Lösungen

Vektorkurven und Richtungsvektoren

1. Beispiel von 2D-Kurven

Remove["Global`*"]

v[t_]:={2Cos[t]+Cos[5t]/3,Sin[t]+Sin[5t]/3};

pl1 = 
        ParametricPlot[v[t], {t, -Pi, 2  Pi}, 
          AspectRatio -> Automatic];

ParametricPlot[{2  Cos[t] + Cos[6  t]/4, 
          Sin[t] + Sin[6  t]/4}, {t, -Pi, 2  Pi}, 
        AspectRatio -> Automatic];

ParametricPlot[{Cos[t]+Cos[8t]/6,Sin[t]+Sin[8t]/6},{t,-Pi,2Pi},AspectRatio->Automatic];

ParametricPlot[{Cos[t]+Cos[8t]/6,Sin[t]+Cos[8t]/6},{t,-Pi,2Pi},AspectRatio->Automatic];

2. Ein Richtungsvektor

v'[t]

D[{2  Cos[t] + Cos[5  t]/3, Sin[t] + Sin[5  t]/3}, t]

vv = 
      D[{2  Cos[t] + Cos[5  t]/3, Sin[t] + Sin[5  t]/3}, t] /. 
        t -> 1.2

g[t_]:=v[1.2]+t vv;
pl2 = ParametricPlot[Evaluate[g[u]],{u,-1,1}];

Show[pl1,pl2,Graphics[{PointSize[0.03],Point[v[1.2]]}]];

An einer Spitze entsteht ein Extremum mindestens einer Komponente des Tangentenvektors. Es existiert dort manchmal auch keine eindeutige Tangente.  
Beispiel:

Solve[Evaluate[D[2Cos[t]+Cos[5t]/3,t]]==0,{t}]//N

vv=D[{2Cos[t]+Cos[5t]/3,Sin[t]+Sin[5t]/3},t]/.t->0.854718

vv={0,-0.051656283416512405};

g1[t_]:=v[0.854718]+t vv;
pl3 = ParametricPlot[Evaluate[g1[t]],{t,-20,10},AspectRatio->Automatic];

Show[pl1,pl3,Graphics[{PointSize[0.03],Point[v[0.854718]]}]];

3. Ein 3D-Plot

Remove["Global`*"]

ParametricPlot3D[{2  Cos[t] + Cos[5  t]/3, Sin[t] + Sin[5  t]/3, 
          Sqrt[(t - Pi)^2]}, {t, -Pi, 2  Pi}];

4. Geradenspiegelung

Remove["Global`*"]

a = {3, 1};
λ1 = 1;  λ2 = -1; 
v1 = a;  
    v2 = {-a[[2]], a[[1]]};
OP = {1, 5};
OO = {0, 0};

eV = Transpose[{v1, v2}];  eV // MatrixForm

eD = {{λ1, 0}, {0, λ2}};   eD // 
      MatrixForm

M = eV . eD . Inverse[eV];   M // MatrixForm

OQ = M . OP

Show[
        Graphics[{
Line[{OO, a}], Line[{OO, {3, 0}}], 
            Line[{OO, {0, 3}}]
, PointSize[0.04], 
            Point[OP], Point[OQ], Line[{OP, OQ}]
}], 
        AspectRatio -> Automatic];

4. Geradenspiegelung

Remove["Global`*"]

a = {3, 1};
λ1 = 1;  λ2 = -1; 
v1 = a;  
    v2 = {-a[[2]], a[[1]]};
OP = {1, 5};
OO = {0, 0};
dM5[
          k_] = {{Cos[k 2  Pi/5], -Sin[k 2  Pi/5]}, {Sin[k 2  Pi/5], 
            Cos[k 2  Pi/5]}};
p[k_] := dM5[k] . OP

eV = Transpose[{v1, v2}];  eV // MatrixForm

eD = {{λ1, 0}, {0, λ2}};   eD // 
      MatrixForm

M = eV . eD . Inverse[eV];   M // MatrixForm

OQ = M . OP

q[k_] := M . p[k]

