Arc Length, Curvature and The TNB Frame

Arc Length, Curvature and The TNB Frame Introduction and Goals: The main goal of this lab will help you visualize the tools we use to describe the geometry of vector-valued functions. Moreover, we want you to begin to view the tangent, normal and binormal vectors of a curve and their relationship to the movement of the curve. We will also investigate arc length, curvature, normal planes, osculating planes and osculating circles. Before You Start: Make sure that you read and understand the mathematics from the corresponding sections in your textbook. It would also help if you looked over the lab on Lines and Planes. Textbook Correspondence: Stewart 5 th Edition: 4.3. Stewart 5 th Edition Early Transcendentals: 3.3. Thomas Calculus 0 th Edition Early Transcendentals: 0.7. Maple Commands and Packages Used: Packages: plots, linalg, VectorCalculus. Commands: ArcLength, Curvature, simplify, TangentVector, evalf, plot, PrincipalNormal, Binormal, subs, norm,., &x, read. User Defined Commands: plotvvf, plotvvftn, plotvvft, plotvvftseq, plotvvfnn, plotvvfn, plotvvfnseq, plotvvfbn, plotvvfb, plotvvfbseq, plotvvftnbn, plotvvftnb, plotvvftnbseq, plotvvftnbn, plotvvftnb, plotvvftnbseq, plotvvftn, plotvvft, plotvvftseq, plotvvfnn, plotvvfn, plotvvfnseq, plotvvfbn, plotvvfb, plotvvfbseq, normplot, plotvvfocn, plotvvfoc, plotvvfocseq, plotvvfnpn, plotvvfnp, plotvvfnpseq, plotvvfopn, plotvvfop, plotvvfopseq. History & Biographies: Maple Commands: The new Maple commands are a combination between special commands in the Vector Calculus package and some user-defined commands we have built for the more complex graphs. Most of the commands in this lab are in or need the three packages plots, linalg and VectorCalculus, load these into your worksheet..

> with(plots): Warning, the name changecoords has been redefined > with(linalg): Warning, the protected names norm and trace have been redefined and unprotected > with(vectorcalculus): Warning, the assigned names <,>, < > and Wronskian now have a global binding Warning, these protected names have been redefined and unprotected: *, +,., Vector, diff, int, limit, series We will start out with some of the commands in the Vector Calculus package that calculate some of the standard quantities of interest. Let s begin with defining a couple vector valued functions. > w:=t-><t,t^,t^3>; w := t VectorCalculus:-`<,>` ( t, t, t 3 ) > v:=t-><sin(t),cos(t),t>; v := t VectorCalculus:-`<,>` ( sin( t ), cos( t), t ) > r:=t-><t*sin(t),t*cos(t),t/0>; r := t VectorCalculus:-`<,>` VectorCalculus:-`*` ( t, sin( t )), VectorCalculus:-`*` ( t, cos( t) ), VectorCalculus:-`*` t, 0 To find the arc length we use the ArcLength command which takes the curve and parameter range as inputs. For example, > ArcLength(r(t),t=-5*Pi..5*Pi); π 0 + 500 π 0 + ln ( 50 0 π + 5500 π + 00 ) 00 0 ln( 50 0 π + 5500 π + 00 ) 00 > evalf(arclength(r(t),t=-5*pi..5*pi)); 50.73896 The curvature is calculated simply using the Curvature command, for example, > Curvature(v(t));

4 sin( t) + cos( t ) > simplify(%,assume=real); > Curvature(r(t)); 0 000 ( sin( t) + t cos( t )) t 0 ( cos( t ) t sin( t) ) + ( 0 + 00 t ) ) 0 + 00 t 000 ( cos( t) t sin( t )) t 0 ( sin( t ) t cos( t) ) + + + ( 0 + 00 t ) ) 0 + 00 t ( / ) 0 + 00 t 0000 t ( 0 + 00 t ) 3 ^ > cr:=simplify(%,assume=real); 00 00 t 4 + 40 t + 404 cr := ( 0 + 00 t ) ) > plot(cr,t=-0..0); To find the general tangent, normal and binormal vector equations we use the TangentVector, PrincipalNormal and Binormal commands. For example, > TangentVector(w(t)); e + t e + 3 t e x y z > PrincipalNormal(w(t));

