E_2

E_1

O

O'

E'_1

E'_2

Zobraziť pohyblivý bod P a jeho obraz P'

SAM

SAM

Zobraziť množinu samodružných bodov

Zobraziť kružnicu k; pohybujte kružnicou alebo bodmi S, R, M

Zobraziť samodružné smery

\text{Nemá samodružné body}

C = (0, 1)
B = (1, 0)
A = (0, 0)
A' = (1.5, 1.5)
B' = (1.5, 0.5)
C' = (0.5, 1.5)
A2: x(A)
A3: 0
A4: x(B)
A5: 0
A6: x(C)
A7: 0
B2: 0
B3: x(A)
B4: 0
B5: x(B)
B6: 0
B7: x(C)
C2: y(A)
C3: 0
C4: y(B)
C5: 0
C6: y(C)
C7: 0
D2: 0
D3: y(A)
D4: 0
D5: y(B)
D6: 0
D7: y(C)
E2: 1
E3: 0
E4: 1
E5: 0
E6: 1
E7: 0
F2: 0
F3: 1
F4: 0
F5: 1
F6: 0
F7: 1
G2: x(A')
G3: y(A')
G4: x(B')
G5: y(B')
G6: x(C')
G7: y(C')
a_x = SurdText(x(A'))
a_y = SurdText(y(A'))
b_x = SurdText(x(B'))
b_y = SurdText(y(B'))
c_x = SurdText(x(C'))
c_y = SurdText(y(C'))
text2: "\text{Afinné zobrazenie, ktoré zobrazuje body }\\\\ " + (LaTeX[A]) + "  →\left( " + (LaTeX[a_x]) + "," + (LaTeX[a_y]) + "  \right)    , \; \;    " + (LaTeX[B]) + " → \left(" + (LaTeX[b_x]) + "," + (LaTeX[b_y]) + "   \right) , \; \;    " + (LaTeX[C]) + "  → \left(  " + (LaTeX[c_x]) + "," + (LaTeX[c_y]) + " \right) "
t1, c_4, a_4, b_4 = Polygon(A, B, C)
t2, c', a', b' = Polygon(A', B', C')
e = Vector(A, A')
f = Vector(B, B')
g = Vector(C, C')
m1: {{A2, B2, C2, D2, E2, F2, G2}, {A3, B3, C3, D3, E3, F3, G3}, {A4, B4, C4, D4, E4, F4, G4}, {A5, B5, C5, D5, E5, F5, G5}, {A6, B6, C6, D6, E6, F6, G6}, {A7, B7, C7, D7, E7, F7, G7}}
m2: {{A2, B2, C2, D2, E2, F2}, {A3, B3, C3, D3, E3, F3}, {A4, B4, C4, D4, E4, F4}, {A5, B5, C5, D5, E5, F5}, {A6, B6, C6, D6, E6, F6}, {A7, B7, C7, D7, E7, F7}}
m3: {{G2}, {G3}, {G4}, {G5}, {G6}, {G7}}
a11 = SurdText(Element[(m2^(-1) * m3), 1, 1])
a22 = SurdText(Element[(m2^(-1) * m3), 4, 1])
a12 = SurdText(Element[(m2^(-1) * m3), 3, 1])
a21 = SurdText(Element[(m2^(-1) * m3), 2, 1])
o1 = SurdText(Element[(m2^(-1) * m3), 5, 1])
o2 = SurdText(Element[(m2^(-1) * m3), 6, 1])
A13 = Element((m2^(-1) * m3), 1, 1)
B13 = Element((m2^(-1) * m3), 2, 1)
A14 = Element((m2^(-1) * m3), 3, 1)
B14 = Element((m2^(-1) * m3), 4, 1)
mA: {{A13, B13}, {A14, B14}}
A_t = Transpose(mA)
mA_xmA_t: (mA * A_t)
text4: "\text{Maticový súčin } \mathcal{A}×\mathcal{A}^T:\\\\\\
  \;  \left(  \begin{array}{}  " + (LaTeX[a11]) + "   &  " + (LaTeX[a21]) + "   \\  " + (LaTeX[a12]) + "   &  " + (LaTeX[a22]) + "   \\ \end{array} \right) ×\left( \begin{array}{}  " + (LaTeX[a11]) + " &" + (LaTeX[a12]) + "  \\ " + (LaTeX[a21]) + " &  " + (LaTeX[a22]) + " \\ \end{array}   \right) =" + (LaTeX[mA_xmA_t]) + " "
x'(x, y) = (Element[(m2^(-1) * m3), 1, 1] * x) + (Element[(m2^(-1) * m3), 3, 1] * y) + Element[(m2^(-1) * m3), 5, 1]
y'(x, y) = (Element[(m2^(-1) * m3), 2, 1] * x) + (Element[(m2^(-1) * m3), 4, 1] * y) + Element[(m2^(-1) * m3), 6, 1]
text3: "\text{Transformačné rovnice}\\\\\\
\;x'=  " + (LaTeX[x']) + "
\\\\
\;y'=  " + (LaTeX[y']) + ""
O': (Element[(m2^(-1) * m3), 5, 1], Element[(m2^(-1) * m3), 6, 1])
E'_1: ((Element[(m2^(-1) * m3), 1, 1] * 1) + (Element[(m2^(-1) * m3), 3, 1] * 0) + Element[(m2^(-1) * m3), 5, 1], (Element[(m2^(-1) * m3), 2, 1] * 1) + (Element[(m2^(-1) * m3), 4, 1] * 0) + Element[(m2^(-1) * m3), 6, 1])
E'_2: ((Element[(m2^(-1) * m3), 1, 1] * 0) + (Element[(m2^(-1) * m3), 3, 1] * 1) + Element[(m2^(-1) * m3), 5, 1], (Element[(m2^(-1) * m3), 2, 1] * 0) + (Element[(m2^(-1) * m3), 4, 1] * 1) + Element[(m2^(-1) * m3), 6, 1])
