Chapter 2: Space Curves
Section 2.6: Binormal and Torsion
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Example 2.6.2
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For , the curve defined by in Example 2.5.7,
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Calculate the torsion by both formulas on the right in Table 2.6.1.
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c)
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Verify the equality .
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d)
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Graph , along with the TNB-frame at .
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Solution
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Mathematical Solution
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Part (a)
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The vectors T, N, and B are respectively
Table 2.6.2(a) provides a path through the manual calculations of the TNB-frame. The overdot represents differentiation with respect to ; the prime, with respect to arc length . The calculations are done in the following order: three down the left-hand column then three down the right-hand column, and finally, the calculation across the bottom.
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=
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Table 2.6.2(a) Manual calculation of the TNB-frame
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There are other ways to obtain the TNB-frame. The student taught a different path might want to modify Table 2.6.2(a) to reflect one of those different methods.
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Part (b)
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Torsion by first formula:
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Torsion by second formula:
= = 36
= = ⇒ =
=
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Part (c)
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Left-hand side:
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Right-hand side:
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Part (d)
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Figure 2.6.2(a) contains a graph of , along with the TNB-frame at .
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is represented by the black arrow.
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is represented by the red arrow.
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is represented by the green arrow.
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The animation in the
tutor can be used to verify that as T advances with increasing arclength, the osculating plane twists about the tangent line clockwise (as viewed in the direction of the advance of T),
a twist consistent with the positive value of the torsion.
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>
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use plots, Student:-VectorCalculus in
module()
local p1,p2,p3,R,V,TNB;
R:=<p,3*p^2,p^3>;
TNB:=TNBFrame(R,p);
V:=map(ConvertVector,eval([TNB],p=1),rooted,[1,3,1]);
p1:=PlotVector(V,color=[black,red,green],width=.1);
p2:=PlotPositionVector(ConvertVector(R,position),p=.5..1.5, curveoptions=[color=blue]);
p3:=display(p1,p2,scaling=constrained,axes=frame,labels=[x,y,z], tickmarks=[[0,1],[0,1],4],orientation=[-65,70,0]);
print(p3);
end module:
end use:
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Figure 2.6.2(a) and TNB-frame at
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Maple Solution - Interactive
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Part (a)
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Tools≻Load Package: Student Vector Calculus
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Loading Student:-VectorCalculus
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Execute the BasisFormat command at the right, or use the
task template.
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Enter the vector notation for as per Table 1.1.1.
Context Panel: Assign Name
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Write R
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Frenet Formalism≻TNB Frame≻
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Context Panel: Simplify≻Assuming Real
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Context Panel: Assign to a Name≻TNB
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=
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Figure 2.6.2(a) is a screenshot of the
tutor adjusted to display an animation of a single TNB-frame traversing a portion of the curve.
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The Plot Options button is used to impose constrained scaling and the frame style for the axes.
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The default number of frames, 5, is changed to 30, and the Animate button is pressed.
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The Display Options drop-down box provides other options: The individual vectors of the TNB-frame can be separately graphed, and graphs of the curvature and torsion can be displayed.
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In Figure 2.6.2(b), the animation of a single TNB-frame traversing the curve was generated by the TNBFrame command, which allows for greater control of all aspects of the animation. (The animation shown in Figure 2.6.2(a) would be written to the worksheet upon pressing the Close button in the tutor.) The colors black, red, and green are used respectively for T, N, and B.
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Figure 2.6.2(a) Space Curves tutor
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>
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Student:-VectorCalculus:-TNBFrame(<p,3*p^2,p^3>,p,output=animation,tangentoptions=[color=black,width=.1],
normaloptions=[color=red,width=.1],
binormaloptions=[color=green,width=.1],
scaling=constrained,
axes=frame,
range=0..1.5,caption="",frames=30,
curveoptions=[labels=[x,y,z],orientation=[45,45,0],tickmarks=[[0,1],4,3],lightmodel=none]);
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Figure 2.6.2(b) TNB-frame animation
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The tutor is based on the TNBFrame command whose output can be a graph of the curve along with a specified number of TNB-frames, an animation such as shown in Figure 2.6.2(b), or the algebraic representation of the vectors of the TNB-frame.
