Chapter 2: Space Curves
Section 2.6: Binormal and Torsion
Obtain τ=R′R″R‴/κ2, the formula in the lower-left portion of Table 2.6.1.
Notice that although this formula is given, it has not been used directly in any Example because it is the rare space curve that can be parametrized with arc length. By tracing the connections between derivatives of R taken with respect to s relate to derivatives taken with respect to some other parameter p, the formula in the lower-right portion of Table 2.6.1 is obtained.
With the prime taken to mean differentiation with respect to arc length, consider the calculations summarized in Table 2.6.7(a).
N=1κT′ = 1κR′′ = 1κR″
B=T×N = R′×1κR″ = 1κR′×R″
B′=dds1κR′×R″ = 1κR′×R″+1κR′×R‴
τ=−N·B′ = −1κR″·dds1κR′×R″
= −1κ2 R″R′R″+R″R′R‴
Table 2.6.7(a) Derivation of the formula τ=R′R″R‴/κ2
In the third-last line, the expressions of the form U·V×W are recognized as triple scalar (i.e., box) products UVW. In the second-last (i.e., the penultimate) line, the first box product is zero because the vector R″ appears twice. A moment's thought to realize that a box product is a determinant makes this vanishing transparent. In that penultimate line also, the sign change takes place because the first and second rows of a determinant have been interchanged.
From this expression for the torsion, the formula τ=R.R..R...R.×R..2, where now R is a function of some other parameter p, and the overdots represent differentiation with respect to p, can be derived.
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