Position Vectors,
Tangent Vectors,
and
Space Curves

An Example

Background Mathematica Stuff

Needs["VisualLA`"]

Off[ParametricPlot3D::"ppcom"];
Off[General::"spell1"];
Off[Power::"infy"];
Off[∞::"indet"];
Off[ParametricPlot3D::"pplr"];
Off[ParametricPlot3D::"ppx"];
Off[Show::"gcomb"]

The Vectors and Required Functions.

The Position Vector

The Parabola

The Helix

The Sin(x) Curve.

A Trefoil Knot

r[t_] = {(2 + Cos[(3/2) t]) Cos[t], (2 + Cos[(3/2) t]) 
          Sin[t], Sin[(3/2) t]}; 
    WINDOW = {{-3.5, 3.5}, {-3.5, 3.5}, {-5, 2  π + 1}}; 
    tRange = {t, 0, 6  π, 0.2}; xMin = 0;

A circle Thingie (Slanted?)

The Derivative Vector and Norm of the Derivative Vector

rprime[t_] = D[r[t], t];
normrprime[t_] = (Sqrt[Simplify])[rprime[t] . rprime[t]];

The Tangent Vector

tanvec[t_] = rprime[t]/normrprime[t];
normtanvec[
          t_] = (Sqrt[tanvec[t] . tanvec[t]]);
tanvecprime[t_] = D[tanvec[t], t];
normtanvecprime[
          t_] = (Sqrt[tanvecprime[t] . tanvecprime[t]]);

The Normal Vector

normal[t_] = tanvecprime[t]/normtanvecprime[t];

The Binormal Vector

binormal[t_] = Cross[tanvec[t], normal[t]];

The Curvature

κ[t_] = normtanvecprime[t]/normrprime[t];

The Radius

radius[t_] = (1/κ)[t];

The Osculating Circle

kissingcircle[t_, θ_] := 
        r[t] + normal[t] radius[t] + 
          radius[t] (tanvec[t] Cos[θ] + 
                normal[t] Sin[θ]);

Graphing Functions

PosVec[t_] := 
        DrawVector3D[r[t], HeadLength -> 0.05, PlotRange -> WINDOW, 
          Ticks -> False, 
          DisplayFunction -> Identity];
TanVec[t_] := 
        DrawVector3D[{{r[t], r[t] + tanvec[t]}}, HeadLength -> 0.05, 
          PlotRange -> WINDOW, Ticks -> False, 
          DisplayFunction -> Identity];
NorVec[t_] := 
        DrawVector3D[{{r[t], r[t] + normal[t]}}, HeadLength -> 0.05, 
          PlotRange -> WINDOW, Ticks -> False, 
          DisplayFunction -> Identity];
BiNorVec[t_] := 
        DrawVector3D[{{r[t], r[t] + binormal[t]}}, HeadLength -> 0.05, 
          PlotRange -> WINDOW, Ticks -> False, 
          DisplayFunction -> Identity];
KissingCircle[t_] := 
        ParametricPlot3D[kissingcircle[t, θ], {θ, 0, 2  π}, 
          PlotRange -> WINDOW, Ticks -> False, 
          DisplayFunction -> Identity];
SpaceCurve[t_] := 
        ParametricPlot3D[r[s], {s, xMin - 0.1, t}, 
          DisplayFunction -> Identity];

The Position Vector

Table[Show[{PosVec[t]}, DisplayFunction -> $DisplayFunction], 
        Evaluate[tRange]];

The Position Vector and Space Curve

Table[
        Show[{ParametricPlot3D[r[s], {s, - .01, t}, 
              PlotRange -> WINDOW, Ticks -> False, 
              DisplayFunction -> Identity], PosVec[t]}, 
          DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

The Space Curve

Table[
        Show[{ParametricPlot3D[r[s], {s, - .01, t}, 
              PlotRange -> WINDOW, Ticks -> False, 
              DisplayFunction -> Identity]}, 
          DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

The Position and Tangent Vectors

Table[
        Show[{PosVec[t], TanVec[t]}, 
          DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

The Position and Tangent Vectors along with the Space Curve

Table[
        Show[{ParametricPlot3D[r[s], {s, - .01, t}, 
              PlotRange -> WINDOW, Ticks -> False, 
              DisplayFunction -> Identity], PosVec[t], TanVec[t]}, 
          DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

The Position, Tangent, and Normal Vectors along with the Space Curve

Table[
        Show[{ParametricPlot3D[r[s], {s, - .01, t}, 
              PlotRange -> WINDOW, Ticks -> False, 
              DisplayFunction -> Identity], PosVec[t], TanVec[t], 
            NorVec[t]}, DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

The Position, Tangent, Normal, and Binormal Vectors along with the Space Curve

Table[
        Show[{ParametricPlot3D[r[s], {s, - .01, t}, 
              PlotRange -> WINDOW, Ticks -> False, 
              DisplayFunction -> Identity], PosVec[t], TanVec[t], 
            NorVec[t], BiNorVec[t]}, 
          DisplayFunction -> $DisplayFunction, 
          ViewPoint -> {2.607, Sin[((-t)/8)],  3.000 + Sin[(t/8)]}, 
          AspectRatio -> 1], Evaluate[tRange]];

The Kissing Circle

Table[
        Show[{KissingCircle[t], SpaceCurve[t]}, 
          DisplayFunction -> $DisplayFunction], 
        Evaluate[tRange]];