12.3 Curvature, torsion and the TNB frame

Transcription

1 1.3 Curvature, torsion and the TNB frame Acknowledgments: Material from a Georgia Tech worksheet by Jim Herod, School of Mathematics, herod@math.gatech.edu, is incorporated into the section on curvature, torsion and the TNB frame. Definition [Curvature] If T is the unit tangent vector of a smooth curve, the dt curvature function of the curve is κ = ds. Definition [Principal unit normal vector] At a point where normal vector for a curve in the plane is N= 1 dt κ ds. κ 0, the principal unit Definition [The binormal vector] The binormal vector of a smooth curve is B = T x N. Definition [Torsion] Let B = T x N. The torsion function of a smooth curve is τ= db. N. Torsion measures how the curve twists. ds Definition [The tangential and normal components of acceleration] a = a T T + a N N, where a T = d s dt = d dt v, and a N = κ ds = κ v are the dt tangential and normal scalar components of acceleration. a N = a a T. A vector formula for curvature vxa κ =. v 3 Formulas for curves in space

2 Unit tangent vector: T = v v. Principal unit normal vector: N = dt dt / dt dt Binormal vector: B = T x N dt Curvature: κ = ds = vxa. v 3 Torsion: τ= db. x' y' z' ds N = det x" y" z" x"' y"' z"' vxa ( ) Tangential and normal scalar components of acceleration: a = a T T + a N N a T = d dt v a N = κ v = a a T > The velocity vector and unit tangent Perhaps as much as any other place, the computations for the tangent and the normal to a curve are best accomplished with a computer. The calculus to obtain these geometric notions for even simple functions can be formidable. Because these ideas arise so prominently in the task of resolving both motion and the forces causing motion, it is well that the ideas be understood at the beginnings of a study of multidimensional calculus. We will give computational tools and illustrate the ideas with examples. Contrast the graphs of R 1 (t) = [ t, f( t )] and R (t) = [ t, f( t )]. The graphs have exactly the same appearance. A unit tangent for R 1 or R should be the same, even though speed along the two graphs, as parameterized, is different. While the tangent vectors for the two should have the same directions, you should expect the tangent vector for the graphs to have different lengths. Below, we draw two graphs for comparison with their tangent vectors. The point will be that while the two graphs have the same appearance, they sweep out the shapes at different rates and draw a particular section in different time spans. > plot({[t,sin(t),t=0..*pi],[t^,sin(t^),t=0..(*pi)^(1/)]}) ;

3 Look at the animation below to compare the speed of the two functions. > restart:with(plots): h:=(t,u)->piecewise(u<t,sin(u)); Cc:=animate({h(t^,u)},u=0..17*Pi,t=0..17*Pi,frames=100,color =green,numpoints=400): Dd:=animate({h(t,u)},u=0..17*Pi,t=0..17*Pi,frames=100,numpoin ts=400,thickness=3,color=red): display({cc,dd},axes=normal,view=[0..50,-1..1]); Warning, the name changecoords has been redefined h := ( t, u ) piecewise ( u < t, sin( u) )

4 Below, we compare the derivatives of R 1 and R. We expect these two to have the same direction, but different lengths. > restart:with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > x:=t->t; y:=t->sin(t); R1:=t->vector([x(t),y(t)]); x := t t y := sin R1 := t [ x( t ), y( t) ] We compute R 1 prime -- the tangent to the graph of R 1. > assume(t,real); > R1p:=t->map(diff,R1(t),t); R1p(t); sqrt(dotprod(r1p(t),r1p(t))); R1p := t map ( diff, R1( t) [ 1, cos( t~ )] 1 + cos( t~ ) Here is the computation for R. > R:=t->vector([x(t^),y(t^)]); R := t [ x( t ), y( t )]

5 > Rp:=t->map(diff,R(t),t); Rp(t); sqrt(dotprod(rp(t),rp(t))); Rp := t map ( diff, R( t) [ t~, cos( t~ ) t~ ] t~ + cos( t~ ) t~ Note that the derivatives -- the tangent vectors -- for R 1 and for R are different. they have the same directions, but different lengths. > simplify(%); 1 + cos( t~ ) t~ The graph below shows the curves and the tangent vectors (velocity vectors), R 1 '(t) and R '(t). The longer tangent vector is the tangent for R. > with(plots):with(plottools): Warning, the name changecoords has been redefined Warning, the name arrow has been redefined > s1 := plot({[t,sin(t),t=0..16*pi],[t^,sin(t^),t=0..(16*pi)^(1/)] }): A:=plots[display](seq(PLOT(arrow(R(s/0),subs(t=s/0,evalm(R p(t))),.1,.3,.1, color=green)), s=0..00), insequence=true): > B:=plots[display](seq(PLOT(arrow(R1(s/0),subs(t=s/0,evalm(R 1p(t))),.1,.3,.1, color=pink)), s=0..00), insequence=true): display({a,b,s1},axes=normal,view=[0..50, ]);

