Math 2060 - Section 13.3

1824 days ago by jimlb

Section 13.3

Here we graph the tangent vector, normal vector, and binormal vectors to the curve given by ${\bf r}(t) = \cos(t) {\bf i} + \sin(t) {\bf j} + t {\bf k}$ at $t = \pi/2$.

var('s,t,y'); 
       
(s, t, y)
(s, t, y)

First we define the functions needed for ${\bf r}(t)$.

f(x) = cos(x); g(x) = sin(x); h(x) = x; 
       

Since we will also need ${\bf r}'(t)$ and ${\bf r}''(t)$, we take first and second derivatives of these functions.

Df = f.derivative(); Dg = g.derivative(); Dh= h.derivative(); D2f(x)=Df.derivative(); D2g=Dg.derivative();D2h=Dh.derivative(); 
       

We now define the relevant vectors: $T$ is the unit tangent vector at $t =\pi/2$, $N$ is the normal vector at $t =\pi/2$, and $B$ is the binormal vector at $t = \pi/2$.  Note that $T$ is given in green, $N$ in red, and $B$ in yellow.

v=vector([Df(pi/2), Dg(pi/2), Dh(pi/2)]);T = v/v.norm();Nv=v.norm(); w= vector([D2f(pi/2), D2g(pi/2), D2h(pi/2)]);N=w/w.norm();B=T.cross_product(N); 
       
T; N; B; 
       
(-1/2*sqrt(2), 0, 1/2*sqrt(2))
(0, -1, 0)
(1/2*sqrt(2), 0, 1/2*sqrt(2))
(-1/2*sqrt(2), 0, 1/2*sqrt(2))
(0, -1, 0)
(1/2*sqrt(2), 0, 1/2*sqrt(2))
P=parametric_plot3d([f(s),g(s),h(s)], (s,0,pi), color='blue',aspect_ratio=true, thickness=3); P+=arrow((f(pi/2),g(pi/2),h(pi/2)), (f(pi/2)+(Df(pi/2)/Nv), g(pi/2)+(Dg(pi/2)/Nv), h(pi/2)+(Dh(pi/2)/Nv)), color='green', width=2); P+=arrow((f(pi/2),g(pi/2),h(pi/2)), (f(pi/2), g(pi/2) - 1, h(pi/2)), color='red', width=2); P+=arrow((f(pi/2),g(pi/2),h(pi/2)), (f(pi/2)+1/sqrt(2), g(pi/2), h(pi/2)+1/sqrt(2)), color='yellow', width=2); P.show() 
       

We can also graph the osculating plane with the curve.  Note the plane is given by the semi-transparent blue plane.  It is best to make the plot interactive and rotate it around.

P+=plot3d(pi/2 - x, (x,-1,1), (y,0,2),color='blue', opacity=0.2);P.show()