Graphing Curves in Space in Mathematica
Math 211  Prof. Leise

Please hit shift-enter for each of the following plotting commands in order to see how they work.

Paths described by vector-valued functions (or parametric equations) can be graphed using the ParametricPlot and ParametricPlot3D commands.

ParametricPlot[{3t+1,t^2},{t,-2,2}]

ParametricPlot3D[{3t+1,t^2,Cos[5t]},{t,-2,2}]

As another example, here is the toroidal spiral shown in Section 13.1 of Stewart:

ParametricPlot3D[{(4+Sin[20 t])Cos[t],(4+Sin[20t])Sin[t],Cos[20t]},{t,0,2Pi}]

Example 1. Plot the curve described by r(t) = < sin(t), sin(2t), sin(3t) >.

Example 2. Find and plot parametric equations for the line through the point (1,-3) and parallel to the vector <2,5>.

Example 3. Find and plot parametric equations for the line through the point (1,-3,2) and perpendicular to the plane x-2y+2z=3.

Example 4. Find and plot the plane through the point (1,2,3) that contains the line x=3t, y=1+t, z=2-t.

Example 5. 
(a) Plot the curve described by r(t) = < (1+cos(16t))cos(t), (1+cos(16t))sin(t), 1+cos(16t) >. Be careful with the parentheses.

(b) Prove that this curve lies on the cone z=Sqrt[x^2+y^2].

Example 6. We can plot the circular cylinder x^2+y^2==4 and the parabolic cylinder z==x^2 as shown below.  ContourPlot3D can be used to plot surfaces described by an implicit equation, in contrast to Plot3D which plots a surface in explicit form z=f(x,y).

(a) Find parametric equations for the curve of intersection of these two surfaces.
(b) Plot the curve of intersection.

Show[ContourPlot3D[z==x^2,{x,-3,3},{y,-3,3},{z,-1,5}],ContourPlot3D[x^2+y^2==4,{x,-2,2},{y,-2,2},{z,-1,5}]] (* Use this to see the intersecting surfaces in Example 6. *)

Example 7. 
(a) Plot the parabolic cylinder y==x^2 and the top half of the ellipsoid x^2+4y^2+4z^2==16 (similar to the plotting command given above in Example 6).  Make sure that the curve of intersection is shown in your plot.
(b) Find parametric equations for the curve of intersection of these two surfaces.
(c) Plot the curve of intersection.