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7 The Unit Tangent Vector T We identity R '(t) as speed and T(t) = R'(t)/ R '(t) as the unit tangent vector. Definition: Speed(t) = R( t ) and T(t) = t t R( t ) speed( t ). For a three dimensional curve, R(t) = [x(t), y(t), z(t)], speed(t) = x( t ) + y( t) + t t t z( t). and T(t) = [x '(t), y '(t), z '(t)] / speed(t).

8 Example: > x:=t->cos(t); y:=t->sin(t); z:=t->t; R3:=t->vector([x(t),y(t),z(t)]); assume(t,real); x := cos y := sin z := t t R3 := t [ x( t ), y( t ), z( t) ] > R3p:=t->map(diff,R3(t),t); R3p(t); speed:=sqrt(dotprod(r3p(t),r3p(t))); T:=t->R3p(t)/speed; simplify(evalm(t(t))); R3p := t map ( diff, R3( t) [ sin( t~ ), cos( t~ ), 1] speed := 1 + sin( t~ ) + cos( t~ ) T := t R3p( t ) speed 1 sin( t~ ), 1 cos( t~ ), 1 > evalm(%); 1 sin( t~ ), 1 cos( t~ ), 1 Of course, this velocity vector R '(t) points in the direction of the tangent vector, T(t). R ' is a multiple of T: (*) R ' (t) = speed(t) * T(t). THE NORMAL VECTOR In thinking about how to define the unit normal vector, we ask what properties it should have? Here are three: (1) The normal vector should be perpendicular to the tangent vector. () The unit normal vector should have length one. (3) The normal vector should lie in the plane of R ' and R ' '. Toward getting such a vector, consider T '. This vector is perpendicular to the unit tangent for < T, T > = 1,

9 and < T ', T > + < T, T ' > = 0. (Prove this.) Thus, T ' meets Condition 1. However T ' does not meet condition (). Thus, we define the unit normal (**) N(t) = T '(t) / T '(t). This N meets Conditions (1) and (). What about Condition (3)? Recall (*). Take the derivative of both sides: (***) R ' '(t) = speed '(t) T(t) + speed(t) T '(t). Define curvature, κ to be the number valued function κ = T '(t) / speed(t). From (**), T '(t) = T '(t) N = κ speed(t) N(t). Substitute this into (***) and we have R ' '(t) = speed '(t) T(t) + speed κ N. Thus, the unit normal as defined above meets the Conditions (1), (), and (3). Computational summary: R(t) is the curve, speed = R '(t). Unit Tangent = T(t) = R( t ) t speed( t).

10 Unit Normal = T( t) t T( t) t. curvature = κ = T( t) t speed( t) = R ' x R ' ' / speed 3. It seems appropriate to work some problems. Illustrations Find the unit tangent, the unit normal, and curvature for each of the following. Example 1. R(t) = [ 3 cos(t), 3 sin(t), 0] > restart: > with(linalg):assume(t,real); Warning, the protected names norm and trace have been redefined and unprotected Define > x:=t->3*cos(t); y:=t->3*sin(t); z:=t->0; R:=t->vector([x(t),y(t),z(t)]); x := t 3 cos( t) y := t 3 sin( t) z := 0 R := t [ x( t ), y( t ), z( t) ] > Rp:=t->map(diff,R(t),t); Rp(t); Rp := t map ( diff, R( t) [ 3 sin( t~ ), 3 cos( t~ ), 0] > sqrt(dotprod(rp(t),rp(t))); simplify(%); > > speed:=unapply(%,t); 3 sin( t~ ) + cos( t~ ) 3

