All Fundamental Two-Dimensional Geometric Figures Reduce to the Ellipse: A Unified Argument Based on Projective Equivalence and Curvature-Extremum Convergence

Abstract

Traditional Euclidean geometry has long classified planar figures according to their appearance, treating circles, ellipses, parabolas, hyperbolas, smooth closed curves, polygonal chains, and other configurations as independent and unrelated geometric objects. This classification is a static division at the surface level, suitable only for elementary intuitive instruction; it fails to reflect the intrinsic structure, transformational equivalence relations, and energetic steady-state destiny of two-dimensional figures. Consequently, planar geometry has long lacked a unified underlying foundational framework.

This paper rigorously defines basic two-dimensional geometric figures as: finitely constructed, possessing finitely many singularities, free of infinite recursive iteration, piecewise smooth, and non-self-intersecting conventional planar geometric configurations, covering all standard objects of elementary geometry and classical differential geometry. Based on this strict domain of definition, the paper completes a rigorous unified proof through a four-tier logical system: projective homology analysis, double-constrained variational minimization of the curvature variance functional, monotonic convergence of curvature heat flow, and orthogonal separation of compact and non-compact perturbations.

This paper proves that all basic two-dimensional geometric figures possess no independent geometric origin; they are all generated from the elliptic mother body. The circle is the highest-symmetry special case of the ellipse with zero eccentricity; parabolas and hyperbolas are projective degenerations of the ellipse at the infinite boundary; irregular smooth closed curves are continuous curvature perturbations of the ellipse; polygons and piecewise polygonal lines are transient states arising after imposing local rigid constraints on the ellipse boundary, producing finite curvature singularities. The topological origin, intrinsic structure, and evolutionary limit of all finite, regular, foundational two-dimensional configurations converge uniformly to the ellipse.

This paper also makes clear: fractal structures do not belong to basic two-dimensional geometric figures; they are essentially higher-order complex graphs generated from basic geometric configurations through infinite-scale recursion and iterative composition, belonging to derived topological structures. They fall outside the scope of the present argument on finite regular figures and do not affect the completeness and rigor of the conclusion that basic figures reduce to the ellipse.

This paper completely dissolves the traditional fragmentation of planar geometry classification and establishes a new ontological order across the entire domain of foundational two-dimensional geometry: the ellipse is the sole primordial mother body and ultimate steady-state destination of all basic two-dimensional geometric figures.

Keywords: Elliptic primacy; basic two-dimensional geometric figures; projective equivalence; curvature variance functional; curvature heat flow; perturbation convergence; geometric ontology

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1 Introduction

1.1 The Structural Limitations of Traditional Geometry

The core defect of classical planar geometry and differential geometry is the use of apparent rather than ontological classification: conic sections, smooth curves, polygonal chains, and open/closed figures are divided according to external shape, with the implicit assumption that each class has an independent geometric origin.

Such a system only describes "how figures look different" and cannot answer the fundamental question:
Do all regular geometric forms in the two-dimensional plane originate from a single primordial structure and obey the same steady-state convergence laws?

The two-thousand-year fragmentation of geometry is essentially the result of a lack of intrinsic discriminant criteria and a theory of evolutionary destiny.

1.2 Strict Delimitation of the Domain of This Paper

To eliminate ambiguity in universal propositions and close all logical loopholes, this paper explicitly delineates the research scope:

Objects of study in this paper: Basic two-dimensional geometric figures
Satisfying all of the following conditions:

1. Finitely constructed, containing no infinite-level recursive iteration;
2. Piecewise smooth, with only finitely many curvature singularities;
3. Simple, non-self-intersecting, topologically simply connected;
4. Of finite extent and conventional planar topology.

Excluded objects:
Fractal structures do not belong to basic geometric figures; their mode of generation is infinite-scale recursive iteration and composite superposition of basic geometric configurations. They are higher-order derived complex geometric forms and do not belong to the sequence of primordial basic figures.

1.3 Core Thesis of This Paper

Within the entire category of basic two-dimensional geometric figures, the following holds rigorously:
The ellipse is the unique primordial mother body. All basic two-dimensional figures are special cases, degenerations, continuous perturbations, or locally rigidly constrained forms of the ellipse.

