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I read a paper about robotics. It inform me 'To ensure constant plane curvature, the curvature and unit binormal vector of the curve must possess constant values as given in the following.'

I don't understand that the meaning of binormal vector is constant.

As far as I know, the binormal vector $B$ is a vector vertical to osculating plane which is configured of the tangent vector, $T$, and normal vector $N$ by $B=T\times N$.

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1 Answer

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Yes, and if $B$ is constant, the curve lies in a plane with that normal vector. The osculating plane never changes, and so the curve stays in that fixed plane. Note that if the curve is parametrized by $g(t)$, then indeed $g(t)\cdot B$ has derivative $0$ and is therefore constant.

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