11 speed := 3 The Unit Tangent > unitt:=t->evalm(rp(t)/speed(t)); unitt(t); Rp( t) evalm speed( t) [ sin( t~ ), cos( t~ ), 0] unitt := t > Tp:=t->map(diff,unitT(t),t); map(simplify,tp(t)); Tp := t map ( diff, unitt( t) [ cos( t~ ), sin( t~ ), 0 ] The Unit Normal > N:=t->evalm(Tp(t)/sqrt(dotprod(Tp(t),Tp(t)))); map(simplify,n(t)); Tp( t) N := t evalm dotprod ( Tp( t ), Tp( t) ) [ cos( t~ ), sin( t~ ), 0 ] > Rpp:=t->map(diff,Rp(t),t); Rpp(t); Rpp := t map ( diff, Rp( t) [ 3 cos( t~ ), 3 sin( t~ ), 0 ] Curvature > topk:=crossprod(rp(t),rpp(t)); kappa:=simplify(sqrt(dotprod(topk,topk))/speed(t)^3); topk := [ 00,, 9 sin( t~ ) + 9 cos( t~ ) ] 1 κ := 3 > Example. R(t) = t t,, 0. > restart: > with(linalg):assume(t,real); Warning, the protected names norm and trace have been redefined and unprotected > x:=t->t; y:=t->t^/; z:=t->0; R:=t->vector([x(t),y(t),z(t)]); x := t t

12 1 y := t t z := 0 R := t [ x( t ), y( t ), z( t) ] > Rp:=t->map(diff,R(t),t); Rp(t); Rp := t map ( diff, R( t) [ 1, t~, 0 ] > sqrt(dotprod(rp(t),rp(t))); simplify(%); > > speed:=unapply(%,t); > 1 + t~ 1 + t~ speed := t~ 1 + t~ The Unit Tangent > unitt:=t->evalm(rp(t)/speed(t)); unitt(t); Rp( t) unitt := t evalm speed( t) 1 t~,, t~ 1 + t~ > Tp:=t->map(diff,unitT(t),t); map(simplify,tp(t)); Tp := t map ( diff, unitt( t) t~ 1,, 0 ( 1 + t~ ) ( 3 / ) ( 1 + t~ ) ( 3 / ) The Unit Normal > N:=t->evalm(Tp(t)/sqrt(dotprod(Tp(t),Tp(t)))); map(simplify,n(t)); Tp( t) N := t evalm dotprod ( Tp( t ), Tp( t) ) t~ 1,, t~ 1 + t~ > Rpp:=t->map(diff,Rp(t),t);

13 Rpp(t); Rpp := t map ( diff, Rp( t) [ 010,, ] Curvature > topk:=crossprod(rp(t),rpp(t)); kappa:=simplify(dotprod(topk,topk)/speed(t)^3); topk := [ 001,, ] 1 κ := ( 1 + t~ ) ( 3 / ) > Example 3:R(t) = [ exp(t) cos(t), exp(t) sin(t), exp(t) ] We also draw the unit tangent and the unit normal for this last graph. > restart: > with(linalg): with(plots): with(plottools):assume(t,real); Warning, the protected names norm and trace have been redefined and unprotected Warning, the name changecoords has been redefined Warning, the name arrow has been redefined > x:=t->exp(t)*cos(t); y:=t->exp(t)*sin(t); z:=t->exp(t); x := t e t cos( t) y := t e t sin( t ) z := exp > x:=t->cos(t); y:=t->sin(t); z:=t->1/*t; x := cos y := sin 1 z := t t > R:=t->vector([x(t),y(t),z(t)]); R := t [ x( t ), y( t ), z( t) ] > Rp:=t->map(diff,R(t),t); map(simplify,rp(t)); Rp := t map ( diff, R( t) [ e t~ ( cos( t~ ) sin( t~ )), e t~ ( sin( t~ ) + cos( t~ )), e t~ ] > sqrt(dotprod(rp(t),rp(t))); simplify(%);

