1

2

3

4

A

B

C

D

E

F

V BRNĚ

SK.

V

M

LDF MENDELU

ZS 2019

KG-L

N_1(5)

B_1(2)

P_1(0)

C_1(5)

M_1(4)

A_1(1)

P_1(0)

1.\ \textrm{Body }A,B,C\textrm{ určují rovinu }\alpha. \
\hspace{0.6cm}\textrm{Hledáme její půdorysnou stopu }p_1^\alpha\textrm{ a hlavní přímku o kótě }5.\
\hspace{0.6cm}\textrm{Pro sestrojení půdorysné stopy musíme nalézt dva půdorysné stopníky.}\
\hspace{0.6cm}\textrm{Na polopřímce }BA\textrm{ je půdosný stopník }P_1\textrm{ s kótou 0}\
\hspace{0.6cm}\textrm{umístěn tak, že }|B_1A_1|=|A_1P_1|.

(C)

(B)

2.\ \textrm{Na polopřímce }CB\textrm{ najdeme půdorysný stopník pomocí sklápění.}\
\hspace{0.6cm}\textrm{Určíme bod }(B)\textrm{ tak, že }B_1(B)\perp B_1C_1,\textrm{ přičemž je }|B_1(B)|=2\,\textrm{cm.}\
\hspace{0.6cm}\textrm{(volíme 1\,dílek jako 1\,cm)}\
\hspace{0.6cm}\textrm{Stejně tak }C_1(C)\perp B_1C_1\textrm{ a }|C_1(C)|=5\,\textrm{cm.}\

P_1'(0)

3.\ P_1'=\ \rightarrow\! C_1B_1\ \cap\ \rightarrow\! (C)(B)

p_1^\alpha

4.\ p_1^\alpha=\ \leftrightarrow\! P_1P_1'\dots\textrm{půdorysná stopa roviny }\alpha

h_1^\alpha(5)

5.\ C_1\!\in h_1^\alpha\parallel p_1^\alpha\dots\textrm{hlavní přímka roviny }\alpha\textrm{ o kótě 5}

6.\ \textrm{Body }M,N,P\textrm{ určují rovinu }\beta.\
\hspace{0.6cm}\textrm{Hledáme její půdorysnou stopu }p_1^\beta\textrm{ a hlavní přímku o kótě 5.}\
\hspace{0.6cm}\textrm{Pro sestrojení hlavní přímky o kótě 5 potřebujeme dva body o kótě 5.}\
\hspace{0.6cm}\textrm{Na polopřímce }PM\textrm{ najdeme bod }Q\textrm{ s kótou 5 pomocí stupňování.}\
\hspace{0.6cm}\textrm{Bodem }P_1\textrm{ vedeme libovolnou polopřímku,}\
\hspace{0.6cm}\textrm{na níž naneseme 5 stejných dílků (libovolně velkých).} 

7.\ \textrm{Čtvrtý dílek spojíme s bodem }M_1.\
\hspace{0.6cm}\textrm{Pátým dílkem vedeme rovnoběžku,}\ 
\hspace{0.6cm}\textrm{která protne polopřímku }P_1M_1\textrm{ v bodě }Q_1\textrm{ s kótou 5.}

Q_1(5)

h_1^\beta(5)

8.\ h_1^\beta=\ \leftrightarrow\! Q_1N_1\dots\textrm{hlavní přímka roviny }\beta\textrm{ o kótě 5}

p_1^\beta

9.\ P_1\!\in p_1^\beta\parallel h_1^\beta\dots\textrm{půdorysná stopa roviny }\beta

P_1(0)

−

H_1(5)

10.\ \overline{\!P}_1=p_1^\alpha\cap p_1^\beta\
\hspace{0.8cm}H_1=h_1^\alpha\cap h_1^\beta

r_1

11.\ r_1=\ \leftrightarrow\overline{\!P}_1H_1\dots\textrm{průsečnice rovin }\alpha\textrm{ a }\beta

X_1

Y_1

12.\ \textrm{Na průsečnici }r_1\textrm{ je z hlediska záseku trojúhelníků}\
\hspace{0.8cm}\textrm{podstatná pouze úsečka }X_1Y_1,\textrm{ která leží uvnitř obou trojúhelníků.}

13.\ \textrm{Pro posouzení viditelnosti vezmeme v úvahu bod }C_1\textrm{ s kótou 5,}\
\hspace{0.8cm}\textrm{který leží v rovině }\alpha=(A,B,C),\
\hspace{0.8cm}\textrm{a bod se stejným půdorysem,\,který leží v rovině }\beta=(P,M,N).\
\hspace{0.8cm}\textrm{Bod }C\textrm{ má nutně větší kótu,\,a proto je viditelný. }