The Display Options in the tutor provides for graphs of the individual tangent, principal normal, and binormal vectors, and normalized versions of these vectors. That is because the underlying commands TangentVector, PrincipalNormal, and Binormal, originally did not have options for returning normalized vectors. Surprisingly, the vectors in the TNB-frame returned by the TNBFrame command are normalized by default.
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Part (b)
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Obtain the torsion via the Context Panel system
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Write R
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻
Frenet Formalism≻Torsion≻
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Obtain the torsion by the upper-right formula in Table 2.6.1
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Keyboard the norm bars.
Calculus palette: Differentiation operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻rho
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Extract B and N from the TNB-frame obtained in Part (a).
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Calculus palette: Differentiation operator
Common Symbols palette: Dot product operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Simplify
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Obtain the torsion by the lower-right formula in Table 2.6.1
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Calculus palette: Differentiation operator
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Context Panel: Assign Name
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Calculus palette: Differentiation operator
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Context Panel: Assign Name
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Calculus palette: Differentiation operator
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Context Panel: Assign Name
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Keyboard the norm bars, and use Atomic Identifiers when referencing the derivatives of R.
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Common Symbols palette: Dot and cross product operators
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻tau
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Part (c)
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Evaluate the right-hand side of the given identity
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Write R
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Frenet Formalism≻Curvature≻
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Context Panel: Simplify≻Assuming Real
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Context Panel: Assign to a Name≻kappa
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Write the expression.
Context Panel: Evaluate and Display Inline
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Evaluate the left-hand side of the given identity
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Calculus palette: Differentiation operator; extract T and B from the TNB-frame in Part (a)
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Common Symbols palette: Dot product operator
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Context Panel: Evaluate and Display Inline
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Context Panel: Simplify≻Simplify
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Part (d)
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As a first recourse, use the
tutor to explore the behavior of the TNB-frame along .
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Figure 2.6.2(b) shows the state of the tutor after it has been applied to , the number of frames changed from the default 5 to 1, the axes changed to frame-style, and scaling set to constrained.
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If the Animate option is selected, then the selected number of frames will be seen to traverse the curve.
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Figure 2.6.2(b) Space Curve tutor
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The following interactive construction results in an approximation to Figure 2.6.2(a).
Evaluate the TNB-frame at
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Expression palette: Evaluation template
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Context Panel: Evaluate and Display Inline
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Context Panel: Assign to a Name≻TNBp
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Graph each vector in the evaluated TNB-frame
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Type , to reference respectively
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Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus≻Conversions≻To Free Vector
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Context Panel: Plots≻Arrow from point≻ (for , respectively)
Context Panel: Color≻black, red, green, respectively
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Graph R and add the vectors of the TNB-frame
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Write R
Context Panel: Evaluate and Display Inline
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Context Panel: Student Vector Calculus: Conversions≻To List
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Context Panel: Plots≻Plot Builder
Set
Options: Constrained Scaling, color = blue, Axes = frame
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Copy/paste the graphs of the three vectors
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Unfortunately, there is no provision as yet to vary the "heft" of vectors graphed interactively.
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Maple Solution - Coded
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Part (a)
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Install the Student VectorCalculus package.
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Define as the position vector R.
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Part (b)
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Obtain the torsion by the upper-right formula in Table 2.6.1
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Use the diff and Norm commands to obtain .
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Obtain the torsion by the lower-right formula in Table 2.6.1
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Apply the DotProduct, CrossProduct, Norm, and simplify commands as appropriate.
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Part (c)
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Obtain by applying the Curvature and simplify commands to R.
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Form the product on the right-hand side.
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Obtain the left-hand side by applying the diff, DotProduct, and simplify commands, extracting T and B from the TNB-frame obtained in Part (a).
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Part (d)
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The TNBFrame command can return the TNB-frame, a graph of along with a specified number of individual frames, or an animation of a frame moving along the curve. The following implementation of the command returns one frame at the point corresponding to .
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The following implementation of the TNBFrame command returns an animation of a single frame moving along .
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A graph of , along with the TNB-frame at , can be constructed with the following commands. The TNB-frame is evaluated at , but the resulting vectors are rooted at the symbolic point . Hence, these vectors are "converted" to vectors rooted at the contact point determined by . Then they can be graphed with the PlotVector command. The PlotPositionVector is used to graph (after R is converted to a position vector with the ConvertVector command), and the display command merges the two graphs.
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