exercício-1-c-seção-21.sagews

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## Exercício 1 (c) da Seção 21

Exercício 1 (c) da Seção 21

A = 9
B = -12
C = 16
D = 10
E = -55
F = 171

x, y = var('x y')
phi(x,y) = A*x^2 + 2*B*x*y + C*y^2 + D*x + E*y + F
phi

(x, y) |--> 9*x^2 - 24*x*y + 16*y^2 + 10*x - 55*y + 171

c = 0

phi(x,y) == c

9*x^2 - 24*x*y + 16*y^2 + 10*x - 55*y + 171 == 0

xmin = -10
xmax = 15
ymin = -10
ymax = 15
pcurva = implicit_plot(phi(x,y) - c, (x, xmin, xmax), (y, ymin, ymax))
pcurva.show(axes_labels=['$X$', '$Y$'], axes=True)

s, t = var('s t')
h, k = var('h k')
X = s + h
Y = t + k

phibarra(s,t) = phi(X,Y).simplify_full()
phibarra

(s, t) |--> 9*h^2 - (24*h + 55)*k + 16*k^2 + 2*(9*h - 12*k + 5)*s + 9*s^2 - (24*h - 32*k + 24*s + 55)*t + 16*t^2 + 10*h + 171

A*C - B^2

0

Dlinha = 2*A*h + 2*B*k + D
Elinha = 2*B*h + 2*C*k + E

Dlinha

18*h - 24*k + 10

Elinha

-24*h + 32*k - 55

N = Matrix([[18, -24, -10], [-24, 32, 55]])
N

[ 18 -24 -10] [-24 32 55]

N.echelon_form()

[ 6 -8 80] [ 0 0 125]

solve(Elinha == 0, k)

[k == 3/4*h + 55/32]

h = 0
k = 3/4*h + 55/32

X = s + h
Y = t + k

X

s

Y

t + 55/32

phibarra(s,t) = phi(X,Y).simplify_full()
phibarra

(s, t) |--> 9*s^2 - 24*s*t + 16*t^2 - 125/4*s + 7919/64

pcurvaOST = implicit_plot(phibarra(s,t) - c, (s, xmin, xmax), (t, ymin, ymax))
pcurvaOST.show(axes_labels=['$S$', '$T$'], axes=True)

t = var('t')
r1 = parametric_plot([h, t], (t, -10, 15), color='black', linestyle='--')
r2 = parametric_plot([t, k], (t, -10, 15), color='black', linestyle='--')
tS = text("S", (14, 3), color='black')
tT = text("T", (1, 14), color='black')
p = pcurva + r1 + r2 + tS + tT
p.show(axes_labels=['$X$', '$Y$'], axes=True)

M = Matrix([[A, B], [B, C]])
M

[ 9 -12] [-12 16]

z = var('z')
solve(z^2 - (A+C)*z + A*C - B^2 == 0, z)

[z == 0, z == 25]

M.eigenvalues()

[25, 0]

lambda1, lambda2 = M.eigenvalues()

lambda1

25

lambda2

0

D, P = M.eigenmatrix_right()

D

[25 0] [ 0 0]

P

[ 1 1] [-4/3 3/4]

w1 = vector(P[:,0])
w2 = vector(P[:,1])

w1

(1, -4/3)

w2

(1, 3/4)

u1 = (1/w1.norm())*w1
u2 = (1/w2.norm())*w2

u1

(3/5, -4/5)

u2

(4/5, 3/5)

u1.dot_product(u2)

0

p1 = arrow((0,0), u1, color='green', legend_label='u1', legend_color='black')
p2 = arrow((0,0), u2, color='red', legend_label='u2', legend_color='black')
pP = arrow((0,0), 3*u1, linestyle='--', width=1, color='black')
pQ = arrow((0,0), 3*u2, linestyle='--', width=1, color='black')
tP = text("P", 3.2*u1, color='black')
tQ = text("Q", 3.2*u2, color='black')
p = p1 + p2 + pcurvaOST + pP + pQ + tP + tQ
p.show(aspect_ratio=1,axes_labels=['$S$', '$T$'], xmin=-2, xmax=8, ymin=-3, ymax=8, axes=True, gridlines=True)

a = u1[0]
b = u1[1]

a

3/5

b

-4/5

theta = var('theta')
solve([cos(theta) == a, sin(theta) == b], theta)

[[theta == 2*pi*z66 - arctan(4/3)]]

-arctan(4/3).n()

-0.927295218001612

(-arctan(4/3)*180/pi).n()

-53.1301023541560

p, q = var('p q')

S = a*p - b*q
T = b*p + a*q

phibarra2(p,q) = phibarra(S,T).simplify_full()
phibarra2

(p, q) |--> 25*p^2 - 75/4*p - 25*q + 7919/64

phibarra2(p,q) == c

25*p^2 - 75/4*p - 25*q + 7919/64 == 0

pcurvaPQ = implicit_plot(phibarra2(s,t) - c, (s, xmin, xmax), (t, ymin, ymax))
pcurvaPQ.show(axes_labels=['$P$', '$Q$'], axes=True)

%latex
$$25p^2 - \frac{75}{4}p - 25q + \frac{7919}{64} = 0$$
$$25 \left( p^2 - \frac{3}{4}p \right) - 25q + \frac{7919}{64} = 0$$
$$25 \left( p - \frac{3}{8} \right)^2 -25 \left( \frac{3}{8} \right)^2 - 25q + \frac{7919}{64} = 0$$
$$25 \left( p - \frac{3}{8} \right)^2 - 25q + \frac{7694}{64} = 0$$
$$\left( p - \frac{3}{8} \right)^2 = q - \frac{3847}{800}$$

m, n = var('m n')

P = m + 3/8
Q = n + 3847/800

phibarra3(m,n) = phibarra2(P,Q).simplify_full()
phibarra3

(m, n) |--> 25*m^2 - 25*n

pcurvaMN = implicit_plot(phibarra3(m,n) - c, (m, xmin, xmax), (n, ymin, ymax))
pcurvaMN.show(axes_labels=['$M$', '$N$'], axes=True)

X

s

Y

t + 55/32

S

3/5*p + 4/5*q

T

-4/5*p + 3/5*q

P

m + 3/8

Q

n + 3847/800

Xf(s,t) = X
Yf(s,t) = Y
Sf(p,q) = S
Tf(p,q) = T
Pf(m,n) = P
Qf(m,n) = Q

Xf

(s, t) |--> s

Yf

(s, t) |--> t + 55/32

Sf

(p, q) |--> 3/5*p + 4/5*q

Tf

(p, q) |--> -4/5*p + 3/5*q

Pf

(m, n) |--> m + 3/8

Qf

(m, n) |--> n + 3847/800

Sf(Pf(m,n), Qf(m,n))
Tf(Pf(m,n), Qf(m,n))

3/5*m + 4/5*n + 509/125 -4/5*m + 3/5*n + 10341/4000

X1f = Xf(Sf(Pf(m,n), Qf(m,n)), Tf(Pf(m,n), Qf(m,n)))
Y1f = Yf(Sf(Pf(m,n), Qf(m,n)), Tf(Pf(m,n), Qf(m,n)))
X1f
Y1f

3/5*m + 4/5*n + 509/125 -4/5*m + 3/5*n + 538/125

phibarra(m,n) = phi(X1f,Y1f).simplify_full()
phibarra

(m, n) |--> 25*m^2 - 25*n