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Chapter 2: Space Curves

Section 2.6: Binormal and Torsion

Example 2.6.3

For $C$ , the curve defined by $R (p) = \ln (\cos (p)) i + \ln (\sin (p)) j + \sqrt{2} p k$ , $p \in (0, π / 2)$ , in Example 2.5.8,

a)	Obtain the TNB-frame.

b)	Calculate the torsion $τ$ by both formulas on the right in Table 2.6.1.

c)	Verify the equality $\overset{&period;}{T} (p) \cdot \overset{&period;}{B} (p) = - κ τ ρ^{2}$ .

d)	Graph $C$ , along with the TNB-frame at $p = π / 4$ .

Solution

Mathematical Solution

Part (a)

The vectors T, N, and B are respectively

$[\begin{array}{c} - \sin^{2} (p) \\ \cos^{2} (p) \\ \sqrt{2} \cos (p) \sin (p) \end{array}]$ , $[\begin{array}{c} - \sqrt{2} \cos (p) \sin (p) \\ - \sqrt{2} \cos (p) \sin (p) \\ 2 {\cos (p)}^{2} - 1 \end{array}]$ , $[\begin{array}{c} \cos^{2} (p) \\ - \sin^{2} (p) \\ \sqrt{2} \cos (p) \sin (p) \end{array}]$

Table 2.6.3(a) provides a path through the manual calculations of the TNB-frame. The overdot represents differentiation with respect to $p$ ; the prime, with respect to arc length $s$ . 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.

$\overset{&period;}{R} = [\begin{array}{c} - \frac{\sin (p)}{\cos (p)} \\ \frac{\cos (p)}{\sin (p)} \\ \sqrt{2} \end{array}]$

$T' = \overset{&period;}{T} / ρ = [\begin{array}{c} - 2 \sin^{2} (p) \cos^{2} (p) \\ - 2 \sin^{2} (p) \cos^{2} (p) \\ \sin (p) \cos (p) \sqrt{2} (2 \cos^{2} (p) - 1) \end{array}]$

$ρ = ∥\overset{&period;}{R}∥ = \frac{1}{\cos (p) \sin (p)}$

$κ = ∥ T' ∥$ = $\sqrt{2} \cos (p) \sin (p)$

$T = \overset{&period;}{R} / ρ = [\begin{array}{c} - \sin^{2} (p) \\ \cos^{2} (p) \\ \sqrt{2} \cos (p) \sin (p) \end{array}]$

$N = T' / κ = [\begin{array}{c} - \sqrt{2} \cos (p) \sin (p) \\ - \sqrt{2} \cos (p) \sin (p) \\ 2 \cos^{2} (p) - 1 \end{array}]$

$B = T \times N$	$= \|\begin{array}{c} i & j & k \\ - \sin^{2} (p) & \cos^{2} (p) & \sqrt{2} \sin (p) \cos (p) \\ - \sqrt{2} \cos (p) \sin (p) & - \sqrt{2} \cos (p) \sin (p) & 2 \cos^{2} (p) - 1 \end{array}\|$
	$= [\begin{array}{c} \cos^{2} (p) \\ - \sin^{2} (p) \\ \sqrt{2} \cos (p) \sin (p) \end{array}]$

Table 2.6.3(a) Manual calculation of the TNB-frame

There are other ways to obtain the TNB-frame. The student taught a different path might want to modify Table 2.6.3(a) to reflect one of those different methods.

Part (b)

Torsion by first formula:

$τ = - (\overset{&period;}{B} / ρ) \cdot N$	$= - \frac{1}{1 / (\cos (p) \sin (p))} ([\begin{array}{c} - 2 \sin (p) \cos (p) \\ - 2 \sin (p) \cos (p) \\ \sqrt{2} (2 {\cos (p)}^{2} - 1) \end{array}]) \cdot ([\begin{array}{c} - \sqrt{2} \cos (p) \sin (p) \\ - \sqrt{2} \cos (p) \sin (p) \\ 2 \cos^{2} (p) - 1 \end{array}])$
	$= - 4 \sqrt{2} \sin^{3} (p) \cos^{3} (p) - \sin (p) \cos (p) \sqrt{2} {(2 \cos^{2} (p) - 1)}^{2}$
	$= - \sqrt{2} \cos (p) \sin (p)$