Show[
        Graphics[{
Line[{-2 a, 2  a}], 
            Line[{OO, {3, 0}}], Line[{OO, {0, 3}}]
, 
            PointSize[0.04], Point[OP], Point[p[1]], Point[p[2]], 
            Point[p[3]], Point[p[4]], 
            Point[p[5]], 
Point[OQ], Point[q[1]], 
            Point[q[2]], Point[q[3]], Point[q[4]], 
            Point[q[5]], 
Line[{OP, OQ}], 
            Line[{p[1], q[1]}], Line[{p[2], q[2]}], Line[{p[3], q[3]}], 
            Line[{p[4], q[4]}], 
            Line[{p[5], q[5]}], 
Line[{p[1], p[2], p[3], 
                p[4], p[5], p[1]}], 
Line[{q[1], q[2], 
                q[3], q[4], q[5], q[1]}]
}], 
        AspectRatio -> Automatic];

5. Ebenenspiegelung

Remove["Global`*"]

r0 = {1, 2, 3};  a = {1, 2, -1};  
    b = {-1, 1, 
        1};
    
x[λ_, μ_] = 
        r0 + λ a  +  μ b;

a  Translation, Ebenenspiegelung, Rücktranslation

tr[p_] := p - r0;

eV = Transpose[{a, b, Cross[a, b]}];  eV // MatrixForm

λ1 = 1;  λ2 = 
      1;  λ3 = -1;
eD = {{λ1, 0, 0}, {0, λ2, 0}, {0, 
          0, λ3}};   eD // MatrixForm

M = eV . eD . Inverse[eV];   M // MatrixForm

sp[p_] := M . p

rtr[p_] := p + r0;

b  Berechnung von M siehe a

ϕSpiegel[p_] := rtr[M . tr[p]]

c  Spiegelung des Tetraeders {p1,p2,p3,p4}

p1 = {1, 5, 2};
p2 = {2, 6, 5};
p3 = {4, 3, 1};
p4 = {-1, 1, 5};

q1 = ϕSpiegel[p1]

q2 = ϕSpiegel[p2]

q3 = ϕSpiegel[p3]

q4 = ϕSpiegel[p4]

c  Visualisierung

OO = {0, 0, 0}

Show[
        Graphics3D[{
Line[{OO, {3, 0, 0}}], 
            Line[{OO, {0, 3, 0}}], 
            Line[{OO, {0, 0, 3}}], 
Line[{OO, r0}], 
            Line[{r0, r0 + a}], 
            Line[{r0, r0 + b}], 
PointSize[
              0.04], 
Point[p1], 
            Point[p2], 
Point[p3], 
            Point[p4], 
Point[q1], 
            Point[q2], 
Point[q3], 
            Point[q4], 
Line[{p1, p2, p3, p1, p4, p2, p3, 
                p4}],   
              Line[{p1, q1}], Line[{p2, q2}], 
            Line[{p3, q3}], 
Line[{p4, 
                q4}]
}], AspectRatio -> Automatic, 
        ViewPoint -> {-2.539,  3.502,  1.037}];

Show[
        Graphics3D[{
Line[{OO, {3, 0, 0}}], 
            Line[{OO, {0, 3, 0}}], 
            Line[{OO, {0, 0, 3}}], 
Line[{OO, r0}], 
            Line[{r0, r0 + a}], 
            Line[{r0, r0 + b}], 
PointSize[
              0.04], 
Point[p1], 
            Point[p2], 
Point[p3], 
            Point[p4], 
Point[q1], 
            Point[q2], 
Point[q3], 
            Point[q4], 
Line[{p1, p2, p3, p1, p4, p2, p3, 
                p4}], Line[{q1, q2, q3, q1, q4, q2, q3, q4}], Line[{p1, q1}], 
            Line[{p2, q2}], 
            Line[{p3, q3}], 
Line[{p4, 
                q4}]
}], AspectRatio -> Automatic, 
        ViewPoint -> {-2.539,  3.502,  1.037}];

Show[
        Graphics3D[{
Line[{OO, {3, 0, 0}}], 
            Line[{OO, {0, 3, 0}}], 
            Line[{OO, {0, 0, 3}}], 
Line[{OO, r0}], 
            Line[{r0, r0 + a}], 
            Line[{r0, r0 + b}], 
PointSize[
              0.04], 
Point[p1], 
            Point[p2], 
Point[p3], 
            Point[p4], 
Point[q1], 
            Point[q2], 
Point[q3], 
            Point[q4], 
Line[{p1, p2, p3, p1, p4, p2, p3, 
                p4}], Line[{q1, q2, q3, q1, q4, q2, q3, q4}], Line[{p1, q1}], 
            Line[{p2, q2}], 
            Line[{p3, q3}], 
Line[{p4, 
                q4}]
}], 
        AspectRatio -> Automatic];