Abstract
We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.
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Appendix: : Proof of Lemma 5.1
Appendix: : Proof of Lemma 5.1
Validity of the lemma is supported by the two plots in Fig. ??. However, they show merely the evaluation on a finite grid using standard double precision arithmetic, and do not constitute a credible proof. While an analytic treatment is cumbersome, interval arithmetic provides a convenient way to settle the issue in a numerical, though rigorous way. Here, we used Intlab [38] to verify the claim for bounds γ1 = .9 and γ2 = .999. Smaller constants are possible, but lead to quite long computation times without providing deeper insight.
We start with considering the ratio Γ1 according to (??). By symmetry, it satisfies
We employ polar coordinates \((\beta _{0},\beta _{1}) =(r {\cos \limits } \phi , r {\sin \limits } \phi )\) to represent the parameter vector β = (β0,β1). By the above symmetry, it suffices to verify
to cover the whole domain D = {(β0,β1) : r ≤ π/2}. To this end, we partition \(D_{1}^{*}\) into a uniform rectangular grid of 9 × 9 subcells. On each of these subcells, a rigorous upper bound of the function \({\Gamma }_{1}^{*}\) is determined using Intlab, where the range of the integrals appearing in the formula for S1 is enclosed by upper and lower sums corresponding to a partition of the domain of integration into 200 equal pieces. Figure 12 shows a piecewise constant upper bound on the function \({\Gamma }_{1}^{*}\) over the domain \(D_{1}^{*}\) based on this computation. The maximal value .872 is less than γ1, what verifies the claim for the shrinkage of secant lengths.
Establishing the contraction of angles is more involved since the ratio Γ2(β0,β1) is not defined for β0 = β1 = 0. To settle this issue, we distinguish two cases. First, we consider pairs of angles βJ in the annulus A2 := {βJ : π/4 ≤ r ≤ π/2}, thus staying away from the singularity at the origin. Γ2 has the same symmetry properties as Γ1. Hence, as before, we have to show that
Evaluation on a uniform grid is posssible, but not efficient since it would have to be quite fine. Instead, we implemented a recursive algorithm that computes an upper bound on a given rectangle. If it is less than γ2, the rectangle is accepted. Otherwise, it is split into four equal parts, and the procedure is repeated for all of them. Starting from \(A^{*}_{2}\), the algorithm terminates, and Fig. 13(right) shows the resulting partition of the domain \(A^{*}_{2}\), consisting of 2011 rectangles.
The remaining case of angles βJ in the disk D2 := {βJ : r ≤ π/4} is treated as follows: We want to show that the function
is nonnegative. A direct verification by means of interval arithmetic is not possible since the true range is necessarily over-estimated, and Δ(0) = 0. Instead, we set Δ∗(r,φ) := Δ(βJ) and consider
Unlike Δ∗ and \(\partial _{r} {\Delta }^{*}\), this function is strictly positive, and indeed, a recursive Intlab algorithm analogous to the one described above confirms that \({\partial _{r}^{2}} {\Delta }^{*}(r,\varphi ) > 0\) on \(D_{2}^{*}\). Figure 12(left) shows the resulting partition of the domain, consisting of 616 rectangles.
Using \({\Delta }^{*}(0,\varphi ) = \partial _{r} {\Delta }^{*}(0,\varphi ) = 0\), we integrate twice to find
for \((r,\varphi ) \in D_{2}^{*}\). Hence, Δ(βJ) ≥ 0 for pairs of angles in the right half plane, i.e., βJ ∈{(β0,β1) ∈ D2 : β0 ≥ 0}. However, by symmetry, Δ(−βJ) = Δ(βJ) so that Δ(βJ) ≥ 0 for βJ ∈ D2. Consequently,
Concerning \(\beta ^{\prime }_{1,J}\), we flip the angles β0,β1 and set \(\tilde \beta :=(\beta _{1},\beta _{0})\). Again by symmetry,
This yields
and the proof is complete.
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Reif, U., Weinmann, A. Clothoid fitting and geometric Hermite subdivision. Adv Comput Math 47, 50 (2021). https://doi.org/10.1007/s10444-021-09876-5
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DOI: https://doi.org/10.1007/s10444-021-09876-5
Keywords
- Geometric Hermite subdivision
- Non-linear subdivision
- Circle-preserving scheme
- Clothoid fitting
- 2D curve design