Torsion by second formula:

$[\overset{&period;}{R} \overset{..}{R} \overset{.. &period;}{R}] = \overset{&period;}{R} \cdot (\overset{..}{R} \times \overset{.. &period;}{R})$ = $| \begin{array}{c} - \frac{\sin (p)}{\cos (p)} & \frac{\cos (p)}{\sin (p)} & \sqrt{2} \\ - \frac{1}{\cos^{2} (p)} & - \frac{1}{\sin^{2} (p)} & 0 \\ - \frac{2 \sin (p)}{\cos^{3} (p)} & \frac{2 \cos (p)}{\sin^{3} (p)} & 0 \end{array} |$ = $- \frac{2 \sqrt{2}}{\sin^{3} (p) \cos^{3} (p)}$

$\overset{&period;}{R} \times \overset{..}{R}$ = $| \begin{array}{c} i & j & k \\ - \frac{\sin (p)}{\cos (p)} & \cos (p) & \sqrt{2} \\ - \frac{1}{\cos^{2} (p)} & - \frac{1}{\sin^{2} (p)} & 0 \end{array} |$ = $[\begin{array}{c} \frac{\sqrt{2}}{{\sin (p)}^{2}} \\ - \frac{\sqrt{2}}{{\cos (p)}^{2}} \\ \frac{2}{\cos (p) \sin (p)} \end{array}]$

⇒ ${∥ \overset{&period;}{R} \times \overset{..}{R} ∥}^{2}$ = $\frac{2}{{\sin (p)}^{4} {\cos (p)}^{4}}$ ⇒ $τ = \frac{\overset{&period;}{R} \cdot (\overset{..}{R} \times \overset{.. &period;}{R})}{{∥\overset{&period;}{R} \times \overset{..}{R}∥}^{2}}$ = $- \sqrt{2} \cos (p) \sin (p)$

Part (c)

Left-hand side:

$\overset{&period;}{T} (p) \cdot \overset{&period;}{B} (p)$	$= ([\begin{array}{c} - 2 \sin (p) \cos (p) \\ - 2 \sin (p) \cos (p) \\ \sqrt{2} (2 {\cos (p)}^{2} - 1) \end{array}]) \cdot ([\begin{array}{c} - 2 \sin (p) \cos (p) \\ - 2 \sin (p) \cos (p) \\ \sqrt{2} (2 {\cos (p)}^{2} - 1) \end{array}])$
	$= 8 \sin^{2} (p) \cos^{2} (p) + {(- \sqrt{2} \sin^{2} (p) + \sqrt{2} \cos^{2} (p))}^{2}$
	$= 2$

Right-hand side:

$- κ τ ρ^{2} = - (\sqrt{2} \cos (p) \sin (p)) (- \sqrt{2} \cos (p) \sin (p)) {(\frac{1}{\cos (p) \sin (p)})}^{2} = 2$

Part (d)

•	Figure 2.6.3(a) contains a graph of $C$ , along with the TNB-frame at $p = π / 4$ .

•	$T (π / 4)$ is represented by the black arrow.

•	$N (π / 4)$ is represented by the red arrow.

•	$B (π / 4)$ is represented by the green arrow.

•	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 counterclockwise (as viewed in the direction of the advance of T), a twist consistent with the negative value of the torsion.

use plots, Student:-VectorCalculus in
module()
local p1,p2,p3,R,V,TNB;
R:=<ln(cos(p)),ln(sin(p)),sqrt(2)*p>;
TNB:=TNBFrame(R,p);
V:=map(ConvertVector,eval([TNB],p=Pi/4),rooted,[-ln(sqrt(2)),-ln(sqrt(2)),sqrt(2)*Pi/4]);
p1:=PlotVector(V,color=[black,red,green],width=.1);
p2:=PlotPositionVector(ConvertVector(R,position),p=0..Pi/2, curveoptions=[color=blue]);
p3:=display(p1,p2,scaling=constrained,axes=frame,labels=[x,y,z], tickmarks=[2,2,2],orientation=[-65,70,0]);
print(p3);
end module:
end use:

Figure 2.6.3(a) $C$ and TNB-frame at $p = π / 4$

Maple Solution - Interactive

Part (a)

•	Tools≻Load Package: Student Vector Calculus

Loading Student:-VectorCalculus

•	Execute the BasisFormat command at the right, or use the task template.

