Chapter 2: Space Curves
Section 2.6: Binormal and Torsion
Obtain τ=R.R..R...R.×R..2, the formula in the lower-right portion of Table 2.6.1.
Let R′=ddsRs and R.=ddpRp. Start with the formulas τ=R′R″R‴/κ2 and κ=R.×R../ρ3. The intermediate calculations, heavily dependent on the chain rule for differentiation of composite functions, are summarized in Table 2.6.8(a).
=α R.+β R..+1ρ3R...
Table 2.6.8(a) Relating the derivatives in R′R″R‴ to those in R.R..R...
The last row in Table 2.6.8(a) is obtained by observing that in the box product, only rows that are independent will "survive." This becomes clear in the following concluding steps where the additive properties of a determinant are essential.
If ABC represents the determinant of matrix whose columns are the vectors A, B, and C, then
U+VWQ = UWQ+VWQ
In other words, if a column is the sum of two other column vectors, then the determinant splits into the sum of two determinants. And what is true for columns is true for rows, because the determinant of a matrix has the same value as the determinant of the transpose of the matrix.
Write R′R″R‴ as the determinant
|R.ρ(−ρ.ρ3R.+R..ρ2α R.+β R..+1ρ3R...|
By the additivity property of determinants, this splits into a sum of determinants, but any of the resulting determinants that have two columns proportional will be zero. Hence, the "surviving" determinant is the box product
R.ρ R..ρ2 R...ρ3=1ρ6R.R..R...
Consequently, τ=R′R″R‴/κ2 becomes
R.R..R.../ρ6(R.×R../ρ3)2 = R.R..R...R.×R..2
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