14.1 Vector Functions and Space Curves

Example

Plot the curve r(t) = ti + (t^2)j + 2tk.

ParametricPlot3D[{t, t^2, 2  t}, {t, -2, 2},  
        ViewPoint -> {-1,  3,  2}];

The curve is the interesection of the parabolic sheet y  =  (x^2) and the plane z  =  2  x.

s1  =  
        ParametricPlot3D[{u, u^2, v}, {u, -2, 2}, {v, -4, 
            4}];

s2  =  
        ParametricPlot3D[{u, v, 2  u}, {u, -2, 2}, {v, -4, 
            4}];

Show[s1, s2,  ViewPoint -> {-1,  3,  2}];

Example

Plot the curve r(t) = sin(t)i + sin(t)j +(Sqrt[2])cos(t)k.

ParametricPlot3D[{Sin[t], Sin[t], (Sqrt[2]) Cos[t]}, {t, 0, 
          2  π}];

The curve is the intersection of the sphere (x^2) +  (y^2) +  (z^2) =  2 and the vertical plane y  =  x.

s1 = 
        ParametricPlot3D[{2 Sin[p] Sin[t], 2 Sin[p] Cos[t], 
            2 Cos[p]}, {t, 0, 2  π},  {p, 0, π}];

s2  =  
        ParametricPlot3D[{u, u, v},  {u, -2, 2}, {v, -2, 
            2}];

Show[s1, s2];

Example

Sketch the intersection of the cylinder (x^2)  +  (y^2)  =  4 and the parabolic sheet z  =  (x^2). Find parametric equations for this curve.

s1  =  
        ParametricPlot3D[{2  Cos[t],  2  Sin[t], v},  {t, 0, 
            2  π},  {v, -2, 4.5}];

s2  =  
        ParametricPlot3D[{u,  v, (u^2)},  {u, -2.5, 
            2.5},  {v, -2.5, 2.5}];

Show[s1, s2];

Parametric equations for the curve of intersection are given by x  =  2  cos(t), y  =  2  sin(t), and z  =  4  (cos^2)(t).

ParametricPlot3D[{2  Cos[t], 2 Sin[t], 4 Cos[t]^2}, {t, 0, 
        2  π}]