R_3 = (20.85, 29.7)
R_1'' = (1.3, 0.1)
R_2'' = (20.85, 0.1)
S_2 = Midpoint(R_3, R_2'')
R_4'' = (1.3, 29.7)
S_4 = Midpoint(R_4'', R_1'')
m = Segment(S_4, S_2)
U_1 = Point(m)
r_2 = OrthogonalLine(U_1, m)
U_1' = Point(r_2)
s_3 = OrthogonalLine(U_1', r_2)
h_3 = Circle(U_1', 8.3)
R_2 = (21, 0)
R_1 = (0, 0)
R_4 = (0, 29.7)
a = Segment(R_4, R_3)
b = Segment(R_3, R_2)
c = Segment(R_2, R_1)
d = Segment(R_4, R_1)
R_1' = (1.8, 0.6)
R_2' = (20.35, 0.6)
R_3' = (20.35, 29.2)
R_4' = (1.8, 29.2)
e = Segment(R_4', R_3')
f = Segment(R_3', R_2')
g = Segment(R_2', R_1')
h = Segment(R_4', R_1')
i = Segment(R_4'', R_3)
j = Segment(R_3, R_2'')
k = Segment(R_4'', R_1'')
l = Segment(R_1'', R_2'')
S_1 = Midpoint(R_1'', R_2'')
S_3 = Midpoint(R_4'', R_3)
n = Segment(S_3, S_1)
p = Circle(S_4, 1.5)
q = Circle(S_2, 1.5)
r = Circle(S_3, 1.5)
S_4' = Intersect(p, m)
S_2' = Intersect(q, m)
S_3' = Intersect(r, n)
t = Segment(S_4, S_4')
S_1' = Intersect(g, n)
s = Segment(S_3, S_3')
a_1 = Segment(S_2', S_2)
b_1 = Segment(S_1, S_1')
c_1 = Circle(S_3, 5.2)
A_6, A_5 = Intersect(c_1, i)
d_1 = Circle(S_2, 5.2)
A_3, A_2 = Intersect(d_1, j)
A_1 = Mirror(S_2, A_2)
A_4 = Mirror(S_2, A_3)
e_1 = OrthogonalLine(A_1, j)
f_1 = OrthogonalLine(A_2, j)
g_1 = OrthogonalLine(A_3, j)
h_1 = OrthogonalLine(A_4, j)
i_1 = OrthogonalLine(A_5, i)
j_1 = OrthogonalLine(A_6, i)
A_1' = Intersect(e_1, f)
A_2' = Intersect(f_1, f)
A_3' = Intersect(g_1, f)
A_4' = Intersect(h_1, f)
A_5' = Intersect(i_1, e)
A_6' = Intersect(j_1, e)
k_1 = Segment(A_6, A_6')
l_1 = Segment(A_5, A_5')
m_1 = Segment(A_4', A_4)
n_1 = Segment(A_3, A_3')
p_1 = Segment(A_2, A_2')
q_1 = Segment(A_1, A_1')
r_1 = Circle(R_1', 4)
T_1 = Intersect(r_1, h)
s_1 = Circle(R_2', 4)
T_1' = Intersect(s_1, f)
t_1 = Segment(T_1, T_1')
c_2 = Circle(R_1', 1)
d_2 = Circle(R_2', 1)
T_2 = Intersect(c_2, h)
T_2' = Intersect(d_2, f)
a_2 = Segment(T_2, T_2')
T_3' = Mirror(R_2', T_2')
b_2 = OrthogonalLine(T_3', j)
e_2 = Circle(T_3', 6)
T_3, T_3'' = Intersect(e_2, b_2)
f_2 = Segment(T_3, T_3')
g_2 = OrthogonalLine(T_3, g)
T_4 = Intersect(t_1, g_2)
T_4' = Intersect(g, g_2)
h_2 = Segment(T_4, T_4')
k_2 = Circle(T_3, 3.5)
T_5 = Intersect(k_2, f_2)
i_2 = OrthogonalLine(T_5, g)
T_5' = Intersect(g, i_2)
j_2 = Segment(T_5, T_5')
T_6' = Midpoint(T_3, T_4)
l_2 = OrthogonalLine(T_6', f)
T_6 = Intersect(l_2, h)
m_2 = Segment(T_6, T_6')
p_2 = Circle(T_4, 2.5)
T_7, T_7'' = Intersect(p_2, t_1)
n_2 = OrthogonalLine(T_7, t_1)
T_7' = Intersect(m_2, n_2)
q_2 = Segment(T_7, T_7')
s_2 = Circle(T_2, 2.5)
T_8 = Intersect(s_2, a_2)
t_2 = OrthogonalLine(T_8, m_2)
T_8' = Intersect(g, t_2)
a_3 = Segment(T_8, T_8')
d_3 = Circle(U_1', 5)
e_3 = Circle(U_1', 3)
U_2', U_2 = Intersect(d_3, r_2)
c_3 = Circle(U_1', 3.9)
N_1', N_1 = Intersect(c_3, r_2)
U_3, U_3' = Intersect(e_3, r_2)
