The THIRD of the seven beauties. The integral of unit normal over a curve.

[@zubetto85]

at 00:55 it should probably be noted that a sufficient condition for the orthogonality of two vector sums under consideration is that vectors of the same length must be summed: if A1 • B1 = 0, A2 • B2 = 0 and a1 = sqrt(A1•A1), a2 = sqrt(A2•A2),..., then |(A1 + A2) • (B1 + B2)| = |A1 • B2 + A2 • B1| = |cosφ * (a1*b2 - a2*b1)|, where cosφ = A1 • B2 / (a1*b2) = -A2 • B1 / (a2*b1). Therefore, in general, the vector sum (A1 + A2) is not always orthogonal to the vector sum (B1 + B2).

[@hannahnowxyz]

what happens with space curves in 3 dimensions? I'm trying to imagine it. is it M*ΔR where the rotation matrix M can be decomposed as M = (a 90 degree rotation)*(another rotation related to the torsion)?

[@MathTheBeautiful]

Good question. The 3D case is different and more difficult, but yours truly cracked it! See this video: kzbin.info/www/bejne/qqe5ha2eppqXrNE

[@Pluralist]