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The Frenet-Serret Formulae Relate the Derivatives of the Frenet Vectors to Linear Combinations of Them

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The Frenet-Serret Formulae Relate the Derivatives of the Frenet Vectors to Linear Combinations of Them The Frenet-Serret formulae relate the derivatives of the Frenet vectors to linear combinations of them. The Frenet frame is the triplet of Frenet vectors which form the orthonormal basis of vectors following the trajectory of a point along the parametrized path of its motion in R3. The equations are named after Jean Fr¶ed¶ericFrenet (1816 - 1900), who was a French mathematician, astronomer, and meteorologist; and Joseph Alfred Serret (1819 - 1885) who was a french mathematician, born in Paris and died in Versailles. The Formulae 3 Specify C to be a space-curve in R which has a parametrization r(t) = (r1(t); r2(t); r3(t)) on [®; ¯]. Suppose that s = s(º) is the distance parameter along C (beginning at t = ®), then Z ° ° º ° ° °dr(t)° s(º) = ° ° dt : ® dt We now de¯ne the unit tangent along the curve to be dr Tb = : ds From the ¯rst fundamental theorem of calculus we have Z ° ° ° ° º ° ° ° ° ds(º) d °dr(t)° °dr(º)° = ° ° dt = ° ° : dº dº ® dt dº Thus we have Á ° ° dr dr dº dr(º) °dr(º)° Tb = = = ° ° : ds dº ds dº ° dº ° The normal vector is de¯ned to be dTb N = ; ds and thus the unit normal is ° ° Á ° ° dTb °dTb ° Nb = ° ° : ds ° ds ° That may be rearranged to give ° ° ° ° dTb °dTb ° = ° ° Nb ; ds ° ds ° dTb and when we de¯ne the curvature, ·, to be · = ds (the radius of curvature ½ is ½ = 1=·), the equation becomes dTb = ·Nb : ds The plane which contains Tb and Nb is called the osculating plane, and the circle with tangent Tb and radius equal to the radius of curvature ½ is called the osculating circle. The binormal is de¯ned to be Bb = Tb £ Nb ; 1 from which we also have Nb = Bb £ Tb; and Tb = Nb £ Bb : Di®erentiation of these expressions yields the Frenet-Serre formulae. The derivative of Bb = Tb £ Nb yields Bb 0 = Bb 0 £ Nb + Tb £ Nb 0, but Tb 0 is parallel to Nb and thus the ¯rst term vanishes. Thus the binormal must be orthogonal to Tb, but also to Nb 0, and, consequently, Bb 0 = ¿Nb ° ° ° dBb ° dNb in which ¿, called the torsion, is simply ¿ = ° ds °. The derivative of ds will yield a linear com- b b dNb b b 0 b 0 b bination of T and B. We di®erentiate to get ds = B £ T + B £ T. Upon substitution, we get Nb 0 = ·Bb £ Nb + ¿Nb £ Tb or Nb 0 = ¡·Tb ¡ ¿Bb . The Frenet-Serre formulae are therefore dTb = ·Nb ; ds dBb = ¡·Tb ¡ ¿Bb ; and ds dBb = ¿Nb : ds In matrix form, they are 0 1 0 1 0 1 Tb 0 0 · 0 Tb B C B C @Nb 0A = @¡· 0 ¡¿A @Nb A Bb 0 0 ¿ 0 Bb The unit tangent, Tb, unit normal, Nb , and binormal unit vector, Bb , are collectively called the Frenet vectors. Together, they form an orthonormal basis for R3, a reference frame which follows the trajectory of a point. The reference frame is called a Frenet frame, and it is neither static nor inertial because the Frenet frame moves tangentially to the (non-straight) curve, so it is constantly accelerating. The curve can be assumed to be parametrically dependent on time, i.e. the Frenet frame can be visualised kinematically. Exercise For the circular helix, x(t) = a cos ti+a sin tj+btk with a; b > 0 and 0 · t < 1, derive the following: p s(t) = a2 + b2 t ; Tb(t) = c(¡a sin ti + a cos tj + bk) ; Nb (t) = ¡ cos ti ¡ sin tj) ; Bb (t) = c(b sin ti ¡ b cos tj + ak) ; ·(t) = a=(a2 + b2) ; ½(t) = (a2 + b2)=a ; and ¿(t) = ¡b=(a2 + b2) ; p in which c = 1= a2 + b2. 2.
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