Chapter 2: Space Curves
Section 2.6: Binormal and Torsion
Example 2.6.3
For C, the curve defined by Rp=lncosp i+lnsinp j+2p k, p∈0,π/2, in Example 2.5.8,
Obtain the TNB-frame.
Calculate the torsion τ by both formulas on the right in Table 2.6.1.
Verify the equality T.p·B.p=−κ τ ρ2.
Graph C, along with the TNB-frame at p=π/4.
Solution
Mathematical Solution
Part (a)
The vectors T, N, and B are respectively
−sin2pcos2p2cosp⁢sinp, −2cosp⁢sinp−2cosp⁢sinp2⁢cos⁡p2−1,cos2p−sin2p2cosp⁢sinp
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.
R.=−sin⁡pcos⁡pcos⁡psin⁡p2
T′=T./ρ=−2 sin2p⁢cos2p−2 sin2p⁢cos2psin⁡p⁢cos⁡p⁢2⁢2⁢cos2p−1
ρ=R.=1cos⁡p⁢sin⁡p
κ=∥T′∥ = 2cosp⁢sinp
T=R./ρ=−sin2pcos2p2cosp⁢sinp
N=T′/κ=−2cosp⁢sinp−2cosp⁢sinp2⁢cos2p−1
B=T×N
=ijk−sin2pcos2p2sinpcosp−2cospsinp−2cospsinp2 cos2p−1
=cos2p−sin2p2cosp⁢sinp
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= −B./ρ·N
=−11/cospsinp−2⁢sin⁡p⁢cos⁡p−2⁢sin⁡p⁢cos⁡p2⁢2⁢cos⁡p2−1·−2cosp⁢sinp−2cosp⁢sinp2⁢cos2p−1
= −42 sin3p⁢cos3p−sinp⁢cosp⁢2⁢2⁢cos2p−12
=−2cosp⁢sinp
Torsion by second formula:
R.R..R...=R.·R..×R... = |−sinpcospcospsinp2−1cos2p−1sin2p0−2 sinpcos3p2 cospsin3p0| = −2⁢2sin3p⁢cos3p
R.×R.. = |ijk−sinpcospcosp2−1cos2p−1sin2p0| = 2sin⁡p2−2cos⁡p22cos⁡p⁢sin⁡p
⇒ ∥R.×R..∥2 = 2sin⁡p4⁢cos⁡p4 ⇒ τ=R.·R..×R...R.×R..2 = −2cospsinp
Part (c)
Left-hand side:
T.p·B.p
=−2⁢sin⁡p⁢cos⁡p−2⁢sin⁡p⁢cos⁡p2⁢2⁢cos⁡p2−1·−2⁢sin⁡p⁢cos⁡p−2⁢sin⁡p⁢cos⁡p2⁢2⁢cos⁡p2−1
=8⁢sin2p⁢cos2p+−2⁢sin2p+2⁢cos2p2
=2
Right-hand side:
−κ τ ρ2=−2cosp⁢sinp−2cosp⁢sinp1cos⁡p⁢sin⁡p2=2
Part (d)
Figure 2.6.3(a) contains a graph of C, along with the TNB-frame at p=π/4.
Tπ/4 is represented by the black arrow.
Nπ/4 is represented by the red arrow.
Bπ/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
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Execute the BasisFormat command at the right, or use the task template.
BasisFormatfalse:
Enter the vector notation for C as per Table 1.1.1. Context Panel: Assign Name
R=lncosp,lnsinp,2p→assign
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
Rln⁡cos⁡pln⁡sin⁡p2⁢p = →TNB frame−sin⁡p1cos⁡p2⁢sin⁡p2⁢cos⁡pcos⁡p1cos⁡p2⁢sin⁡p2⁢sin⁡p21cos⁡p2⁢sin⁡p2,−21cos⁡p2⁢sin⁡p2−21cos⁡p2⁢sin⁡p22⁢cos⁡p2−1cos⁡p⁢sin⁡p⁢1cos⁡p2⁢sin⁡p2,cos⁡p2−sin⁡p22⁢sin⁡p⁢cos⁡p→simplify symbolic−sin⁡p2cos⁡p22⁢sin⁡p⁢cos⁡p,−2⁢sin⁡p⁢cos⁡p−2⁢sin⁡p⁢cos⁡p2⁢cos⁡p2−1,cos⁡p2−sin⁡p22⁢sin⁡p⁢cos⁡p→assign to a nameTNB
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.
Obtain the torsion via the Context Panel system
Context Panel: Student Vector Calculus≻Frenet Formalism≻Torsion≻p
R = ln⁡cos⁡pln⁡sin⁡p2⁢p→torsion−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: Assign to a Name≻rho
ⅆⅆ p R = 1cos⁡p2⁢sin⁡p2→simplify symbolic1cos⁡p⁢sin⁡p→assign to a nameρ
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
−ⅆⅆ p TNB3·TNB2/ρ
−4⁢cos⁡p2⁢sin⁡p2⁢2+2⁢cos⁡p2−2⁢sin⁡p2⁢2⁢cos⁡p2−1⁢cos⁡p⁢sin⁡p
= simplify
−2⁢sin⁡p⁢cos⁡p
Obtain the torsion by the lower-right formula in Table 2.6.1
Form the name R. as an Atomic Identifier.
Calculus palette: Differentiation operator
Context Panel: Assign Name
R.=ⅆⅆ p R→assign
Form the name R.. as an Atomic Identifier.
R..=ⅆ2ⅆp2 R→assign
Form the name R... as an Atomic Identifier.
R...=ⅆ3ⅆp3 R→assign
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: Assign to a Name≻tau
R.·R..×R...∥R.×R..∥2
12⁢2⁢−1−sin⁡p2cos⁡p2⁢2⁢cos⁡psin⁡p+2⁢cos⁡p3sin⁡p3−−1−cos⁡p2sin⁡p2⁢−2⁢sin⁡pcos⁡p−2⁢sin⁡p3cos⁡p3⁢sin⁡p4⁢cos