$BasisFormat (false) &colon;$

•	Enter the vector notation for $C$ as per Table 1.1.1. Context Panel: Assign Name

$R = ⟨\ln (\cos (p)), \ln (\sin (p)), \sqrt{2} p⟩$ $\overset{assign}{\to}$

•	Write R Context Panel: Evaluate and Display Inline

•	Context Panel: Student Vector Calculus≻Frenet Formalism≻TNB Frame≻ $p$

•	Context Panel: Simplify≻Symbolic

•	Context Panel: Assign to a Name≻TNB

$R$ $[\begin{array}{c} \ln (\cos (p)) \\ \ln (\sin (p)) \\ \sqrt{2} p \end{array}]$ = $\overset{TNB frame}{\to}$ $[\begin{array}{c} - \frac{\sin (p)}{\sqrt{\frac{1}{{\cos (p)}^{2} {\sin (p)}^{2}}} \cos (p)} \\ \frac{\cos (p)}{\sqrt{\frac{1}{{\cos (p)}^{2} {\sin (p)}^{2}}} \sin (p)} \\ \frac{\sqrt{2}}{\sqrt{\frac{1}{{\cos (p)}^{2} {\sin (p)}^{2}}}} \end{array}], [\begin{array}{c} - \frac{\sqrt{2}}{\sqrt{\frac{1}{{\cos (p)}^{2} {\sin (p)}^{2}}}} \\ - \frac{\sqrt{2}}{\sqrt{\frac{1}{{\cos (p)}^{2} {\sin (p)}^{2}}}} \\ \frac{2 {\cos (p)}^{2} - 1}{\cos (p) \sin (p) \sqrt{\frac{1}{{\cos (p)}^{2} {\sin (p)}^{2}}}} \end{array}], [\begin{array}{c} {\cos (p)}^{2} \\ - {\sin (p)}^{2} \\ \sqrt{2} \sin (p) \cos (p) \end{array}]$ $\overset{simplify symbolic}{\to}$ $[\begin{array}{c} - {\sin (p)}^{2} \\ {\cos (p)}^{2} \\ \sqrt{2} \sin (p) \cos (p) \end{array}], [\begin{array}{c} - \sqrt{2} \sin (p) \cos (p) \\ - \sqrt{2} \sin (p) \cos (p) \\ 2 {\cos (p)}^{2} - 1 \end{array}], [\begin{array}{c} {\cos (p)}^{2} \\ - {\sin (p)}^{2} \\ \sqrt{2} \sin (p) \cos (p) \end{array}]$ $\overset{assign to a name}{\to}$ $TNB$

•	Figure 2.6.3(a) is a screen-shot of the tutor adjusted to display an animation of a single TNB-frame traversing a portion of the curve. (Note the avoidance of $p = 0$ in the tutor.)

•	The Plot Options button is used to impose constrained scaling and the frame style for the axes.

•	The default number of frames, 5, is changed to 30, and the Animate button is pressed.

•	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.

•

In Figure 2.6.3(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 2.6.3(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.

Figure 2.6.3(a) Space Curves tutor

Student:-VectorCalculus:-TNBFrame(<ln(cos(p)),ln(sin(p)),sqrt(2)*p>,p,output=animation,tangentoptions=[color=black,width=.1],
normaloptions=[color=red,width=.1],
binormaloptions=[color=green,width=.1],
scaling=constrained,
axes=frame,
range=0..Pi/2,caption="",frames=30,
curveoptions=[labels=[x,y,z],orientation=[45,45,0],tickmarks=[3,4,3],lightmodel=none]);

Figure 2.6.3(b) TNB-frame animation

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 2.6.3(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.

