*sin(phi)^2))-1 = 0, x = -20 .. 20, y = -20 .. 20, color = blue, thickness = 3); Hyp := implicitplot((b*x*cos(phi)+a*y*sin(phi)-a . b)^2/(b^2*(b^2*cos(phi)^2+a^2*sin(phi)^2))+(a*x*sin(phi)-b*y*cos(phi)-c^2*sin(phi)*cos(phi))^2/(-c^2*sin(phi)^2*(b^2*cos(phi)^2+a^2*sin(phi)^2))-1 = 0, x = -20 .. 20, y = -20 .. 20, color = black); dF1 := plottools[disk](F1, .3, color = red); dF2 := plottools[disk](F2, .3, color = red); cir1 := implicitplot(x^2+y^2 = (a+b)^2, x = -20 .. 20, y = -18 .. 18, color = pink); cir2 := implicitplot(x^2+y^2 = (a-b)^2, x = -10 .. 10, y = -4 .. 4, color = coral); asym1 := implicitplot((b*x*cos(phi)+a*y*sin(phi)-a . b)/b+(a*x*sin(phi)-b*y*cos(phi)-c^2*sin(phi)*cos(phi))/(c*sin(phi)) = 0, x = -20 .. 20, y = -18 .. 18, color = black, linestyle = DOT); asym2 := implicitplot((b*x*cos(phi)+a*y*sin(phi)-a . b)/b-(a*x*sin(phi)-b*y*cos(phi)-c^2*sin(phi)*cos(phi))/(c*sin(phi)) = 0, x = -20 .. 20, y = -18 .. 18, color = black, linestyle = DOT); tp := textplot([[M[1], M[2]+.8, "M"], [F1[1]-.8, F1[2], "F1"], [F2[1]+.8, F2[2]+.3, "F2"], [5, 15, "axe P"], [8, -10, "axe Q"]]); display([Ell, vec, P, Q, F1F2, cir1, cir2, ELL, Hyp, dF1, dF2, asym1, asym2, tp], scaling = constrained, axes = normal, axis = [gridlines = [1, color = blue]], xtickmarks = 0, ytickmarks = 0, view = [-20 .. 20, -20 .. 20], size = [500, 500]); #Eléments fixes : Ell, cir1, cir2, O #Parties mobiles : ELL, Hyp, P,Q, M,F1, F2, # FIGURE MOBILE n := 100; dt := 2*Pi/n; Phi := 0; P := b*x*cos(phi+dt)+a*y*sin(phi+dt)-a . b = 0; Q := a*x*sin(phi+dt)-b*y*cos(phi+dt)-c^2*sin(phi+dt)*cos(phi+dt) = 0; M := [cos(phi+dt)*(sin(phi+dt)^2*a*c^2+Typesetting[delayDotProduct](a . b, b, true))/(a^2*sin(phi+dt)^2+cos(phi+dt)^2*b^2), sin(phi+dt)*(-cos(phi+dt)^2*b*c^2+Typesetting[delayDotProduct](a . b, a, true))/(a^2*sin(phi+dt)^2+cos(phi+dt)^2*b^2)]; ELL := (b*x*cos(phi+dt)+a*y*sin(phi+dt)-a . b)^2/(a^2*(a^2*sin(phi+dt)^2+cos(phi+dt)^2*b^2))+(a*x*sin(phi+dt)-b*y*cos(phi+dt)-c^2*sin(phi+dt)*cos(phi+dt))^2/(c^2*cos(phi+dt)^2*(cos(phi+dt)^2*b^2+a^2))-1 = 0; NULL; display([Ell, cir1, cir2], scaling = constrained);
unprotect(D); f := proc (x, y) options operator, arrow; (295849/5841396)*x^2-(29441/324522)*y*x+(33995/216348)*y^2-(5989/14751)*x+(3635/4917)*y+1 end proc; 295849 2 29441 33995 2 5989 f := (x, y) -> ------- x - ------ y x + ------ y - ----- x 5841396 324522 216348 14751 3635 + ---- y + 1 4917 coeffs(f(x, y)); -5989 3635 295849 33995 -29441 1, -----, ----, -------, ------, ------ 14751 4917 5841396 216348 324522 A, B, C, D, E, F := %; -5989 295849 33995 -29441 3635 A, B, C, D, E, F := 1, -----, -------, ------, ------, ---- 14751 5841396 216348 324522 4917

I would like to draw an ellipse by orthonal affinity of a circle. Thank you.

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