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We know that the Frenet-Serret equation implies that the coefficient matrix of $\dot t,\dot n,\dot b$ is anti symmetric wrt $t,n,b$. But is there any geometric intuition that immediately gives this result? Thanks!

Golbez
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2 Answers2

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Antisymmetric matrices describe linear maps involving oriented planes. In this case, the plane is the one that describes the instantaneous rotation (and dilation*) of the frame. The system rotation is described in two parts--a part the $tn$-plane, describing how the curve bends, and a part in the $nb$-plane, describing how the frame twists around the path of the curve. Non-unit coefficients describe how the frame also dilates or shrinks in these planes.

(*) I say "dilation" here. That is to say, the "natural" description of a frame of basis vectors is not necessarily a frame of unit vectors, and the coefficients reflect that.

Muphrid
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  • "Antisymmetric matrices describe linear maps involving oriented planes", that's a new viewpoint I haven't learnt before. Can you describe it more clearly? Thanks for your attention and effort! – Golbez Sep 27 '13 at 05:32
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    Sure. An antisymmetric matrix used as a linear map takes a vector input and returns a vector in the plane that is orthogonal to the original vector's projection onto the plane. Whether the output vector is clockwise or counterclockwise from the input is dependent on the plane's innate "orientation" (in 3d, this is equivalent to a choice between two opposing normal vectors). This description is entirely equivalent to taking a normal vector and doing a cross product. – Muphrid Sep 27 '13 at 05:49
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If $v$ is a vector $= [v_x v_y v_z]^T$ in the frame described by versors $i, j, k$ then skew-symetric matrix $S(v)$ assigned to the vector $v$ is constructed as follows $$S(v) = \begin{pmatrix} v \times i \ \ v \times j \ \ v \times k \end{pmatrix}$$ (cross products of vector $v$ with versors)

From this construction geometric properties of skew-symmetric matrix follow, e.g. that columns of $S(v)$ are coplanar, vectors (columns) lie in the plane perpendicular to the vector $v$.

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