b_3 = OrthogonalLine(U_2, r_2)
f_3 = Circle(U_2, 3.5)
B_1', B_1 = Intersect(f_3, b_3)
g_3 = Circle(U_1', 7)
C_1', C_1 = Intersect(g_3, s_3)
i_3 = OrthogonalLine(U_3, r_2)
k_3 = Circle(U_3, 1.4)
p_3 = Circle(U_3, 5.5)
M_1', M_1 = Intersect(p_3, i_3)
A', A = Intersect(k_3, i_3)
j_3 = Segment(B_1, A)
P_1', P_1 = Intersect(h_3, s_3)
q_3 = Segment(N_1, M_1)
n_3 = Segment(P_1, N_1)
r_3 = Segment(P_1, M_1)
m_3 = Segment(C_1, B_1)
l_3 = Segment(A, C_1)
W_1: (-17, 29.3)
W_2: (0, 29.3)
W_3: (-17, 2)
pos = 14 (slider from 0 to 14, step 1)
t_3 = Segment(W_1, W_2)
a_4 = Segment(W_1, W_3)
b_4 = Ray(B_1, A)
P_2 = Mirror(B_1, A)
Z_1 = Point(b_4)
c_4 = Segment(A, Z_1)
d_4 = Ray(C_1, B_1)
e_4 = OrthogonalLine(C_1, d_4)
f_4 = OrthogonalLine(B_1, d_4)
X_1 = Point(e_4)
X_2 = Mirror(C_1, X_1)
X_3 = Mirror(X_1, X_2)
X_4 = Mirror(X_2, X_3)
X_5 = Mirror(X_3, X_4)
g_4 = OrthogonalLine(X_1, e_4)
h_4 = OrthogonalLine(X_2, e_4)
X_7 = Intersect(h_4, f_4)
X_6 = Intersect(g_4, f_4)
i_4 = Segment(X_5, C_1)
j_4 = Segment(X_7, B_1)
k_4 = Ray(X_5, X_7)
P_2' = Intersect(k_4, d_4)
Z_2 = Point(d_4)
Z_3 = Point(k_4)
l_4 = Segment(B_1, Z_2)
m_4 = Segment(X_5, Z_3)
n_4 = Line(P_2', P_2)
Z_4 = Point(n_4)
Z_5 = Point(n_4)
p_4 = Segment(Z_4, Z_5)
q_4 = Line(C_1, p_4)
Z_6 = Point(q_4)
Z_7 = Point(q_4)
r_4 = Segment(Z_6, Z_7)
Y_1 = (7.3558075524268345, 15.631772107156198)
Y_2 = Mirror(P_1, Y_1)
Y_3 = Mirror(Y_1, Y_2)
Y_4 = Mirror(Y_2, Y_3)
Y_5 = Mirror(Y_3, Y_4)
s_4 = Segment(P_1, Y_5)
t_4 = Ray(P_1, M_1)
Z_8 = Point(t_4)
a_5 = Segment(M_1, Z_8)
b_5 = Segment(Y_4, M_1)
c_5 = Line(Y_5, b_5)
Q_1 = Intersect(c_5, a_5)
Z_9 = Point(c_5)
d_5 = Segment(Y_5, Z_9)
Y_3' = Mirror(Y_2, Y_3)
Y_4' = Mirror(Y_3, Y_3')
e_5 = Line(N_1, Q_1)
Z_{10} = Point(e_5)
Z_{11} = Point(e_5)
f_5 = Segment(Z_{10}, Z_{11})
g_5 = Line(P_1, f_5)
Z_{12} = Point(g_5)
Z_{13} = Point(g_5)
h_5 = Segment(Z_{12}, Z_{13})
P_3 = Intersect(p_4, h_5)
H_1 = Intersect(f_5, r_4)
i_5 = Line(P_3, H_1)
Z_{14} = Point(i_5)
Z_{15} = Point(i_5)
j_5 = Segment(Z_{14}, Z_{15})
X_0 = Intersect(n_3, j_5)
Y_0 = Intersect(l_3, j_5)
k_5 = Segment(X_0, Z_{14})
l_5 = Segment(X_0, Y_0)
m_5 = Segment(Y_0, Z_{15})
n_5 = Segment(C_1, B_1)
E_1 = Intersect(n_3, m_3)
E_2 = Intersect(n_3, j_3)
E_3 = Intersect(q_3, j_3)
E_4 = Intersect(l_3, q_3)
p_5 = Segment(B_1, E_2)
q_5 = Segment(E_3, A)
r_5 = Segment(E_2, E_3)
s_5 = Segment(C_1, Y_0)
t_5 = Segment(Y_0, E_4)
a_6 = Segment(E_4, A)
b_6 = Segment(M_1, N_1)
c_6 = Segment(P_1, E_1)
d_6 = Segment(E_1, X_0)
e_6 = Segment(X_0, N_1)
f_6 = Segment(P_1, M_1)
mnohoúhelník1, y_0, c_7, b_7, e_7, x_0 = Polygon(Y_0, C_1, B_1, E_2, X_0)
mnohoúhelník2, a_7, e_8, e_9 = Polygon(E_4, E_3, A)
mnohoúhelník3, e_{10}, p_6, m_6, n_6, x_1, y_1, c_8 = Polygon(E_1, P_1, M_1, N_1, X_0, Y_0, C_1)