3D Projections

Definitions of Projections and Rotations

proj[w_, o_]  :=  {w . o[[2]], w . o[[3]]};

rot1[
          u_] := {{Cos[u], -Sin[u], 0}, {Sin[u], Cos[u], 0}, {0, 0, 
            1}};
rot2[
          u_] := {{Cos[u], 0, -Sin[u]}, {0, 1, 0}, {Sin[u], 0, 
            Cos[u]}};
rot3[
          u_] := {{1, 0, 0}, {0, Cos[u], -Sin[u]}, {0, Sin[u], 
            Cos[u]}};
rot[t_] :=  
      rot1[t[[1]]] . rot2[t[[2]]] . rot3[t[[3]]]

threerots  =  
        If[u < 1, {2  π u, 0, 0}, 
If[
            u < 2, {0, 2  π (u - 1), 0}, 
If[
              u <= 3, {0, 0, 2  π (u - 2)}]]];

Twisted Cubic

w  =  {x, (x^2), (x^3)};

c = 
      ParametricPlot3D[Evaluate[w], {x, -2, 2}, Axes -> False, 
        ViewPoint -> {5, 0, 1.5},  
        BoxRatios -> {1, 1, 1.5}]

<< Graphics`Animation`

SpinShow[c]

Twsited Cubic Projected into a Plane

Rotate simultaneously through three directions in (R^3).

o  =  rot[{u, u, u}];

Do[ParametricPlot[Evaluate[proj[w, o]], {x, -2, 2}, 
        Axes -> False, 
        PlotRange -> {{-4, 4}, {-4, 4}}], {u, 0, 
        2  π - π/16, π/16}]

Rotate separately through three directions in (R^3).

o  :=  rot[threerots];

Do[ParametricPlot[Evaluate[proj[w, o]], {x, -2, 2}, 
        Axes -> False, 
        PlotRange -> {{-4, 4}, {-4, 4}}], {u, 0, 3, 
        1/16}]

Helix

w  =  {Cos[t], Sin[t], t/π};

c = 
      ParametricPlot3D[Evaluate[w], {t, -2 π, 2  π}, 
        Axes -> False,  BoxRatios -> {1, 1, 1.5}]

SpinShow[c]

Helix Projected into a Plane

Rotate simultaneously through three directions in (R^3).

o  =  rot[{u, u, u}];

Do[ParametricPlot[Evaluate[proj[w, o]], {t, -2 π, 2  π}, 
        Axes -> False, 
        PlotRange -> {{-2, 2}, {-2, 2}}], {u, 0, 
        2  π - π/16, π/16}]

Rotate separately through three directions in (R^3).

o  :=  rot[threerots];

Do[ParametricPlot[Evaluate[proj[w, o]], {t, -2 π, 2  π}, 
        Axes -> False, 
        PlotRange -> {{-2, 2}, {-2, 2}}], {u, 0, 3, 
        1/16}]