14 ( e t~ cos( t~ ) e t~ sin( t~ )) + ( e t~ sin( t~ ) + e t~ cos( t~ )) + ( ) 3 e t~ > speed:=unapply(%,t); speed := t~ 3 e t~ The Unit Tangent > unitt:=t->evalm(rp(t)/speed(t)); unitt(t); Rp( t) unitt := t evalm speed( t) 1 3( e t~ cos( t~ ) e t~ sin( t~ )) 1 3( e t~ sin( t~ ) + e t~ cos( t~ )) 1,, 3 e t~ 3 e t~ 3 > Tp:=t->map(diff,unitT(t),t); map(simplify,tp(t)); Tp := t map ( diff, unitt( t) 1 3( sin( t~ ) + cos( t~ )), 1 3( cos( t~ ) + sin( t~ )), The Unit Normal > N:=t->evalm(Tp(t)/sqrt(dotprod(Tp(t),Tp(t)))); map(simplify,n(t)); Tp( t) N := t evalm dotprod ( Tp( t ), Tp( t) ) 1 ( sin( t~ ) + cos( t~ )), 1 ( cos( t~ ) + sin( t~ )), 0 > Rpp:=t->map(diff,Rp(t),t); Rpp(t); Rpp := t map ( diff, Rp( t) [ e t~ sin( t~ ), e t~ cos( t~ ), e t~ ] The Binormal vector > B:=t->evalm(crossprod(unitT(t),N(t))); map(simplify,b(t)); B := t evalm ( crossprod ( unitt( t ), N( t) ) ) 1 6( cos( t~ ) + sin( t~ )), 1 1 6( sin( t~ ) + cos( t~ )), Curvature > topk:=crossprod(rp(t),rpp(t)); kappa:=simplify(dotprod(topk,topk)/speed(t)^3); e t~ 6 3

15 topk := [ ( e t~ sin( t~ ) + e t~ cos( t~ )) e t~ ( e t~ ) cos( t~ ), > ( e t~ ) sin( t~ ) ( e t~ cos( t~ ) e t~ sin( t~ )) e t~, ( e t~ cos( t~ ) e t~ sin( t~ )) e t~ cos( t~ ) + ( e t~ sin( t~ ) + e t~ cos( t~ )) e t~ sin( t~ )] κ := 3 e t~ 3 Here, we draw the curve, the unit tangent, and the unit normal > base:=convert(map(evalf,subs(t=pi/4,r(t))),list); directiont:=convert(map(evalf,subs(t=pi/4,unitt(t))),list); directionn:=convert(map(evalf,subs(t=pi/4,n(t))),list); directionb:=convert(map(evalf,subs(t=pi/4,b(t))),list); base := [ , , ] directiont := [ 0., , ] directionn := [ , , 0. ] directionb := [ , , ] > sqrt(dotprod(directiont,directionb));digits:=0; 0. Digits := 0 > K:=spacecurve([x(t),y(t),z(t)],t=0..Pi/,axes=NORMAL,color=GR EEN): J:=arrow(base, directiont, [1/4, 1/4, 1/4],.,.4,1/10,axes=NORMAL,color=red): L:=arrow(base, directionn, [1/4, 1/4, 1/4],.,.4,1/10,axes=NORMAL,color=BLACK): M:=arrow(base, directionb, [1/4, 1/4, 1/4],.,.4,1/10,axes=NORMAL,color=blue): > display({j,k,l,m},axes=normal,orientation=[-10,80],scaling=co nstrained,color=black);

16 > n:=0:lastpt:=pi/:jj:=array(1..n):ll:=array(1..n):mm:=array( 1..n):base:=array(1..n):pic:=array(1..n): K:=spacecurve([x(t),y(t),z(t)],t=0..lastpt,scaling=constraine d,axes=normal,color=green): for i from 1 to n do base[i]:=convert(map(evalf,subs(t=lastpt*i/n,r(t))),list): Jj[i]:=arrow(base[i], convert(map(evalf,subs(t=lastpt*i/n,unitt(t))),list), [1/4, 1/4, 1/4],.,.4,1/10,axes=NORMAL,color=red): Ll[i]:=arrow(base[i], convert(map(evalf,subs(t=lastpt*i/n,n(t))),list), [1/4, 1/4, 1/4],.,.4,1/10,axes=NORMAL,color=BLACK): Mm[i]:=arrow(base[i], convert(map(evalf,subs(t=lastpt*i/n,b(t))),list), [1/4, 1/4, 1/4],.,.4,1/10,axes=NORMAL,color=blue): pic[i]:=display({k,jj[i],ll[i],mm[i]},axes=normal,orientation =[-10,80],scaling=constrained,color=BLACK): od: > display(seq(pic[i], i=1..n), scaling=constrained,insequence=true);

17 > > Exercise for the student. Find the unit tangent, the unit normal, and curvature for the ellipse x y + = 1, a > b. a b Determine where the curvature is maximum and where it is minimum. Note that a parametric representation for this ellipse is x(t)=a cos(t), y(t)=b sin(t), z(t)=0.

18 > >