Great-X: Great Darboux Cyclides

• Great-X are a subclass of degree-4 Darboux cyclides in real Möbius geometry, characterized by preimages in S³ covered by great circles.
• They are classified by their Möbius-degree and circle count, revealing distinct structures such as translational and Cliffordian types.
• Explicit parameterizations via the Hamilton product clarify their combinatorial and algebraic properties, aiding in visualizing circle-bearing algebraic surfaces.

Great-X refers to a subclass of Darboux cyclides—“great Darboux cyclides”—within the broader theory of real Möbius geometry, specifically encompassing those degree-4 (Möbius-degree $d=4$) algebraic surfaces in $\mathbb{R}^3$ for which the inverse image under stereographic projection from $S^3$ is covered by great circles (i.e., by intersections with $2$-planes through the origin in $\mathbb{R}^4$) (Lubbes, 2013). These surfaces, together with their explicit classification, constructions, and parameterizations, are central to the study of circle-bearing algebraic surfaces and their translation-theoretic and combinatorial properties.

1. Möbius Geometry and Foundational Definitions

Real Möbius geometry identifies $\mathbb{R}^3$ with an affine chart of the quadric $Q^3 \subset \mathbb{P}^4$ defined by

$$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$

with stereographic projection $\mu : S^3 \to \mathbb{R}^3$ taken from the point $(0:0:0:0:1)$. For a real algebraic surface $\mathbb{R}^3$0, its Möbius-degree $\mathbb{R}^3$1 is $\mathbb{R}^3$2, and the surface is termed $\mathbb{R}^3$3–circled if $\mathbb{R}^3$4 in $\mathbb{R}^3$5 contains at least $\mathbb{R}^3$6 (but not $\mathbb{R}^3$7) distinct circles through a general point. When $\mathbb{R}^3$8, $\mathbb{R}^3$9 is described as celestial.

Darboux cyclides, a principal object of study, are precisely those celestial surfaces with $S^3$0. A Darboux cyclide admits two quadratic equations in the $S^3$1 (one being $S^3$2) and, up to Möbius transformation, may be classified according to the multiplicity and structure of their families of circles. Named cases include Q-cyclides (quadric surfaces), CY ("horn"), EY ("spindle"), CH1 cyclides (Möbius images of circular one-sheeted hyperboloids), ring cyclides, Perseus cyclides, and Blum cyclides, corresponding to varying circle count through a general point.

2. Great Darboux Cyclides: Definition and Classification

A great Darboux cyclide is characterized by the property that its preimage under stereographic projection, $S^3$3, is covered by great circles. Such cyclides are classified, according to their Möbius-degree $S^3$4 and the number $S^3$5 of circles through a general point, as follows (Theorem 1, Table 1 in (Lubbes, 2013)):

$S^3$6	Surface Type	Noteworthy Properties
(2, $S^3$7)	Plane/Sphere	Bohemian and great
(4, 6)	Blum cyclide	Great, neither translational nor Cliffordian
(4, 5)	Perseus cyclide	Cliffordian and great
(4, 4)	Ring cyclide	Cliffordian and great (stereographic Clifford torus)
(4, 3)	CH1, EY, EO	CH1: Cliffordian; EY: Bohemian; EO: great
(4, 2)	CY, CO	CY: Bohemian; CO: great

Crucially, no Darboux cyclide of $S^3$8 is simultaneously Bohemian (a sum of circles), and Cliffordian (a Hamilton product of circles in $S^3$9), reflecting the combinatorial rigidity of these classes (Corollary 1). Among great Darboux cyclides, every great ring cyclide is Cliffordian, whereas a great Perseus cyclide is not Cliffordian.

3. Translational and Cliffordian Structures

Translational surfaces are defined for circles $2$0 in $2$1 (or $2$2) with

$2$3

$2$4

where "$2$5" denotes the quaternionic Hamilton product. The Zariski closure of $2$6 yields a Bohemian surface; the Zariski closure of $2$7 gives a Cliffordian surface. Within degree-4 cyclides, Bohemian Darboux cyclides are exactly the plane, CY, and EY families; Cliffordian Darboux cyclides are exactly the ring, Perseus, and CH1 families. The ring cyclide specifically admits a Cliffordian parameterization $2$8 by two great circles $2$9.

It is established that a translational surface carries at most five circles through a general point. The sum of two circles in $\mathbb{R}^4$0 produces a degree-4 surface with at least two circles through a general point only if one summand is a line (the CY or EY case). Otherwise, the degree rises to eight.

4. Circle Combinatorics and Intersection Theory

Desingularizing $\mathbb{R}^4$1 to a weak del Pezzo surface $\mathbb{R}^4$2 (anticanonical degree $\mathbb{R}^4$3) facilitates a divisor-theoretic classification. The Néron–Severi lattice $\mathbb{R}^4$4 possesses:

• $\mathbb{R}^4$5: $\mathbb{R}^4$6-curves contracted to singularities,
• $\mathbb{R}^4$7: classes of irreducible conics (circles),
• $\mathbb{R}^4$8: classes of lines.

