Minimal-Area Ellipses Circumscribed About Convex Quadrilaterals: Generalizing Steiner's Theorem — E8 Intelligence Research

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For a convex quadrilateral, the minimal-area circumscribed ellipse is achieved when its center is the quadrilateral's centroid and its axes align with the affine image of the square, generalizing Steiner's theorem. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Andrew Stewart Caldin

doi:10.5281/zenodo.22701567

September 11, 2026Open Access

Minimal-Area Ellipses Circumscribed About Convex Quadrilaterals: Generalizing Steiner's Theorem — E8 Intelligence Research

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Key Points

To identify the geometric properties and minimal-area configurations of ellipses circumscribed about convex quadrilaterals by generalizing Steiner's theorem.
Applied affine transformations and algebraic geometry to characterize the family of ellipses circumscribing convex quadrilaterals.
Analyzed common conjugate directions and centroid alignments to determine minimal area and minimal eccentricity conditions.
Demonstrated that the circumscribed ellipse reaches minimal area when its center aligns with the quadrilateral centroid and its axes match the affine image of a square.
Showed that the minimal-eccentricity circumscribed ellipse degenerates to a circle if and only if the quadrilateral is cyclic, otherwise following the relationship e² = 1 − (b²/a²).

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Implication

Geometric analysis demonstrates minimal-area and minimal-eccentricity circumscribed ellipses around convex quadrilaterals, generalizing Steiner's theorem for broader geometric configurations.

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Cite This Study

Andrew Stewart Caldin (2026) studied this question.

synapsesocial.com/papers/6aa7b2f70926e14a848b18ea https://doi.org/10.5281/zenodo.22701567

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Minimal-Area Ellipses Circumscribed About Convex Quadrilaterals: Generalizing Steiner's Theorem — E8 Intelligence Research

Key Points

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Implication

Discussion

Cite This Study

Minimal-Area Ellipses Circumscribed About Convex Quadrilaterals: Generalizing Steiner's Theorem — E8 Intelligence Research

Key Points

No abstract yet

Implication

Discussion

Cite This Study

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