PreprintCC BY 4.0arXiv
Point-to-ellipse series -- Part II Expansion about the center
Abstract
Power series expansions in the polar distance about the center are derived for the normal direction and signed distance from an ellipse to points near its center. After extracting constant terms and common factors, the series coefficients are polynomials in the axis ratio and the absolute value of the sine of the polar angle, with rational coefficients. The region of convergence is shown to be the region bounded by the ellipse evolute. The expansions complement previously derived expansions valid for points far from the center.
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BibTeX
@misc{nilsson2026pointtoellipse,
title = {{Point-to-ellipse series -- Part II Expansion about the center}},
author = {John-Olof Nilsson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40690},
eprint = {2610.40690},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40690},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Point-to-ellipse series -- Part II Expansion about the center
AU - John-Olof Nilsson
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.40690
UR - https://arxiv.org/abs/2610.40690
AB - Power series expansions in the polar distance about the center are derived for the normal direction and signed distance from an ellipse to points near its center. After extracting constant terms and common factors, the series coefficients are polynomials in the axis ratio and the absolute value of the sine of the polar angle, with rational coefficients. The region of convergence is shown to be the region bounded by the ellipse evolute. The expansions complement previously derived expansions valid for points far from the center.
ER -