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Polar Equations of Conics

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A conic section can be defined in polar coordinates as the set of all points whose distance from a fixed point, known as the focus, bears a constant ratio to their distance from a fixed line, known as the directrix. This constant ratio is called the eccentricity. This definition unifies all types of conic sections—ellipses, parabolas, and hyperbolas—under a single framework. When the focus is positioned at the origin of the polar coordinate system, a single polar equation can...

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Updated: Oct 29, 2025

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Computation of Atmospheric Concentrations of Molecular Clusters from ab initio Thermochemistry

Published on: April 8, 2020

Sequence-specific minimizers via polar sets.

Hongyu Zheng¹, Carl Kingsford¹, Guillaume Marçais¹

¹Computational Biology Department, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA.

Bioinformatics (Oxford, England)

|July 12, 2021

Summary

We introduce polar sets for sequence-specific minimizers, improving k-mer sampling from genomic data. These sets offer superior performance for reference genomes compared to existing methods.

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Area of Science:

Bioinformatics
Computational Biology
Genomics

Background:

Minimizers efficiently sample k-mers from genomic sequences, preserving matches.
Current methods focus on random sequences and universal hitting sets, lacking specialization for specific genomes.
Sequence-specific minimizers are less studied, with limited theoretical understanding and practical methods.

Purpose of the Study:

To address the limitations of existing minimizer construction methods for specific genomic sequences.
To develop a novel approach for constructing efficient sequence-specific minimizers.
To improve k-mer sampling and sketch size optimization for reference genomes.

Main Methods:

We propose polar sets as a novel concept, complementary to universal hitting sets.
Polar sets are designed to be spread out on a reference sequence, enabling specialization.
Link energy is introduced to quantify the spread of polar sets, allowing theoretical bounds on sketch size.

Main Results:

Polar sets provably specialize well to specific sequences, unlike universal hitting sets.
Link energy provides theoretically sound upper and lower bounds for sketch size, enabling direct optimization.
Experiments on the human reference genome demonstrate the practical superiority of polar sets for sequence-specific minimizers.

Conclusions:

Polar sets represent a significant advancement in designing efficient sequence-specific minimizers.
This approach offers improved theoretical understanding and practical performance for genomic sequence analysis.
The proposed methods and heuristics provide a powerful tool for optimizing k-mer sampling from reference genomes.