Space Curve

451 days ago by ketchers

In this first example we see the plane of motion, the TNB frame, and the circle of curvature associated to the helix

\mathbf r(t)=\cos(t)\mathbf i+\sin(t)\mathbf j+t\mathbf k.

We have

\begin{align*}

\mathbf r'(t)&=-\sin(t)\mathbf i+\cos(t)\mathbf j+\mathbf k\\

\|\mathbf r'(t)\|&=\sqrt{\sin^2(t)+\cos^2(t)+1}=\sqrt{2}\\

\mathbf T(t)&=2^{-1/2}(-\sin(t),\cos(t),1)\\

\mathbf r''(t)&=-\cos(t)\mathbf i-\sin(t)\mathbf j\\

\mathbf r'(t)\times\mathbf r''(t) &= \det\left[\begin{matrix}\mathbf i&\mathbf j&\mathbf k\\-\sin(t)&\cos(t)&1\\-\cos(t)&-\sin(t)&0\end{matrix}\right]=(\sin(t),-\cos(t),1)\\

\|\mathbf r'(t)\times\mathbf r''(t)\|&=\sqrt{2}\\

\mathbf B(t)&=2^{-1/2}(\sin(t),-\cos(t),1)\\

\mathbf N(t) &= \mathbf B(t)\times\mathbf T(t)=\frac{1}{2}\det\left[\begin{matrix}\mathbf i&\mathbf j&\mathbf k\\\sin(t)&-\cos(t)&1\\-\sin(t)&-\cos(t)&1\end{matrix}\right]=(-\cos(t),-\sin(t),0)\\

\kappa(t)&=\frac{\|\mathbf r'(t)\times\mathbf r''(t)\|}{\|\mathbf r'(t)\|^3}=\frac{\sqrt{2}}{(\sqrt{2})^3}=\frac{1}{2}

\end{align*}

So the circle of curvature has radius \frac{1}{\kappa}=2 and is parametrized by

C(t,\theta)=(\mathbf r(t)+2\mathbf N(t))+2\cos(\theta)\mathbf T(t)+2\sin(\theta)\mathbf N(t)\quad\text{ for }0\le\theta\le 2\pi

 
reset() var('t,s,k') @interact def _(b=slider(-pi,pi,pi/12,0)): r=vector([cos(t),sin(t),t]) T=(2^(-1/2))*vector([-sin(t),cos(t),1]) N=vector([-cos(t),-sin(t),0]) B=(2^(-1/2))*vector([sin(t),-cos(t),1]) P=parametric_plot3d(r,(t,-pi,pi)) P+=arrow3d(r(t=b),r(t=b)+T(t=b),color='blue') P+=arrow3d(r(t=b),r(t=b)+N(t=b),color='green') P+=arrow3d(r(t=b),r(t=b)+B(t=b),color='red') P+=polygon3d([r(t=b)+3*(T(t=b)+2*N(t=b)),r(t=b)+3*(T(t=b)-.2*N(t=b)), r(t=b)+3*(-T(t=b)-.2*N(t=b)),r(t=b)+3*(-T(t=b)+2*N(t=b))],color=(.6,.6,.6),opacity=.4) P+=parametric_plot3d(r(t=b)+2*N(t=b)+2*cos(s)*T(t=b)*k+2*sin(s)*N(t=b)*k, (k,0,1),(s,-pi,pi),color=(.3,.3,.3),opacity=.5,fill=True) show(P,aspect_ratio=[1,1,1]) 
       
b 

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Now suppose you are moving around on the curve described above an observing an object moving along the curve (parametrized in terms of the same variable t) given by

\mathbf s(t)=2\cos(3t)\mathbf i+3\sin(2t)\mathbf j+cos(2t)\mathbf k

So from your perspective the object appears like \mathbf s(t)-\mathbf r(t)=(2\cos(3t)-\cos(t))\mathbf i+(3\sin(2t)-\sin(t))\mathbf j+(\cos(2t)-t)\mathbf k which written in terms of the TNB-frame is

l(t)=\bigl((\mathbf s(t)-\mathbf r(t))\bullet \mathbf T(t)\bigr) \mathbf T(t) +\bigl((\mathbf s(t)-\mathbf r(t))\bullet \mathbf N(t)\bigr) \mathbf N(t)+\bigl((\mathbf s(t)-\mathbf r(t))\bullet \mathbf B(t)\bigr) \mathbf B(t)