8 t + 36 t 3 t ( 8 t + 36 t e 3 ) + ( + 4 t + 9 t 4 ) ) x + e + ( + 4 t + 9 t 4 ) ) + 4 t + 9 t 4 y 3 t ( 8 t + 36 t 3 ) 6 t + e ( + 4 t + 9 t 4 ) ) + 4 t + 9 t 4 z > Binormal(w(t)); t 3 t ( 8 t + 36 t 3 ) 6 t + ( + 4 t + 9 t 4 ) ) + 4 t + 9 t 4 + 4 t + 9 t 4 3 t t ( 8 t + 36 t 3 ) + ( + 4 t + 9 t 4 ) ) + 4 t + 9 t 4 + 4 t + 9 t 4 e x + 3 t ( 8 t + 36 t 3 ) 6 t + 3 t ( 8 t + 36 t 3 ) ( + 4 t + 9 t 4 ) ) + 4 t + 9 t 4 ( + 4 t + 9 t 4 ) e + 4 t + 9 t 4 + y t ( 8 t + 36 t 3 ) + ( + 4 t + 9 t 4 ) ) + 4 t + 9 t 4 t ( 8 t + 36 t 3 ) + e + 4 t + 9 t 4 ( + 4 t + 9 t 4 ) z One thing to be careful with is that these are not the unit tangent, unit normal or unit binormal vectors, as can be seen by the following calculation which finds the length of the tangent vector to w(t) at t=5, > evalf(subs(t=5,norm(tangentvector(w(t)),))); 75.67033765 Had this been the unit tangent the calculation would have resulted in a. The unit tangent, normal and binormal vectors can be calculated with the above commands if we further divide by the norm of the vector but there is another way to do the same calculation in Maple. We can find all three at once using the command TNBFrame. For example, > TNBFrame(v(t)); + cos( t ) e x sin( t) e y e, sin( t) e cos( t) e, z x y + cos( t) e x sin( t ) e y cos( t ) sin( t ) e z

Note that the output of this command is a list of three vectors. The first is the unit tangent, the second is the unit normal and the third is the unit binormal. To extract the desired vector we simply use the index of the vector. For example, to extract the unit normal vector we execute the command, > TNBFrame(v(t))[]; sin( t ) e x cos( t) e y we can also convert it to a vector valued function using the unapply command. > nv:=unapply(tnbframe(v(t))[],t); nv := t rtable (.. 3, { ( ) = sin( t ), ( ) = cos( t ) }, datatype = anything, subtype = Vector column, storage = rectangular, order = Fortran_order, attributes = [ coords = cartesian ] ) > nv(pi); e y We can easily check that these are in fact unit vectors by taking the norms of each of them. Note that they all simplify to length. > norm(tnbframe(v(t))[],); + cos( t) + sin( t) > simplify(%,assume=real); > norm(tnbframe(v(t))[],); sin( t ) + cos( t) > simplify(%,assume=real); > norm(tnbframe(v(t))[3],); cos( t) + sin( t ) + 4 + cos( t ) sin( t ) > simplify(%,assume=real); There are also a few quick shortcut commands that are loaded with the VectorCalculus package. The dot product can be done with a period, for example,

> a:=<,,3>; > b:=<5,9,6>; > a.b; a := e + e + 3 e x y z b := 5 e + 9 e + 6 e x y z 4 and the cross product can be done with the &x operator, like the following, > a &x b; (-5) e + 9 e e x y z There are several other commands in the VectorCalculus package that will be of interest to us later but we have enough for this lab. The remainder of the commands used for this lab are contained in one of two files TNB.mpl or CalcIIIProcs.mpl. The TNB.mpl file contains all of the commands used in this lab and the CalcIIIProcs.mpl file contains all of the commands we have created for multivariable Calculus. To load these commands we use the Maple read command. The read command simply requires a string that has the path and filename of the file to be loaded. In the path a backslash is created with two backslashes and a forward slash is created with a single forward slash. So if the TNB.mpl file is in the root directory of a PC (c:\) the command would be > read("c:\\tnb.mpl"); and if it were in the directory C:\My Documents\Word\MapleLabs the command would be > read("c:\\my Documents\\Word\\MapleLabs\\TNB.mpl"); We would suggest that you store your maple files on your personal network drive and read them in from there. When you read this file in you will not get any output from Maple but the commands will have been loaded. The following is the list of commands that are in the TNB.mpl file. plotvvf, plotvvftn, plotvvft, plotvvftseq, plotvvfnn, plotvvfn, plotvvfnseq, plotvvfbn, plotvvfb, plotvvfbseq, plotvvftnbn, plotvvftnb, plotvvftnbseq, plotvvftnbn, plotvvftnb, plotvvftnbseq, plotvvftn, plotvvft, plotvvftseq, plotvvfnn, plotvvfn, plotvvfnseq, plotvvfbn, plotvvfb, plotvvfbseq, normplot, plotvvfocn, plotvvfoc, plotvvfocseq, plotvvfnpn, plotvvfnp, plotvvfnpseq, plotvvfopn, plotvvfop, plotvvfopseq

