The trefoil knot

x = (2 + Cos[1.5 t]) Cos[t];
y = (2 + Cos[1.5 t]) Sin[t];
z = Sin[1.5t];
ParametricPlot3D[{x,y,z},{t,0,10}]



Notice that we have specified parameter values 0 < t < 10. We can specify a longer interval of parameter t values:

ParametricPlot3D[ {x,y,z},{t,0,12.5}]




---

• An ellipse

x = Cos[t];
y = Sin[t];
z = 2-Sin[t];
ParametricPlot3D[ {x,y,z},{t,0,6}]




---

• The twisted cubic in red

x = t;
y = t^2;
z = t^3;
ParametricPlot3D[ {x,y,z,RGBColor[1,0,0]}, {t,-2,2} ]

• A toroidal spiral

x = (4+Sin[20t])Cos[t];
y = (4+Sin[20t])Sin[t];
z = Cos[20t];
ParametricPlot3D[ {x,y,z}, {t, 0, 2 Pi} ]



This picture is not good, notice the sharp corners which the true curve does not have. We need to increase the number of points used by default in the plot:

ParametricPlot3D[ {x,y,z},{t,0,2 Pi},PlotPoints->300]

You can use the program Geomview installed in the Silicon Graphics computers in Burton 302 for live 3-D manipulation of the curves:

Using Geomview