Oriented Circles & Inversive Invariants

• Oriented circles are circles with signed curvature that encode both geometric and algebraic properties, establishing a framework for invariance under Möbius transformations.
• Inversive invariants capture key relationships—such as intersection angles and distance–curvature measures—that facilitate precise solutions to classical problems like the Apollonius construction.
• Dynamic invariants seen in bicentric polygons and conformal differential geometry illustrate how the framework unifies inversion, Möbius, and higher-dimensional conformal analyses.

Oriented circles constitute a framework for encoding both the geometric and algebraic features of circles in the plane or higher-dimensional spaces, with additional structure reflecting their orientation (such as the sign of curvature or interior/exterior designation). Inversive invariants characterize pairwise and multiwise relationships among oriented circles under Möbius transformations and inversions, underpinning modern approaches to classical problems (e.g., Apollonius’ problem, Poncelet porisms, conformal differential geometry), and facilitating the development of dynamic metric invariants. This account surveys foundational definitions, construction of key invariants, algebraic encoding and solution of geometric problems, and links to advanced geometric structures.

1. Algebraic Models for Oriented Circles

An oriented circle in the Euclidean plane is specified by its signed curvature $k_0\in\mathbb{R}$ (with $k_0=0$ corresponding to a line), a base point $(x_0,y_0)$, and a tangent direction $\tau_0$ at that point. This data is captured via the quadratic form

$$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$

constrained by $b^2 + c^2 - a d = 1$ and with parameters $(a, b, c, d)$ determined by

$$(a,b,c,d) = \left( k_0,\;-k_0 x_0 + \sin\tau_0,\;-k_0 y_0 - \cos\tau_0,\;k_0(x_0^2 + y_0^2) - 2x_0\sin\tau_0 + 2y_0\cos\tau_0 \right).$$

The orientation (sign of $k_0$) distinguishes “interior” from “exterior” with respect to the circle. Analogous models generalize to higher-dimensional subspheres via Minkowski space: in $\mathbb{R}^5_1$, oriented $k_0=0$0-spheres correspond to points on the de Sitter quadric, and oriented circles correspond to planes or pure $k_0=0$1-vectors in $k_0=0$2 (Langevin et al., 2011).

2. Inversive Invariants and Möbius Geometry

A primary feature of oriented circle frameworks is invariance under Möbius transformations and inversions. The quadratic forms $k_0=0$3 transform up to scale, preserving relative relationships. The principal real-valued invariants for pairs of circles/lines include:

• Intersection angle invariant. For oriented circles of curvatures $k_0=0$4 and centers separated by $k_0=0$5, the intersection angle $k_0=0$6 satisfies

$k_0=0$7

• Distance–curvature invariant. Between a line ($k_0=0$8) and a circle ($k_0=0$9) with signed distance $(x_0,y_0)$0,

$(x_0,y_0)$1

• Homogeneous cross-form. For normalized representations $(x_0,y_0)$2,

$(x_0,y_0)$3

Tangent circles correspond to $(x_0,y_0)$4, counter-tangency to $(x_0,y_0)$5. The algebraic relationships among $(x_0,y_0)$6-values drive multiplicity and configuration in problems such as Apollonius’ construction (Kurnosenko, 9 Jan 2026).

3. Solving Geometric Problems with Oriented Circles

The Apollonius problem in terms of oriented circles seeks all circles tangent (with matching orientation) to three given oriented circles $(x_0,y_0)$7, $(x_0,y_0)$8. Tangency in this context entails identical tangent vectors at the contact point. The problem reduces to solving for a fourth quadratic form $(x_0,y_0)$9 satisfying three algebraic tangency equations or, equivalently, $\tau_0$0 for $\tau_0$1.

Classification proceeds via the triple of pairwise invariants $\tau_0$2, leading to the discriminant

$\tau_0$3

(where the sum is cyclic over $\tau_0$4). If $\tau_0$5 and $\tau_0$6, exactly two (oriented) tangent circles exist; otherwise, real solutions may not exist, or special configurations yield a pencil of circles. The full set of solutions to the classical (non-oriented) problem is produced by reversing the orientation of data circles, systematically permuting $\tau_0$7, and $\tau_0$8 (Kurnosenko, 9 Jan 2026).