You will notice that all of these commands are for graphing and that the names of most of these commands come in triplets, *, *n and *seq. The * commands graph one copy of the item of interest, the *n commands graph n copies of the item of interest evenly spaced along the curve. Finally the *seq commands graph the item of interest at each point in a user-defined list. We will discuss each of the commands and when we encounter the triplet functions we will discuss the three of them together. The plotvvf command is similar to the one created in a previous lab but with an added feature of taking any list of options. The plotvvf command will plot a vector valued function, it requires the input of a vector valued function and parameter range. Further options may be input as well, as long as they are options that apply to spacecurves. > plotvvf(r(t),t=-5*pi..5*pi); > plotvvf(r(t),t=- 5*Pi..5*Pi,color=black,numpoints=00,axes=boxed,labels=[x,y,z]);

The nornplot command plots the norm of the vector valued function over the input range of the parameter, it requires the input of a vector valued function and parameter range. Further options may be input as well, as long as they are options that apply to the plot command. > normplot(w(t),t=-..); > normplot(r(s),s=-..,y=0..0.5,color=black,thickness=3); The plotvvftn, plotvvft and plotvvftseq commands plot a vector valued function along with its tangent vectors. The first, second and fourth parameters of each of these commands are the same. The first parameter is the vector valued function, the second parameter is the range in which to ploy the curve and the fourth is the color of the tangent vectors. With the plotvvft command the third parameter is the position, in terms of the curve s parameter, of the tangent vector. For example, the following command plots the curve from 5 to 5 and a tangent vector at 3. Further options may be input as well, as long as they are options that apply to the spacecurve command and will only affect the vector valued function. > plotvvft(r(t),t=-5..5,3,red);

> plotvvft(r(t),t=-5..5,3,red,axes=boxed,color=black); With the plotvvftn command the third parameter is the number of tangent vectors that will be graphed. For example, the following command plots the curve from 5 to 5 and 0 tangent vectors evenly spaced from 5 to 5. Further options may be input as well, as long as they are options that apply to the spacecurve command and will only affect the vector valued function. > plotvvftn(r(t),t=-5..5,0,red);

> plotvvftn(r(t),t=-5..5,0,red,axes=boxed,color=black); With the plotvvftseq command the third parameter is a list of all of the positions where the tangent vectors that will be graphed. For example, the following command plots the curve from 5 to 5 and tangent vectors at, 0,.5,.9, 3 and 5. Further options may be input as well, as long as they are options that apply to the spacecurve command and will only affect the vector valued function. > plotvvftseq(r(t),t=-5..5,[-,0,.5,.9,3,5],red);

> plotvvftseq(r(t),t=-5..5,[-,0,.5,.9,3,5],red,axes=boxed,color=black); Note that these tangent vectors are not the unit tangent vectors. To graph the unit tangents, use the commands plotvvftn, plotvvft and plotvvftseq using the same syntax as above. For example, > plotvvftn(r(s),s=-5..5,50,red,axes=boxed,color=black);

To graph the normal vectors use the commands plotvvfnn, plotvvfn and plotvvfnseq with the same syntax. To graph the unit normal vectors use the commands plotvvfn, plotvvfnseq, plotvvfbn with the same syntax. To graph the binormal vectors use the commands plotvvfbn, plotvvfb and plotvvfbseq with the same syntax. Finally, to graph the unit binormal vectors use the commands plotvvfbn, plotvvfb and plotvvfbseq again with the same syntax. You can create graphs of the tangent, normal and binormal vectors together by using the display command with the commands we have discussed above but we have created six functions to do that automatically. To graph the tangent, normal and binormal together use the commands plotvvftnbn, plotvvftnb and plotvvftnbseq. To graph the unit tangent, normal and binormal together use the commands plotvvftnbn, plotvvftnb and plotvvftnbseq. The only difference between these and the above commands is that instead of requiring one color for the color of the vector they require three colors in the order of tangent, normal and binormal. For example, > plotvvftnb(r(t),t=- 5..5,3,red,green,blue,axes=boxed,color=black);

> plotvvftnb(r(t),t=- 5..5,3,red,green,blue,axes=boxed,color=black); > plotvvftnbn(r(t),t=- 5..5,50,red,green,blue,axes=boxed,color=black);

