Chapter 2: Space Curves
Section 2.6: Binormal and Torsion
For C, the curve defined by Rp=3 p−p3 i+3 p2 j+3 p+p3 k in Example 2.5.9,
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=−1.
The vectors T, N, and B are respectively
Table 2.6.4(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.
κ=∥T′∥ = 13⁢p2+12
B=T×N=ijk−12⁢2⁢p2−1p2+12⁢pp2+11/2−2⁢pp2+1−p2−1p2+10 = 12⁢2⁢p2−1p2+1−2⁢pp2+112⁢2
Table 2.6.4(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 2.6.4(a) to reflect one of those different methods.
Torsion by first formula:
Torsion by second formula:
R.R..R...=R.·R..×R... = 3−3 p26 p3+3 p2−6 p66 p−600 = 216
R.×R.. = |ijk3 p−3 p26 p3+3 p2−6 p66 p| = 18 p2−1−2 pp2+1 ⇒ ∥R.×R..∥2 = 648 p2+12
τ=R.·R..×R...R.×R..2 = 216648p2+12=13p2+12
−κ τ ρ2
=−13⁢p2+12 13⁢p2+12 3⁢2⁢p2+12
Figure 2.6.4(a) contains a graph of C, along with the TNB-frame at p=−1.
T−1 is represented by the black arrow.
N−1 is represented by the red arrow.
B−1 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 clockwise (as viewed in the direction of the advance of T),
a twist consistent with the positive value of the torsion.
Figure 2.6.4(a) C and TNB-frame at p=1
Maple Solution - Interactive
Tools≻Load Package: Student Vector Calculus
Execute the BasisFormat command at the right, or use the
Enter the vector notation for C as per Table 1.1.1.
Context Panel: Assign Name
R=3 p−p3,3 p2,3 p+ p3→assign
Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Frenet Formalism≻TNB Frame≻p
Context Panel: Simplify≻Assuming Real
Context Panel: Assign to a Name≻TNB
R−p3+3⁢p3⁢p2p3+3⁢p = →TNB frame−12⁢2⁢p2−1⁢csgn⁡p2+1p2+12⁢csgn⁡p2+1⁢pp2+112⁢2⁢csgn⁡p2+1,−12⁢2⁢csgn⁡1,p2+1⁢p4+4⁢csgn⁡p2+1⁢p−csgn⁡1,p2+1csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12⁢p2+122⁢csgn⁡1,p2+1⁢p3−csgn⁡p2+1⁢p2+csgn⁡1,p2+1⁢p+csgn⁡p2+1csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12⁢p2+1212⁢2⁢csgn⁡1,p2+1csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12,p2−1csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12⁢p2+12−2⁢pcsgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12⁢p2+121csgn⁡1,p2+12⁢p4+2⁢csgn⁡1,p2+12⁢p2+csgn⁡1,p2+12+2p2+12⁢p2+1→assuming real−12⁢2⁢p2−1p2+12⁢pp2+112⁢2,−2⁢pp2