reset() var('t,s,k') @interact def _(b=slider(-pi,2*pi,pi/12,0)): r=vector([cos(t),sin(t),t]) w=vector([2*cos(3*t),3*sin(2*t),cos(2*t)]) l=w-r T=(2^(-1/2))*vector([-sin(t),cos(t),1]) N=vector([-cos(t),-sin(t),0]) B=(2^(-1/2))*vector([sin(t),-cos(t),1]) p=vector([l.dot_product(T),l.dot_product(N),l.dot_product(B)]) P=parametric_plot3d(r,(t,-pi,2*pi),color='blue',aspect_ration=1) P+=parametric_plot3d(w,(t,-pi,2*pi),color='green',aspect_ratio=1) #P+=parametric_plot3d(p,(t,-pi,b),color='red',aspect_ratio=1) P+=arrow(r(t=b),w(t=b),color='black',width=2) P+=arrow(r(t=b),r(t=b)+T(t=b),color=(.3,.3,1)) P+=arrow(r(t=b),r(t=b)+N(t=b),color=(.3,1,.3)) P+=arrow(r(t=b),r(t=b)+B(t=b),color=(1,.3,.3)) #P+=arrow((0,0,0),p(t=b),color='black',width=2) P+=circle(r(t=b),.05,thickness=3,color='blue') P+=circle(w(t=b),.05,thickness=3,color='green') #P+=circle(p(t=b),.05,thickness=3,color='red') P+=polygon3d([r(t=b)+3*(T(t=b)+N(t=b)),r(t=b)+3*(T(t=b)-N(t=b)), r(t=b)+3*(-T(t=b)-N(t=b)),r(t=b)+3*(-T(t=b)+N(t=b))],color=(.6,.6,.6),opacity=.2) P+=polygon3d([r(t=b)+3*(T(t=b)+B(t=b)),r(t=b)+3*(T(t=b)-B(t=b)), r(t=b)+3*(-T(t=b)-B(t=b)),r(t=b)+3*(-T(t=b)+B(t=b))],color=(.6,.6,.6),opacity=.2) P+=polygon3d([r(t=b)+3*(N(t=b)+B(t=b)),r(t=b)+3*(N(t=b)-B(t=b)), r(t=b)+3*(-N(t=b)-B(t=b)),r(t=b)+3*(-N(t=b)+B(t=b))],color=(.6,.6,.6),opacity=.2) show(P,apect_ratio=[1,1,1],axes_labels=['$x$','$y$','$z$']) 
       
b 

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reset() var('t,s,k') @interact def _(b=slider(-pi,2*pi,pi/12,0)): r=vector([cos(t),sin(t),t]) w=vector([2*cos(3*t),3*sin(2*t),cos(2*t)]) l=w-r T=(2^(-1/2))*vector([-sin(t),cos(t),1]) N=vector([-cos(t),-sin(t),0]) B=(2^(-1/2))*vector([sin(t),-cos(t),1]) p=vector([l.dot_product(T),l.dot_product(N),l.dot_product(B)]) #P=parametric_plot3d(r,(t,-pi,2*pi),color='blue',aspect_ration=1) #P+=parametric_plot3d(w,(t,-pi,2*pi),color='green',aspect_ratio=1) P=parametric_plot3d(p,(t,-pi,b),color='red',aspect_ratio=1) #P+=arrow(r(t=b),w(t=b),color='black',width=2) #P+=arrow(r(t=b),r(t=b)+T(t=b),color=(.3,.3,1)) #P+=arrow(r(t=b),r(t=b)+N(t=b),color=(.3,1,.3)) #P+=arrow(r(t=b),r(t=b)+B(t=b),color=(1,.3,.3)) P+=arrow((0,0,0),p(t=b),color='black',width=2) P+=circle(r(t=b),.05,thickness=3,color='blue') P+=circle(w(t=b),.05,thickness=3,color='green') P+=circle(p(t=b),.05,thickness=3,color='red') P+=polygon3d([(-4,-4,0),(-4,4,0),(4,4,0),(4,-4,0)],color='gray',opacity=.2) P+=polygon3d([(-4,0,-4),(-4,0,4),(4,0,4),(4,0,-4)],color='gray',opacity=.2) P+=polygon3d([(0,-4,-4),(0,-4,4),(0,4,4),(0,4,-4)],color='gray',opacity=.2) P+=arrow((0,0,0),(1,0,0),color='blue') P+=arrow((0,0,0),(0,1,0),color='green') P+=arrow((0,0,0),(0,0,1),color='red') show(P,apect_ratio=[1,1,1],axes_labels=['$x$','$y$','$z$']) 
       

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