Answer the following about the Frenet–Serret (“freh-NAY sir-AY”) formulas, which describe the dynamics of the TNB frame along a curve. For 10 points each:
[10m] The Frenet–Serret formulas give expressions for the derivatives of the T, N, and B vectors with respect to this quantity, denoted s. For a function f, the [read slowly] “integral of the square root of 1 plus f-prime squared” gives this quantity, which is finite for rectifiable curves.
ANSWER: arc length [prompt on length]
[10h] The Frenet–Serret formulas involve two constants: the curvature, denoted kappa, and this other quantity, computed as the negative dot product of N and B prime. This quantity measures how quickly a curve is escaping its osculating plane.
ANSWER: torsion [prompt on tau]
[10e] The first and third formulas produce multiples of the normal vector N, which is named for having this property with respect to the tangent vector. Two vectors have this property if they meet at an angle of 90 degrees.
ANSWER: orthogonal [or perpendicular; prompt on normal]
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