Part (b)

Obtain the torsion via the Context Panel system

•	Write R Context Panel: Evaluate and Display Inline

•	Context Panel: Student Vector Calculus≻Frenet Formalism≻Torsion≻ $p$

$R$ = $[\begin{array}{c} \ln (\cos (p)) \\ \ln (\sin (p)) \\ \sqrt{2} p \end{array}]$ $\overset{torsion}{\to}$ $- \sqrt{2} \sin (p) \cos (p)$

Obtain the torsion by the upper-right formula in Table 2.6.1

•	Keyboard the norm bars. Calculus palette: Differentiation operator

•	Context Panel: Evaluate and Display Inline

•	Context Panel: Simplify≻Symbolic

•	Context Panel: Assign to a Name≻rho

$∥\frac{&DifferentialD;}{&DifferentialD; p} R∥$ = $\sqrt{\frac{1}{{\cos (p)}^{2} {\sin (p)}^{2}}}$ $\overset{simplify symbolic}{\to}$ $\frac{1}{\cos (p) \sin (p)}$ $\overset{assign to a name}{\to}$ $ρ$

•	Extract B and N from the TNB-frame obtained in Part (a).

•	Calculus palette: Differentiation operator Common Symbols palette: Dot product operator Press the Enter key.

•	Context Panel: Simplify≻Simplify

$- (\frac{&DifferentialD;}{&DifferentialD; p} {TNB}_{3}) \cdot {TNB}_{2} / ρ$

$- (4 {\cos (p)}^{2} {\sin (p)}^{2} \sqrt{2} + (\sqrt{2} {\cos (p)}^{2} - \sqrt{2} {\sin (p)}^{2}) (2 {\cos (p)}^{2} - 1)) \cos (p) \sin (p)$

$\overset{simplify}{=}$

$- \sqrt{2} \sin (p) \cos (p)$

Obtain the torsion by the lower-right formula in Table 2.6.1

•	Form the name $\overset{&period;}{R}$ as an Atomic Identifier.

•	Calculus palette: Differentiation operator

•	Context Panel: Assign Name

$\overset{&period;}{R} = \frac{&DifferentialD;}{&DifferentialD; p} R$ $\overset{assign}{\to}$

•	Form the name $\overset{..}{R}$ as an Atomic Identifier.

•	Calculus palette: Differentiation operator

•	Context Panel: Assign Name

$\overset{..}{R} = \frac{{&DifferentialD;}^{2}}{{&DifferentialD;}^{} p^{2}} R$ $\overset{assign}{\to}$

•	Form the name $\overset{.. &period;}{R}$ as an Atomic Identifier.

•	Calculus palette: Differentiation operator

•	Context Panel: Assign Name

$\overset{.. &period;}{R} = \frac{{&DifferentialD;}^{3}}{{&DifferentialD;}^{} p^{3}} R$ $\overset{assign}{\to}$

•	Keyboard the norm bars, and use Atomic Identifiers when referencing the derivatives of R.

•	Common Symbols palette: Dot and cross product operators Press the Enter key.

•	Context Panel: Simplify≻Simplify

•	Context Panel: Assign to a Name≻tau

$\frac{\overset{&period;}{R} \cdot (\overset{..}{R} \times \overset{.. &period;}{R})}{{&par; \overset{&period;}{R} \times \overset{..}{R} &par;}^{2}}$

$\frac{1}{2} \sqrt{2} ((- 1 - \frac{{\sin (p)}^{2}}{{\cos (p)}^{2}}) (\frac{2 \cos (p)}{\sin (p)} + \frac{2 {\cos (p)}^{3}}{{\sin (p)}^{3}}) - (- 1 - \frac{{\cos (p)}^{2}}{{\sin (p)}^{2}}) (- \frac{2 \sin (p)}{\cos (p)} - \frac{2 {\sin (p)}^{3}}{{\cos (p)}^{3}})) {\sin (p)}^{4} \cos$

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