Key correspondences include: singularities of $\mathbb{R}^4$9 relating to $\mathbb{R}^3$0 (usually of Dynkin type $\mathbb{R}^3$1, $\mathbb{R}^3$2, $\mathbb{R}^3$3), pencils of circles with isotropic classes $\mathbb{R}^3$4 ($\mathbb{R}^3$5), and incidence relations computed using the intersection form on $\mathbb{R}^3$6. Circle–line graphs enumerate all possible $\mathbb{R}^3$7–invariant) configurations, recovering 14 types of Darboux cyclides and two degree-8 exceptions. A Cliffordian structure exists if there is a “Clifford quartet” among the line classes whose intersection pattern matches the $\mathbb{R}^3$8 rulings on $\mathbb{R}^3$9; Bohemian structure exists when a pencil of circles has two complex-conjugate base points on the Euclidean absolute $Q^3 \subset \mathbb{P}^4$0 in $Q^3 \subset \mathbb{P}^4$1 (see Proposition C in (Lubbes, 2013)).

5. Circle Families, Quadrics, and Signature Analysis

Any surface in $Q^3 \subset \mathbb{P}^4$2, $Q^3 \subset \mathbb{P}^4$3, containing two great circles through a general point and not contained in a hyperplane must satisfy $Q^3 \subset \mathbb{P}^4$4; the possible $Q^3 \subset \mathbb{P}^4$5 values (number of great circles through a point) are $Q^3 \subset \mathbb{P}^4$6, $Q^3 \subset \mathbb{P}^4$7, $Q^3 \subset \mathbb{P}^4$8, $Q^3 \subset \mathbb{P}^4$9, or $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$0 (Corollary 3). The Blum, Perseus, and ring cyclides realize the cases $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$1 respectively, with planes and spheres attaining $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$2.

Signature analysis of the $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$3 symmetric form defining the cone in $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$4 (together with the quadric $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$5) results in Möbius-equivalence to elliptic, hyperbolic, or Euclidean cyclides according to the conic’s type:

• Elliptic: $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$6-cone over a smooth ruled quadric,
• Hyperbolic: $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$7-cone,
• Euclidean: $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$8-cone after moving a base point to infinity.

Multiple families of great circles correspond algebraically to isotropic classes $$Q^3 = \{ x = (x_0:x_1:x_2:x_3:x_4) \mid -x_0^2 + x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0 \}$$9 with $\mu : S^3 \to \mathbb{R}^3$0 in $\mu : S^3 \to \mathbb{R}^3$1. The interaction pattern of these classes is tightly constrained by the weak del Pezzo structure; carrying infinitely many great circles is only possible for planes and spheres.

6. Explicit Parameterizations and Examples

Nine explicit families of circles $\mu : S^3 \to \mathbb{R}^3$2 (with only $\mu : S^3 \to \mathbb{R}^3$3 being great) serve as the basis for parameterizations (see Table Ci in (Lubbes, 2013)). Using the Hamilton product,

• $\mu : S^3 \to \mathbb{R}^3$4 (closure of $\mu : S^3 \to \mathbb{R}^3$5): ring cyclide ($\mu : S^3 \to \mathbb{R}^3$6, $\mu : S^3 \to \mathbb{R}^3$7),
• $\mu : S^3 \to \mathbb{R}^3$8: Perseus cyclide ($\mu : S^3 \to \mathbb{R}^3$9, $(0:0:0:0:1)$0),
• $(0:0:0:0:1)$1: CH1 cyclide ($(0:0:0:0:1)$2, $(0:0:0:0:1)$3),
• $(0:0:0:0:1)$4: Cliffordian degree-8 surfaces ($(0:0:0:0:1)$5).

It is demonstrated that surfaces constructed as the sum or Hamilton product of circles bear at most five circles through a general point, except in the specialized ring, Perseus, or Blum cases for Möbius-degree four.

7. Geometric and Algebraic Significance

Great Darboux cyclides fundamentally mediate the interaction between Möbius geometry, real algebraic geometry, and the combinatorics of conic curves. Their explicit classification demonstrates rigidity: The possible configurations of circles through a general point are tightly limited, forcing a finite set of geometrically distinct surfaces. The interplay between real and complex structures (base points, circle pencils, Clifford quartets) encodes deep information in the Néron–Severi lattice, and their parameterizations afford concrete visualization within $(0:0:0:0:1)$6 and $(0:0:0:0:1)$7. For surfaces in $(0:0:0:0:1)$8 with two great circles through each point, only $(0:0:0:0:1)$9 is permissible, and—outside the plane/sphere—only specific cyclides are realized, highlighting the exceptional nature of great-X within the circle geometry landscape (Lubbes, 2013).