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Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface

  • Steven Finch
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Abstract

Let Ω be the curvilinear square bounded by the four circular arcs |ζ∓1|=√2, |ζ∓ i|=√2, and let W=√(1+14ζ4+ζ8). Put θ(ζ)=arcsin(2(1+i)ζ)/(√(1+4iζ2−ζ4)), f=1/4(−iF[θ,1/4]+F[θ,3/4]), g=1/4( iF[θ,1/4]+F[θ,3/4]), h=(2−√3) F[arcsin(i(2+√3)ζ2),(2−√3)4]. These are, verbatim, the three functions of our companion paper [D] on the diamond surface D: the two papers start from the same point, and differ only in which real parts are taken. With X=(κ Ref, κ Reg, 1/2+κ Imh) and κ=3/(2K[1/9]), the map X parametrizes a fundamental patch of Schwarz's primitive surface P, the conjugate of D. Part I builds the algebraic apparatus … the normalizing constant is identified exactly: ϖ:=κ−1=K[−3]=1/2 K[3/4]=2/3 K[1/9]. Part II carries out the elimination and completes the proof of (⋆) sn(ϖ(x+y),−3) sn(ϖ(x−y),−3) =1/3sn(3ϖ(z−1/2),1/9). Part III shows that (⋆) is, after all, additively separable … for an overview of this paper and [D], please see arXiv:2609.14206.

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Cite

BibTeXRIS
BibTeX
@misc{finch2026elimination,
  title = {{Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface}},
  author = {Steven Finch},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.22313},
  eprint = {2609.22313},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.22313},
}
RIS
TY  - UNPB
TI  - Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface
AU  - Steven Finch
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.22313
UR  - https://arxiv.org/abs/2609.22313
AB  - Let Ω be the curvilinear square bounded by the four circular arcs |ζ∓1|=√2, |ζ∓ i|=√2, and let W=√(1+14ζ^4+ζ^8). Put θ(ζ)=arcsin(2(1+i)ζ)/(√(1+4iζ^2−ζ^4)), f=1/4(−iF[θ,1/4]+F[θ,3/4]), g=1/4( iF[θ,1/4]+F[θ,3/4]), h=(2−√3) F[arcsin(i(2+√3)ζ^2),(2−√3)^4]. These are, verbatim, the three functions of our companion paper [D] on the diamond surface D: the two papers start from the same point, and differ only in which real parts are taken. With X=(κ Ref, κ Reg, 1/2+κ Imh) and κ=3/(2K[1/9]), the map X parametrizes a fundamental patch of Schwarz's primitive surface P, the conjugate of D. Part I builds the algebraic apparatus … the normalizing constant is identified exactly: ϖ:=κ^−1=K[−3]=1/2 K[3/4]=2/3 K[1/9]. Part II carries out the elimination and completes the proof of (⋆) sn(ϖ(x+y),−3) sn(ϖ(x−y),−3) =1/3sn(3ϖ(z−1/2),1/9). Part III shows that (⋆) is, after all, additively separable … for an overview of this paper and [D], please see arXiv:2609.14206.
ER  -