Cubic helical splines with Frenet-frame continuity

We present a scheme of interpolating a sequence of points p0,…,pnp0,…,pn in R3R3 by a piecewise-cubic-helical spline with continuous Frenet frame. Each cubic helical segment rk(ξ)rk(ξ) connecting pk−1pk−1 and pkpk is constructed such that its end tangents are symmetric with respect to the displacement vector Δpk=pk−pk−1Δpk=pk−pk−1. The existence condition of such a segment rk(ξ)rk(ξ) is formulated in terms of the configuration involving ΔpkΔpk and the Frenet frame of rk(ξ)rk(ξ) at pk−1pk−1. A Frenet-frame-continuous spline is obtained by matching the Frenet frames of sequentially constructed segments at their juncture points.

► We interpolate a sequence of points with cubic helical splines with Frenet-frame continuity.
► Each cubic helical segment has end tangents symmetric with respect to the displacement vector.
► The existence condition of such segments is formulated in terms of their initial Frenet frame and displacement vector.