Lösungen/ Solutions
Vorversion /Version préliminaire

Aufgabe /Problème 1

Solve[{
x + y + z + w == 
            1, 
y - z + w == 
            1, 
z - w == 
            1, 
y - w == 1
}, {x, 
          y, z, w}]   // Flatten

Aufgabe /Problème 2

Remove["Global`*"]

vecLen[a_] := Sqrt[a . a]
vF[v_] := MatrixForm[Transpose[{v}]]

OA = {2, 1, 4}; OB = {6, 11, 3}; OC = {5, 7, 8};  
    OD = {5, 1, 9};

a

a = OA; b = OB - OA; c = OC - OA; d = OD - OA;

Transpose[{a, b, c, d}] // MatrixForm

V = Det[{b, c, d}]

b

b . d/(vecLen[b]*vecLen[d]) // N

φ = ArcCos[b . d/(vecLen[b]*vecLen[d])]

φN = N[%]

φN / Degree

?Degree

2 N[Pi]/Degree

c

bcOrth = Cross[b, c];  vF[bcOrth]

dcOrth = Cross[d, c];  vF[dcOrth]

φ1 = 
      ArcCos[bcOrth . dcOrth/(vecLen[bcOrth]*vecLen[dcOrth])]

φ1N = N[%]

φ1N/Degree

Ergänzung/ Supplement

180 - φ1N/Degree

d

OM1 = a + b + 1/2 c;
OM3 = a + c + 1/2 b;
OM2 = a + d + b + 1/2 c;

OQ = a + b + c + d;
S1 = 1/2 vecLen[Cross[OQ - OB, OQ - OC]]

N[%]

e

S2 = 1/2 vecLen[Cross[OM1 - OM3, OM2 - OM3]]

N[%]

f

G = vecLen[Cross[c, d]]

dMin = V/G

N[%]

Aufgabe 3 /Problème 3

1a, b

r = 4; OM = {8, 6};
circ = {(x - 8)^2 + (y - 6)^2 == r^2}

OP0 =  {10, 8};   a =  {4, -1};
g[t_] := OP0 + t a;  g[t] // vF

vF[{x, y}] == vF[g[t]]

line = {x == g[t][[1]], y == g[t][[2]]}

equat = Join[circ, line]

Solve[equat, {x, y, t}]

solv3 = Solve[equat, {x, y, t}] // N

OP1 = {x, y} /.  solv3[[2]]

OP2 = {x, y} /.  solv3[[1]]

1e

OR = OM - (OP1 - OM)

Show[
        Graphics[
{PointSize[
              0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[OM, 
              r], 
Point[OP1], Point[OP2], Point[OM], 
            Point[OR], 
Point[
              OP0], 
Line[{OP1, OP2}], 
            Line[{OP1, OR}], 
Line[{OP2, 
                OR}]
}], 
        AspectRatio -> 11.1/13.1];

1d

P2R = OR - OP2

RP1 = OP1 - OR

RP1Orth = {-RP1[[2]], RP1[[1]]}

P2RProjRP1Orth = P2R . RP1Orth / vecLen[RP1Orth]

P2Q = RP1Orth/vecLen[RP1Orth] * P2RProjRP1Orth

OQ = OP2 + P2Q

Show[
        Graphics[
{PointSize[
              0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[OM, 
              r], 
Point[OP1], Point[OP2], Point[OM], 
            Point[OR], 
Point[
              OQ], 
Line[{OP1, OP2}], 
            Line[{OP1, OR}], 
Line[{OP2, 
                OR}]
}], 
        AspectRatio -> 11.1/13.1];

c

OP3 = OP2 + 2  P2Q

Show[
        Graphics[
{PointSize[
              0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[OM, 
              r], 
Point[OP1], Point[OP2], Point[OM], 
            Point[OR], 
Point[OQ], 
            Point[OP3], 
Line[{OP1, OP2}], 
            Line[{OP1, OR}], 
Line[{OP2, OR}], 
            Line[{OP2, OP3}]
}], 
        AspectRatio -> 11.1/13.1];

1f

OS = {xS, yS}

Expand[(OP1 - OS) . (OR - OS)] == (OP2 - OS) . (OP2 - 
            OS)

cond1 = (Expand[(OP1 - OS) . (OR - OS) - (OP2 - 
                    OS) . (OP2 - OS)] == 0)

cond2 = (Expand[(OP1 - OS) . (OR - OS) - (OP3 - 
                    OS) . (OP3 - OS)] == 0)

solf3f = Solve[{cond1, cond2}, {xS, yS}]  //  Flatten

xS = xS /.  solf3f[[1]]; 
    yS = yS /.  solf3f[[2]];
{xS, yS}

Show[
        Graphics[
{PointSize[
              0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[OM, 
              r], 
Point[OP1], Point[OP2], Point[OM], 
            Point[OR], 
Point[OQ], Point[OP3], 
            Point[OS], 
Line[{OP1, OP2}], 
            Line[{OP1, OR}], 
Line[{OP2, OR}], 
            Line[{OP2, OP3}], 
Line[{OS, OR}], 
            Line[{OS, OP2}], Line[{OS, OP3}]
}], 
        AspectRatio -> 11.1/(12 + 5.88)];

Aufgabe 4 /Problème 4

OM = {8, 6};  r = 4;  OP = {1, 1};  OU = {7, -1};

Beispiel einer Tangente/ Exemple d'une tangente:

TDemo = N[{10, 6 + Sqrt[r^2 - (10 - 8)^2]}]

tgDemo = {({x, y} - OM) . (TDemo - OM) == r^2}

solv4 = Solve[tgDemo, {y}]  // Flatten

tgVecDemo[x_] := 
        Evaluate[{x, y} /. solv4];
tgVecDemo[t] // Expand // vF

Show[
        Graphics[
{PointSize[
              0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[OM, 
              r], 
Point[TDemo], Point[OM], Point[OP], 
            Point[OU], 
Line[{tgVecDemo[9], 
                tgVecDemo[11]}], 
Line[{OM, 
                TDemo}]
}], 
        AspectRatio -> 11.1/(13.1)];

Beispiel einer Zentrale/ Exemple d'une centrale -> OM TDemo!

a

OT = {x, y}

Bedingung: PT hat zu MT den gleich grossen Winkel wie UT zu MT/
Condition: PT a le même angle avec MT comme UT acev MT

cond1 = (    (OP - OT) . (OM - OT)/
            vecLen[(OP - OT)] == (OU - OT) . (OM - OT)/
            vecLen[(OU - OT)]    )

cond2 = ((OT - OM) . (OT - OM) == r^2)

solv44 = Solve[{cond1, cond2}, {x, y}] // N // Chop

OT1 = {x, y} /. solv44[[1]]

OT2 = {x, y} /. solv44[[2]]

OTm = OM -  (OT2 - OT1)

Show[
        Graphics[
{PointSize[
              0.03], 
Line[{{0, -1.1}, {0, 1.1}}], 
            Line[{{-1.1, 0}, {1.1, 0}}], 
Circle[OM, 
              r], 
Point[OM], Point[OP], 
            Point[OU], 
Point[OT1], 
            Point[OT2], 
Line[{OTm, 
                OT2}], 
Line[{OP, OT1}], 
            Line[{OU, OT1}]
}], 
        AspectRatio -> 11.1/(13.1)];

b

α = 
      Pi/2 - ArcCos[(OP - OT1) . (OT1 - OM)/(vecLen[(OP - OT1)]*
                vecLen[(OT1 - OM)])]

α/Degree

d1 = vecLen[(OP - OT1)]

d2 = vecLen[(OU - OT1)]

d = d1 + d2