•Osnovni trirob

Krivulja K je podana s parametrizacijo

In[1]:=

r[t_] := {2 Cos[t], 2 Sin[t], t/2}

za  t∈[0, 4π].

In[2]:=

gr = ParametricPlot3D[r[t], {t, 0, 4 π}] ;

[Graphics:HTMLFiles/index_3.gif]

Osnovni trirob

In[3]:=

T0[t_] = r '[t]/(r '[t] . r '[t])^(1/2) // Simplify

Out[3]=

{-(4 Sin[t])/17^(1/2), (4 Cos[t])/17^(1/2), 1/17^(1/2)}

In[4]:=

b0 = r '[t] × r ''[t] // Simplify ; n0 = b0 × r '[t] // Simplify ; N0[t_] = n0/(n0 . n0)^(1/2) // Simplify

Out[6]=

{-Cos[t], -Sin[t], 0}

In[7]:=

B0[t_] = T0[t] × N0[t] // Simplify

Out[7]=

{Sin[t]/17^(1/2), -Cos[t]/17^(1/2), 4/17^(1/2)}

In[8]:=

trirob[t_] := Show[gr, Graphics3D[{RGBColor[0, 0, 1], Thickness[0.01], Line[{r[t], r[t] + T0[t]}], Text["T", r[t] + 1.3 T0[t]], Line[{r[t], r[t] + N0[t]}], Text["N", r[t] + 1.3 N0[t]], Line[{r[t], r[t] + B0[t]}], Text["B", r[t] + 1.3 B0[t]]}], PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}, {0, 7.5}}] ;

In[9]:=

Table[trirob[t], {t, 0, 4 π, 4 π/50}] ;

[Graphics:HTMLFiles/index_12.gif]

Še en primer

In[10]:=

r[t_] := {Sin[t], Cos[t], Sin[2 t]} ; gr = ParametricPlot3D[r[t], {t, 0, 2 π}] ;

[Graphics:HTMLFiles/index_14.gif]

In[12]:=

T0[t_] = r '[t]/(r '[t] . r '[t])^(1/2) // Simplify

Out[12]=

{Cos[t]/(3 + 2 Cos[4 t])^(1/2), -Sin[t]/(3 + 2 Cos[4 t])^(1/2), (2 Cos[2 t])/(3 + 2 Cos[4 t])^(1/2)}

In[13]:=

b0 = r '[t] × r ''[t] // Simplify ; n0 = b0 × r '[t] // Simplify ; N0[t_] = n0/(n0 . n0)^(1/2) // Simplify

Out[15]=

{(-3 Sin[t] + 3 Sin[3 t] + Sin[5 t])/(27 + 4 Cos[4 t] - 6 Cos[8 t])^(1/2), (-3 Cos[t] - 3 Cos[3 t] + Cos[5 t])/(27 + 4 Cos[4 t] - 6 Cos[8 t])^(1/2), -(4 Sin[2 t])/(27 + 4 Cos[4 t] - 6 Cos[8 t])^(1/2)}

In[16]:=

B0[t_] = T0[t] × N0[t] // Simplify

Out[16]=

{(8 Cos[t] + 3 Cos[5 t] - Cos[7 t])/((3 + 2 Cos[4 t])^(1/2) (27 + 4 Cos[4 t] - 6 Cos[8 t])^(1/2)), (8 Sin[t] + 3 Sin[5 t] + Sin[7 t])/((3 + 2 Cos[4 t])^(1/2) (27 + 4 Cos[4 t] - 6 Cos[8 t])^(1/2)), -(3 + 2 Cos[4 t])^(1/2)/(27 + 4 Cos[4 t] - 6 Cos[8 t])^(1/2)}

In[17]:=

trirob[t_] := Show[gr, Graphics3D[{RGBColor[0, 0, 1], Thickness[0.01], Line[{r[t], r[t] + T0[t]}], Text["T", r[t] + 1.3 T0[t]], Line[{r[t], r[t] + N0[t]}], Text["N", r[t] + 1.3 N0[t]], Line[{r[t], r[t] + B0[t]}], Text["B", r[t] + 1.3 B0[t]]}], PlotRange -> 1.5 {{-1, 1}, {-1, 1}, {-1, 1}}] ;

In[18]:=

Table[trirob[t], {t, 0, 2 π, 2 π/100}] ;

[Graphics:HTMLFiles/index_23.gif]

Converted by Mathematica  (December 8, 2002)