e'_1 = Vector(O', E'_1)
e'_2 = Vector(O', E'_2)
P = (-1.5, 1.5)
P': ((Element[(m2^(-1) * m3), 1, 1] * x(P)) + (Element[(m2^(-1) * m3), 3, 1] * y(P)) + Element[(m2^(-1) * m3), 5, 1], (Element[(m2^(-1) * m3), 2, 1] * x(P)) + (Element[(m2^(-1) * m3), 4, 1] * y(P)) + Element[(m2^(-1) * m3), 6, 1])
v = Vector(P, P')
d = false
Diskr: sqrt((Element[(m2^(-1) * m3), 1, 1] + Element[(m2^(-1) * m3), 4, 1])^(2) - (4 * ((Element[(m2^(-1) * m3), 1, 1] * Element[(m2^(-1) * m3), 4, 1]) - (Element[(m2^(-1) * m3), 2, 1] * Element[(m2^(-1) * m3), 3, 1]))))
λ_1: (Element[(m2^(-1) * m3), 1, 1] + Element[(m2^(-1) * m3), 4, 1] + Diskr) / 2
λ_2: (Element[(m2^(-1) * m3), 1, 1] + Element[(m2^(-1) * m3), 4, 1] - Diskr) / 2
text8 = SurdText(λ_1)
text9 = SurdText(λ_2)
sam1: ((Element[(m2^(-1) * m3), 1, 1] - 1) * x) + (Element[(m2^(-1) * m3), 3, 1] * y) + Element[(m2^(-1) * m3), 5, 1] = 0
sam2: (Element[(m2^(-1) * m3), 2, 1] * x) + ((Element[(m2^(-1) * m3), 4, 1] - 1) * y) + Element[(m2^(-1) * m3), 6, 1] = 0
D = Intersect(sam1, sam2)
h_{1} = If(sam1 ≟ sam2, sam1, Intersect[sam1, sam2])
h = If(sam1 ≟ sam2, sam1, Intersect[sam1, sam2])
E = If(sam1 ≠ sam2, Intersect[sam1, sam2])
eq1 = If(sam1 ≟ sam2, sam1)
text6: "\text{ Množina samodružných bodov}\\\\
 \ \;  \; \text{SAM: }   \left\{  " + (LaTeX[sam1]) + "  \right\} "
text10: "\text{ Množina samodružných bodov}\\\\
 \ \;  \; \text{SAM: }    \left\{  " + (LaTeX[E]) + " \right\}  "
text11 = If(sam1 ≟ sam2, text6, text10)
j = false
S = (-0.5, 0)
R = (-1.7752070034005305, 0.09572417509583597)
c = Circle(S, R)
M = Point(c)
M': ((Element[(m2^(-1) * m3), 1, 1] * x(M)) + (Element[(m2^(-1) * m3), 3, 1] * y(M)) + Element[(m2^(-1) * m3), 5, 1], (Element[(m2^(-1) * m3), 2, 1] * x(M)) + (Element[(m2^(-1) * m3), 4, 1] * y(M)) + Element[(m2^(-1) * m3), 6, 1])
GMB1 = Locus(M', M)
u = Vector(M, M')
k = Line(M, S)
F = Point(k)
l = OrthogonalLine(F, k)
G, H = Intersect(c, l)
I, J = Intersect(c, k)
J': ((Element[(m2^(-1) * m3), 1, 1] * x(J)) + (Element[(m2^(-1) * m3), 3, 1] * y(J)) + Element[(m2^(-1) * m3), 5, 1], (Element[(m2^(-1) * m3), 2, 1] * x(J)) + (Element[(m2^(-1) * m3), 4, 1] * y(J)) + Element[(m2^(-1) * m3), 6, 1])
H': ((Element[(m2^(-1) * m3), 1, 1] * x(H)) + (Element[(m2^(-1) * m3), 3, 1] * y(H)) + Element[(m2^(-1) * m3), 5, 1], (Element[(m2^(-1) * m3), 2, 1] * x(H)) + (Element[(m2^(-1) * m3), 4, 1] * y(H)) + Element[(m2^(-1) * m3), 6, 1])
G': ((Element[(m2^(-1) * m3), 1, 1] * x(G)) + (Element[(m2^(-1) * m3), 3, 1] * y(G)) + Element[(m2^(-1) * m3), 5, 1], (Element[(m2^(-1) * m3), 2, 1] * x(G)) + (Element[(m2^(-1) * m3), 4, 1] * y(G)) + Element[(m2^(-1) * m3), 6, 1])
q1, m, h_2, j_1, g_1 = Polygon(M, H, J, G)
q1', n, p, r, s = Polygon(M', H', J', G')
i = false
A16: 1
B16: 0
A17: 0
B17: 1
mid: {{A16, B16}, {A17, B17}}
S1: mA - (λ_1 * mid)
S2: mA - (λ_2 * mid)
o = false
smer1: ((Element[(m2^(-1) * m3), 1, 1] - λ_1) * x) + (Element[(m2^(-1) * m3), 3, 1] * y) = 0
smer2: (Element[(m2^(-1) * m3), 2, 1] * x) + ((Element[(m2^(-1) * m3), 4, 1] - λ_2) * y) = 0
text7: "\text{Korene charakteristickej rovnice }\\\\
 \;  \; λ_ \;  \;1=" + (LaTeX[text8]) + " , \;  \;λ_2=" + (LaTeX[text9]) + " "