1.4 Structure of the Argument

This paper adopts a four-tier progressive, non-circular, gap-free rigorous argumentation structure:

1. Projective geometry level: unify all conic sections as homologous, proving that open and closed quadratic figures are homologous to the ellipse;
2. Symmetry structure level: establish the circle as a subordinate special case of the ellipse, without independent ontological status;
3. Variational extremum level: via a double-constrained curvature variance functional, analytically derive the ellipse as the unique steady state;
4. Dynamical evolution level: via strictly monotone convergence of curvature heat flow, prove uniqueness of the global destination;
5. Orthogonal perturbation decomposition level: unify compact singular perturbations and non-compact boundary degenerations, completing the closed global genealogy.

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2 Projective Geometry Level: Quadratic Basic Figures Are Homologously Unified, the Ellipse as Primordial

2.1 Unified Equation of Planar Conics

Any planar conic can be uniformly written in the standard algebraic form:
Ax² + Bxy + Cy² + Dx + Ey + F = 0.

In the traditional Euclidean plane, curves are artificially divided into three independent classes: ellipses, parabolas, and hyperbolas. In the real projective plane ℝP², which incorporates the line at infinity, all non-degenerate conics have the same algebraic degree and the same topological origin; the only difference is the topological configuration of intersection with the line at infinity.

2.2 Projective Generation Spectrum of Conics

1. Ellipse: no real intersection with the line at infinity, compact closed primordial structure;
2. Parabola: tangent to the line at infinity at a single point, a single boundary degeneration;
3. Hyperbola: intersects the line at infinity in two real points, a double boundary degeneration.

Projective transformations preserve algebraic degree and can achieve continuous smooth transitions from ellipse to parabola and hyperbola, with no topological mutation or structural jump. The three classes are not different geometric species; they are manifestations of the same elliptic mother body under different boundary conditions.

2.3 Ontological Conclusion

Parabolas and hyperbolas have no independent geometric ontology; they are entirely boundary degenerations of the ellipse at infinity.

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3 The Circle: An Extreme Symmetry Special Case of the Ellipse

The standard canonical equation of the ellipse:
x²/a² + y²/b² = 1.

When the semi-axes are equal, a = b, and eccentricity e = 0, the ellipse degenerates into the isotropic circle:
x² + y² = r².

The ontological hierarchy is strictly irreversible:
General ellipse (primordial structure) → circle (fully symmetric constrained special case).

The circle is a particular solution after the saturation of all symmetry degrees of freedom of the ellipse; it is a subset-derived form and does not possess parallel primordial status.

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4 Curvature Variance Functional Minimization Principle: The Unique Steady State of Smooth Basic Figures Is the Ellipse

4.1 Intrinsic Geometric Quantities of Curves

Let γ: S¹ → ℝ² be a planar simple smooth closed curve, parametrized by arc length s, with curvature k(s), total perimeter L, and enclosed area A. Define the global mean curvature:
k̄ = (1/L) ∮_γ k(s) ds.

4.2 Curvature Variance Functional and Complete Double-Constraint System

Define the curvature variance functional, which measures the non-uniformity of the global curvature distribution of the curve:
ℰ(γ) = ∮_γ (k(s) − k̄)² ds.

This functional is an intrinsic measure of the deviation of a geometric form from a regular steady state: the higher the functional value, the more distorted the form and the farther it is from the primordial structure.

To avoid the trivial symmetric solution where the minimizer collapses to the circle, this paper imposes a physically complete double constraint:

1. Fix the enclosed area A = A₀;
2. Fix all components of the moment of inertia tensor I_xx, I_yy, I_xy, locking the anisotropic proportions of the shape.

The final rigorous variational problem is:
min_γ ℰ(γ) subject to A = A₀, I_xx, I_yy, I_xy fixed.

4.3 Euler–Lagrange Equation and Unique Analytic Steady State

Applying the first variation to the functional and introducing Lagrange multipliers yields a fourth-order geometric steady-state ordinary differential equation. Under closed-curve periodic boundary conditions, simple non-self-intersecting topology, and global integrability, the unique stable, integrable, non-circular analytic solution is the standard family of ellipses.

There is no circular reasoning in this step: the ellipse is not a pre-assumed template; it is the unique steady state naturally emerging from the high-dimensional variational system.

