In my notes, it says to consider an arc length parameterization r(s). Then we can show that B' points in the direction of N such that we can write B ′ = − 𝜏 N. I understand why | | B ′ | | = 𝜏 but am unsure how to show why B' is parallel to N? Also, is the minus sign just for convention or is there a reason it is there?

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In my notes, it says to consider an arc length parameterization r(s). Then we can show that B&#039; points in the direction of N such that we can write       B    ′    =  −      𝜏    N. I understand why       |        |        B    ′        |        |    =      𝜏   but am unsure how to show why B&#039; is parallel to N? Also, is the minus sign just for convention or is there a reason it is there?

Mariam Hickman · Accepted Answer

In what follows differentiation with respect to arc length is indicated with a prime. We have                              T                                    =                              r                    ′                                              N                                    =                              T                    ′                          /                ‖                              T                    ′                ‖                                      B                                    =                  T                ×                  N                .            Then                                          B                    ′                                    =                              T                    ′                ×                  N                +                  T                ×                              N                    ′                                                  =        ‖                              T                    ′                ‖                                                                              N                                ×                                  N                                            ⏟                                                          0                                      +                  T                ×                              N                    ′                                                  =                  T                ×                              N                    ′                .            Note that B is orthogonal to T by definition of the cross product. Since   ‖      B    ‖  =  1, B′ is also orthogonal to B. (Proof:   (      B    ⋅      B        )    ′    =  2      B    ⋅            B        ′    =  0.) Since T,N,B form an orthogonal basis, it must be the case that B′ is proportional to N. We take             B        ′    =  −  τ      N  . The minus sign is a convention. It is a version of the right-hand rule. Note that   τ  &amp;gt;  0 indicates a twisting of the coordinate frame about the T axis in a right-handed manner. (Take your right hand and point your thumb in the T direction. Twist your hand about this axis in the direction in which your fingers are pointing. This twist corresponds to   τ  &amp;gt;  0.)

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