The remainder of the supplementary commands deal with osculating circles, osculating planes and normal planes. As before these commands come in triples. For osculating circles we use the commands plotvvfocn, plotvvfoc and plotvvfocseq. Each of these commands take four parameters, the first is the curve, the second is the range, the third is a position, number of vectors or list depending on which command is being used and the fourth parameter is the color of the circles. Additional options can be used for the curve, as in the previous commands. For example, > plotvvfocn(v(t),t=- 5..5,5,red,axes=boxed,color=black,thickness=3); The normal planes are produced from the commands plotvvfnpn, plotvvfnp and plotvvfnpseq. These commands each take three parameters, the curve, the range and either a position, number of vectors or list depending on which command is being used. As with the other commands, further curve options may be used. For example,

> plotvvfnpn(w(t),t=0..,4,axes=boxed,color=black,thickness=3 ); The osculating plane commands plotvvfopn, plotvvfop and plotvvfopseq are similar. > plotvvfopn(w(t),t=0..,3,axes=boxed,color=black,thickness=3 ); Exercises:. The following exercises concern the vector-valued function b ( t) = sin t,cos t,sin t. ( ) ( ) ( ) t b =. a. Define the vector-valued function ( t) sin( t),cos( t),sin ( t) t b. Plot the curve on the interval [0,3].

c. Describe the path made by the curve. Where are the largest changes in direction? How many of those direction changes are there? d. Graph the curve along with 0 evenly spaced normal vectors. e. Where are these vectors the largest? Where are these vectors the smallest? What is the relationship between the size of the normal vector and the geometry of the curve? f. Use the normplot command to plot the PrincipalNormal vector over the interval [0,3]. Where is this function a maximum? Where is this function a minimum? How do the positions of the maximum and minimum relate to your observations above? g. Plot the curvature function of the curve on [0,3]. Is there a relationship between the curvature and the normal vector graphs for this curve? If so, what? Be specific. h. Plot the curve on the interval [0,3] along with 5 osculating circles. Where are the circles large and where are they small? What is the curve doing where the circles are large and what is the curve doing where the circles are small? Be specific. i. Where do you expect the circles to be the largest? Where do you expect the circles to be the smallest? Why?. The following exercises deal with parameterizations with respect to arc length. r ( t) = t,sinh t, cosh t. a. Define the vector valued function ( ) ( ) b. Graph the vector valued function ( t) = t,sinh( t), cosh( t) r on the interval [0,5]. c. Use the command ArcLength(r(t),t=0..s); to get a function for the arc length of s. You will want to simplify the result with the command simplify(%,assume=real);. d. Set the variable c equal to the result of the last exercise and solve the equation for s. e. Define a vector valued function that takes c to the expression of r(t) with the result of the above exercise substituted for t. f. Evaluate the expression evalf(arclength(d(t),t=0..)); what is the result? Do the same with the replaced by, 3, 5, 0 and 5. What are the results? What does this say about the vector valued function d(t)? g. Using the display command plot the two curves r and d together on the same axes, graph r on [0,5] and d on [0,00]. You may wish to make the color of d red and its thickness 3. What do you observe about the two curves? t t r ( t ) = sin e,cos e, e. h. Do a similar analysis (i.e. repeat a-f) with the curve ( ) ( ) t 3. The following exercises deal with parameterizations of curves. More specifically, the motion of a child having fun, and then possibly getting sick. a. We will think of the parameter t as time. Say that a child is sitting on the edge of a merry-go-round in say a city park. The merry-go-round is 5 feet in diameter and is revolving at a rate of one revolution every two seconds. Find a threedimensional vector valued function that describes the position of the child at any time t. Graph the function with its tangent, normal and binormal vectors. Also use the normplot command to graph the norms of the tangent and normal vectors through one revolution of the merry-go-round. Use all of this information to