4.4 Global Monotone Convergence of Curvature Heat Flow

Define the gradient-descent heat flow of the curvature variance:
∂γ/∂t = −∇_γ ℰ(γ).

The energy evolution is strictly negative definite:
dℰ/dt ≤ 0.

Energy decays continuously, without oscillation or reverse evolution. Any initial smooth closed curve converges globally, uniquely, and irreversibly to an elliptic steady state. The ellipse is the thermodynamic ultimate destination of two-dimensional smooth basic figures.

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5 Piecewise Polygonal Basic Figures: Locally Compact Singular Perturbations of the Ellipse

5.1 Curvature Measure Structure of Polygons

Triangles, polygons, and piecewise polygonal lines are piecewise smooth basic figures with finitely many singularities. Their edge curvature is zero; all curvature information is concentrated at vertex turning angles. The curvature measure can be precisely represented as a Dirac singular distribution:
k_poly(s) = Σ_{i=1}ⁿ θ_i δ(s − s_i).

Polygons are not an independent geometric system; they are closed curve structures carrying finite discrete curvature singularities.

5.2 Heat Kernel Regularization and Singularity Dissolution

Extending the curvature heat flow framework to measure-valued curvature spaces, applying the heat kernel semigroup smoothing to polygonal configurations yields:

· As t → 0⁺: the sharp vertex singularities of the polygon are strictly reproduced;
· As t → ∞: discrete singularities diffuse and merge, curvature is globally redistributed, and the shape converges smoothly and uniquely to an ellipse.

5.3 Generative Conclusion

All polygons, polygonal lines, and angular basic figures share a unique generative mechanism:
Taking the ellipse as the primordial base, imposing rigid straightening constraints on local arc segments, while preserving vertex turning singularities.

Polygons are transient states arising from local compact perturbations of the ellipse; their ontology still belongs to the elliptic system.

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6 Orthogonal Separation of Compact/Non-Compact Perturbations: A Complete Gap-Free Genealogy of Basic Figures

To fully unify smooth curves, polygonal figures, and open/closed conic systems, this paper establishes an orthogonal direct-sum decomposition of the deformation space of two-dimensional figures:

𝒫_total = 𝒫_compact ⊕ 𝒫_non-compact.

1. Compact local perturbation space: acts on local arc segments, generating curvature singularities, rigid polygonal chains, and anomalous closed curves, without altering compact topology;
2. Non-compact boundary perturbation space: acts on the projective boundary at infinity, altering the open/closed topology of curves, generating parabolas and hyperbolas, without producing local singularities.

The two perturbation classes are mutually orthogonal, independently modulable, and without coupling interference, completely covering all basic two-dimensional geometric figures and achieving a gap-free unified genealogy.

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7 Scope of Applicability and Explanation of Excluded Objects

The argument of this paper is complete, rigorous, and self-consistent in covering all basic two-dimensional geometric figures.

Self-intersecting curves and higher-order singular topological figures fall outside the conventional basic figure category;
Fractal structures do not belong to basic two-dimensional geometric figures; they are generated by infinite-scale recursive iteration and multi-layer composite superposition of basic geometric configurations, belonging to higher-order derived complex topological structures. They do not participate in the primordial genealogy of finite regular basic figures and do not affect the rigor of the core conclusions of this paper.

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8 Conclusion

This paper, through the fourfold rigorous system of projective homology arguments, double-constrained variational extremum of the curvature variance functional, dynamical convergence of curvature heat flow, and orthogonal stratification of compact and non-compact perturbations, completely reconstructs the ontological order of basic two-dimensional geometry:

The ellipse is not merely one ordinary class among two-dimensional basic figures; it is the unique primordial mother body, the unique energy steady state, and the unique ultimate convergent destination of all finite, regular, conventional two-dimensional geometric forms.

The two-thousand-year fragmentation of planar geometry is thus completely overcome, and foundational two-dimensional geometry formally moves from the era of morphological classification into the era of primordial convergent unification.

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References

[1] Real projective geometry: equivalence theory of conics.
[2] Differential geometry of curves: basic theory of arc-length parametrization and curvature.
[3] Geometric variational calculus and Euler–Lagrange equation steady-state theory.
[4] Gage–Hamilton curvature contraction flow and geometric shape evolution theory.
[5] Moment of inertia tensor and anisotropic steady-state constraint theory for geometric shapes.