The constructive formula for the two (oriented) solutions expresses the quadratic form parameters explicitly via minors $\tau_0$9–$$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$0 of the data matrix and auxiliary polynomial invariants $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$1, as detailed above.

4. Dynamic Invariants in Bicentric and Billiard Polygons

The oriented circle formalism underpins the analysis of polygon families arising in classical geometry, such as the Poncelet–Jacobi bicentric $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$2-gons ($$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$3-sided polygons inscribed in one circle and tangent to another). For non-concentric, non-intersecting oriented circles $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$4, two limiting points $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$5 arise as centers for inversion transforming the pair into concentric circles. Poncelet’s porism guarantees a one-parameter family of such polygons, succinctly described using Jacobi elliptic functions and a closure parameter (Roitman et al., 2021).

Two key dynamic invariants within the bicentric $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$6-gon family:

• The sum of cosines of internal angles $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$7 is shown, via elliptic function residue calculus and Liouville’s theorem, to be constant across the family;
• The perimeter of the pedal polygon at either limiting point $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$8 or $$N(x, y; a, b, c, d) = a(x^2 + y^2) + 2(bx + cy) + d,$$9, defined by orthogonal projections onto sides, is also constant, given explicitly by elliptic integrals in the bicentric parameters.

Under projective duality and oriented inversion, bicentric polygons correspond to periodic orbits in the elliptic billiard, with pedal polygons or their inversive images retaining perimeter and sum-of-cosine invariance. This synthesis generalizes to hyperbolic billiards via inversion at $b^2 + c^2 - a d = 1$0 (Roitman et al., 2021).

5. Oriented Circles and Conformal Differential Geometry

Higher-dimensional analogs—particularly in $b^2 + c^2 - a d = 1$1 or on $b^2 + c^2 - a d = 1$2—utilize oriented circles and spheres encoded as points or planes in Minkowski space $b^2 + c^2 - a d = 1$3. The set of oriented $b^2 + c^2 - a d = 1$4-spheres is identified with the de Sitter quadric $b^2 + c^2 - a d = 1$5, while the Grassmannian of oriented circles $b^2 + c^2 - a d = 1$6 is realized via $b^2 + c^2 - a d = 1$7-vectors of unit norm, satisfying specific Plücker relations. Any Euclidean curve $b^2 + c^2 - a d = 1$8 is associated to its osculating circle or sphere, represented in wedge products.

The Möbius-invariant moving frame formalism defines the three classical conformal invariants of a space curve:

• Conformal arc-length $b^2 + c^2 - a d = 1$9 characterized by $(a, b, c, d)$0, relating to classical curvature and torsion;
• Conformal torsion $(a, b, c, d)$1 and
• Conformal curvature $(a, b, c, d)$2, given via derivative structure in the moving frame and partials of osculating sphere curves.

These invariants are equivariant under $(a, b, c, d)$3 and thus invariant under Möbius group action, providing a geometric and algebraic classification of curves up to conformal equivalence. Expansion into Möbius normal form yields explicit high-order Taylor coefficients in terms of conformal invariants. Furthermore, the characterization of canal surfaces (envelopes of sphere families) is determined by purity (Plücker relations) of the corresponding curve in the space of oriented circles (Langevin et al., 2011).

6. Implications and Unification in Inversive Geometry

The oriented circle framework, with its encoding via signed curvature and algebraic quadratic forms, unifies classical geometric constructions under inversion and Möbius group action. Dynamic invariants—such as sum of cosines and pedal-perimeter in polygonal families—extend the scope of traditional “static” invariants. Oriented inversion facilitates the construction of new poristic families and enables translation of metric invariants between disparate geometric contexts (e.g., bicentric polygons and billiard orbits).

This approach elucidates and streamlines classical constructions (such as the Apollonius and Poncelet-Jacobi problems), connecting them via fundamental invariants to conformal geometry, Grassmannian and Minkowski models, and broader theories of circle configurations. A plausible implication is that future research can exploit oriented circle algebra to diagnose and solve more complex geometric packing, tangency, and configuration problems, including those arising in moduli spaces or integrable systems.