Translational and great Darboux cyclides

A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2, 3, 4, or 5 circles through a general point. A surface in a unit-sphere of any dimension that contains 2 great circles through a general point contains either 4, 5, 6, or infinitely many circles through a general point. These are some corollaries from our classification of translational and great Darboux cyclides. We use the combinatorics associated to the set of low degree curves on such surfaces modulo numerical equivalence.


Introduction
In this article we investigate surfaces in R 3 that contain at least two circles through each point. Such surfaces are algebraic by [22,Theorem 2] and thus with surface we shall mean a real irreducible algebraic surface (see §2).
Surfaces that are a union of circles in two different ways have applications in architecture [20], kinematics [13,17] and geometric modeling in general [1,11,19]. In particular, the "Darboux cyclides" have a long history [5,12], and its various properties are still a topic of recent research [2,8,18,21,27,29]. In order to clarify our main result and its relation to [25], we recall some definitions for the non-expert.
An inversion with respect to a sphere O ⊂ R 3 with center c and radius r is the Such a map exchanges the interior and exterior of O and takes generalized circles to generalized circles, where a generalized circle is either a circle or a line. We call two surfaces in R 3 Möbius equivalent if one surface is mapped to the other by a composition of inversions.
We may identify the unit-sphere S 3 ⊂ R 4 with the unit quaternions and we denote the Hamiltonian product by . We consider the following constructions where A and B are curves in R 3 or S 3 : Suppose that Z ⊂ R 3 is a surface. We call Z Bohemian or Cliffordian if there exists generalized circles A and B such that Z is the Zariski closure of A + B and π(A B), respectively. We remark that Clifford torus in S 3 is a surface of the form A B such that both A and B are great circles. A surface that is either Bohemian or Cliffordian is called translational.
A Darboux cyclide in R 3 is a surface of Möbius degree four. A Q cyclide is a Darboux cyclide that is Möbius equivalent to a quadric Q. For example, a CH1 cyclide is Möbius equivalent to a Circular Hyperboloid of 1 sheet (see Figure 1), where we used the following abbreviations: E = elliptic/ellipsoid P = parabolic/paraboloid O = cone C = circular H = hyperbolic/hyperboloid Y = cylinder A CO cyclide and CY cyclide is also known as a spindle cyclide and horn cyclide, respectively. A ring cyclide, Perseus cyclide or Blum cyclide is a Darboux cyclide without real singularities that is 4-circled, 5-circled and 6-circled, respectively (see Figure 1). See Table 2 for a complete list of names for celestial Darboux cyclides.
CY EY CH1 cyclide ring cyclide Persues cyclide We shall provide necessary conditions using the combinatorics of divisor classes of curves on such surfaces. This method also allows us restrict the possible candidates for the "great" celestial Darboux cyclides; we call a surface Z ⊂ R 3 great if its inverse stereographic projection π −1 (Z) is covered by great circular arcs.
We will use Theorem A and Theorem B and therefore build on [15]. a) The surface Z is Bohemian if and only if Z is either a plane, CY or EY.
b) If Z is Cliffordian, then Z is either a CH1 cyclide, Perseus cyclide or ring cyclide.
c) If Z is great and celestial, then Z is either a plane, sphere, ring cyclide, CO cyclide, EO cyclide or Blum cyclide.
We summarized Theorem 1 in Table 1. See Example 5, Example 6, Example 7, Example 8 and Example 9 for an example for each row and each possible type. See