describe in detail the path and forces on the child. Find minimum and maximum velocities of the child as well as where they occur. b. As in the above exercise, a child is sitting on the edge of a 5 foot diameter merry-go-round revolving at a rate of one revolution every two seconds. This time there is a hobbyhorse bolted to the edge of the merry-go-round. The hobbyhorse is one of those that sits on a single large spring and can sway back and forth in a circular arc. The hobbyhorse is positioned so that it will sway toward and away from the center of the merry-go-round. The horse is three feet tall, rocks a total arc length of 45 degrees and has a period of ½ second. Find a three-dimensional vector valued function that describes the position of the child at any time t. Graph the function with its tangent, normal and binormal vectors. Also use the normplot command to graph the norms of the tangent and normal vectors through one revolution of the merry-go-round. Use all of this information to describe in detail the path and forces on the child. Find minimum and maximum velocities of the child as well as where they occur. c. Now we move on to bigger and better things. We go to an amusement park where there is a true merry-go-round. The merry-go-round is 60 feet in diameter and our child is on a horse that is 5 feet from the edge. The horse goes from a height of four feet to a height of 5.5 feet three times per revolution and the merry-go-round is revolving once every 5 seconds. Find a three-dimensional vector valued function that describes the position of the child at any time t. Graph the function with its tangent, normal and binormal vectors. Also use the normplot command to graph the norms of the tangent and normal vectors through one revolution of the merry-go-round. Use all of this information to describe in detail the path and forces on the child. Find minimum and maximum velocities of the child as well as where they occur. d. Now we get to the sick part. Same scenario as the previous problem but this time we place the hobbyhorse on the back of the horse on the merry-go-round. Again the hobbyhorse sways along a radius of the merry-go-round. Find a threedimensional vector valued function that describes the position of the child at any time t. Graph the function with its tangent, normal and binormal vectors. Also use the normplot command to graph the norms of the tangent and normal vectors through one revolution of the merry-go-round. Use all of this information to describe in detail the path and forces on the child. Find minimum and maximum velocities of the child as well as where they occur. e. If the last one wasn t bad enough, try this. Same scenario as the previous problem but this time we place the hobbyhorse sways along the perpendicular line to a radius of the merry-go-round. Find a three-dimensional vector valued function that describes the position of the child at any time t. Graph the function with its tangent, normal and binormal vectors. Also use the normplot command to graph the norms of the tangent and normal vectors through one revolution of the merrygo-round. Use all of this information to describe in detail the path and forces on the child. Find minimum and maximum velocities of the child as well as where they occur. f. Historically, the merry-go-round was first used to train knights in the art of jousting. A knight would sit on the fake horse and a small brass ring would hang

off to the side of the merry-go-round. The knight would attempt to spear the ring when it came around. Some amusement parks have merry-go-rounds with a brass ring dispenser off to the side in honor of the origins of the merry-go-round. Riders would try to grab one of the rings as they came by. Hence the saying, Grabbing the brass ring. In some cases children can exchange the rings they grab for prize tokens that can be exchanged for prizes at the arcade. If you wish to experience this please go to Knoebels Grove amusement park in Elysburg Pennsylvania. Say that the ring dispenser is seven feet off the ground and 6 inches away from the edge of the merry-go-round. In each of the three previous scenarios find a three-dimensional vector valued function that describes the position of the ring with respect to the child at any time t. What is the velocity and speed of the ring, with respect to the child, when the ring is the closest to the child? 4. The torsion of a curve can be calculated with the equation, ( r' r'' ) r'' ' τ = r' r'' a. Create a new Maple command that finds the torsion of a curve. r ( t) = t, t sin t, t cos t. Graph the torsion function b. Find the torsion of the curve ( ) ( ) on the interval [,4π ] 0. Graph the curve with 0 sets of TBN frame vectors evenly spaced along the curve. What is happening to the TBN frame vectors when the torsion is large? What is happening to the TBN frame vectors when the torsion is small? c. Find the torsion of the curve r ( t) = cos( t),sin( t), t cos( t) function on the interval [,π ]. Graph the torsion 0. Graph the curve with 0 sets of TBN frame vectors evenly spaced along the curve. What is happening to the TBN frame vectors when the torsion is large? What is happening to the TBN frame vectors when the torsion is small? 3 d. Find the torsion of the curve r ( t) = 4 t, t + 7, t 5t + 6. Graph the torsion function on the interval [,5]. Graph the curve with 0 sets of TBN frame vectors evenly spaced along the curve. What is happening to the TBN frame vectors when the torsion is large? What is happening to the TBN frame vectors when the torsion is small? e. Find the torsion of the curve r ( t) = cos( t ),sin( t ), t on the interval [.,]. Graph the torsion function 0. Graph the curve with 0 sets of TBN frame vectors evenly spaced along the curve. What is happening to the TBN frame vectors when the torsion is large? What is happening to the TBN frame vectors when the torsion is small? f. Find the torsion of the curve r ( t) = t,sin( t),sin( t) cos( t) function on the interval [,π ]. Graph the torsion 0. Graph the curve with 0 sets of TBN frame vectors evenly spaced along the curve. Note that you may need to do two graphs and join them together with the display command. What is happening to the TBN

frame vectors when the torsion is large? What is happening to the TBN frame vectors when the torsion is small? g. From your observations above what does torsion measure?