# Math 2060 - Section 13.1

## 2131 days ago by jimlb

Section 13.1

Example:  The following is the curve ${\bf r}(t) = \langle t, \cos(t), \sin(t) \rangle$.

var('s,t')
 (s, t) (s, t)
P=parametric_plot3d([t,cos(t), sin(t)], (t,-2*pi,2*pi), thickness=5, axes=true);P.show()

Group work:  In groups the curve ${\bf r}(t) = \langle 2 \cos(t), 2 \sin(t), t \rangle$ was graphed. It is pictured here in blue.  The red curve is ${\bf r(t) = \langle 2 \cos(t), 2 \sin(t), t^2 \rangle$.

P=parametric_plot3d([2*cos(t), 2*sin(t), t], (t,0,4*pi), thickness=5);Q = parametric_plot3d([2*cos(t), 2*sin(t), t^2], (t, -4*pi, 4*pi), thickness=5, color='red');(P+Q).show()

Example:  The curve of intersection between the upper hemisphere of the sphere of radius 2 centered at the origin and the cylinder $x^2 + y^2 = 1$ is given below.  The sphere is in blue, the cylinder is in red, and the curve we are interested in is given in yellow.  This curve is given by ${\bf r}(t) = \langle \cos(t), \sin(t), \sqrt{3}\rangle$.

S=Spherical('radius', ['inclination', 'azimuth']) r,theta=var('r,theta') Q=plot3d(2, (r,0,pi/2), (theta, 0, 2*pi), transformation=S, aspect_ratio=true); P=parametric_plot3d([cos(s),sin(s), t], (s,0,2*pi), (t,0,2.25), color='red',aspect_ratio=true); R= parametric_plot3d([cos(s), sin(s), sqrt(3)], (s,0,2*pi), color='yellow', thickness=8);(P+Q+R).show();

Group work:  In groups  you were asked to give a vector-valued function giving the curve of intersection of $z = 4x^2 + y^2$ (pictured in blue) and $y = x^2$ (pictured in red).  The equation is ${\bf r}(t) = \langle t, t^2, 4t^2 + t^4 \rangle$, which is given in yellow.

P=parametric_plot3d([s,t, 4*s^2 + t^2], (s,-1,1), (t,-2,2)); Q=parametric_plot3d([s,s^2, t], (s,-2,2), (t,0,5), color='red');R = parametric_plot3d([t,t^2, 4*t^2+t^4], (t,-1,1),color='yellow', thickness=5);(R+Q+P).show()

Group work:  Consider a particle moving along the curve ${\bf r_1}(t) = \langle t^2, 8/3, 1 + t^2 \rangle$ and another particle moving along the curve ${\bf r_2}(s) = \langle 1+2s, s, -6 + 5s \rangle$.  The first question is if these particles cross paths.  We can see from a graph of their paths that they do cross paths.

P=parametric_plot3d([t^2, 8/3, 1+t^2], (t,-5,5), color='red',thickness=5);Q=parametric_plot3d([1+2*s, s, -6+5*s], (s,0,5),color='green',thickness=5);(P+Q).show()

The next question is whether they collide.  These can be seen by animating the paths traced out by the particles.  Unfortunately, it seems at this point SAGE does not support 3d animation. :-(   Instead, we look at this from a side view.  We observe as if we were looking down the $z$-axis, so projecting down into the $xy$-plane.

step=0.1 P=Graphics() Q=Graphics() v=[] for n in srange(15,32): P+=parametric_plot([t^2,8/3], (t,.25,n*step), color='red', thickness=1) P+=parametric_plot([1+2*t, t], (t,.25,n*step), color='green', thickness=1) P+=parametric_plot([0,t],(t,0,5), color='black', thickness=1) P+=parametric_plot([t,0],(t,0,10),color='black',thickness=1) v.append(P) a = animate(v,xmin=0,ymin=0) a.show()