A projective model for Möbius geometry
We define a real variety X to be a complex variety together with an antiholomorphic involution σ : X → X [24, Section I.1] and we denote its real points by Such varieties can always be defined by polynomials with real coefficients [23, Section 6.1]. Points, curves, surfaces and projective spaces P n are real algebraic varieties and maps between such varieties are compatible with σ unless explicitly stated otherwise. In particular, curves and surfaces in this article are by default reduced and irreducible. We assume that the real structure σ : P n → P n sends x to (x 0 : . . . : x n ).
The Möbius quadric is defined as the smooth hyperquadric The elliptic absolute is defined as the smooth quadric E := {x ∈ S 3 | x 0 = 0} and the Euclidean absolute is defined as the quadric cone U : Let ι : R n → P n be the embedding that sends (z 1 , . . . , z n ) to (1 : z 1 : . . . : z n ). The projective model P(Z) ⊂ P n of Z ⊂ R n is defined as the Zariski closure of ι(Z).
For a variety X ⊂ S 3 we define X(R) := γ(X R ). Notice that S 3 = P(S 3 ), S 3 (R) = S 3 , The Möbius transformations of S 3 are defined as the biregular maps Aut S 3 . These transformations are linear so that Aut S 3 ⊂ Aut P 4 . Since π −1 (R 3 ) is an isomorphic copy of R 3 inside S 3 (R), we find that an inversion R 3 R 3 corresponds uniquely to a Möbius transformation S 3 → S 3 .
The Euclidean similarities are defined as {ϕ ∈ Aut S 3 | ϕ(U) = U} and correspond to Euclidean similarities of  We shall refer to the complex lines in U as generators. Proof. Letπ : S 3 P 3 be the linear projection from the vertex in U(R). Notice thatπ is the projective closure of the stereographic projection π : S 3 R 3 . The is a plane, then µ(H) is parallel to H so that the planes P(H) and P(µ(H)) in P 3 intersect in a line L such that L ∩π(U) consists of complex conjugate points p and p.
Since H was chosen arbitrary, we find that α leavesπ(U) pointwise invariant. The Möbius transformation ϕ ∈ Aut S 3 such thatπ • ϕ = α •π is the corresponding Euclidean translation of S 3 . The generators of U are mapped byπ to the complex points ofπ(U) and thus are left invariant by ϕ as was to be shown.

Associated pencils and absolutes
In §2 we considered elliptic and Euclidean geometry as subgroups of the Möbius transformations that preserve some fixed hyperplane section of the Möbius quadric.
We characterize the intersection of Cliffordian and Bohemian surfaces with such hyperplane sections.
A pencil on a surface X ⊂ S 3 is defined as an irreducible hypersurface F ⊂ X × P 1 such that the π 1 (F ) = X and π 2 (F ) = P 1 where π 1 and π 2 are the projections to the factors of X × P 1 . The curve F i := π 1 (F ∩ X × {i}) ⊂ X for i ∈ P 1 is called a member. The complex points in the intersection ∩ i∈P 1 F i are called base points of the pencil. We call F a pencil of circles if F i is a circle for almost all i ∈ P 1 R .
Suppose that A, B ⊂ R 3 are circles such that A + B is a surface. We observe   b) If the right associated pencil F ⊂ P(A B) × P(B) has no base points on E, Proof. a) For almost all points i, j ∈ P(A) the members F i and F j are circles and these circles are related by a left Clifford translation. We know from Bézout's theorem that F i intersects E in two complex conjugate points. It follows from Proposition 1 that the complex conjugate left generators L, L ⊂ E that pass through these points are left invariant. As F is Zariski closed and without base points on E we conclude that L, L ⊂ P(A B). The proof for assertion b) is analoguous.

Divisor classes of curves on Darboux cyclides
We recall from [15] the possible sets of divisor classes of complex low degree curves on Darboux cyclides in S 3 (see Definition 1). Each entry in this classification translates into a diagram that visualizes how complex lines, complex isolated singularities and circles intersect.
The smooth model of a surface X ⊂ P n is a birational morphism ϕ : Y → X from a nonsingular surface Y , that does not contract complex (−1)-curves. See [10, Theorem 2.16] for the existence and uniqueness of the smooth model.
The Néron-Severi lattice N (X) is an additive group defined by the divisor classes on Y up to numerical equivalence. This group comes with an unimodular intersection product · and a unimodular involution σ * : N (X) → N (X) induced by the real structure σ : X → X. We denote by Aut N (X) the group automorphisms that are compatible with both · and σ * . The class that are not contracted to complex points by the morphism ϕ.
We consider the following subsets of N (X): • B(X) denotes the set of divisor classes of complex irreducible curves C ⊂ Y such that ϕ(C) is a complex point in X, • G(X) denotes the set of classes of complex irreducible conics in X, and • E(X) denotes the set of classes of complex lines in X.
We call W ⊂ B(X) a component if it defines a maximal connected subgraph of the graph with vertex set B(X) and edge set The following proposition is an application of intersection theory on surfaces (see . For its proof we assume some background in algebraic geometry, but its assertions are meant to be accessible to non-experts.
Proposition 3. Suppose that X ⊂ S 3 is a celestial Darboux cyclide.
a) The graph of a component W ⊂ B(X) is a Dynkin graph of type A 1 , A 2 or A 3 and corresponds to a complex isolated singularity of X of the same type. Such a singular point is real if and only if σ * (W ) = W .
b) Suppose that C, C ⊂ X are complex lines and/or circles such that C = C . We Proof. Let ϕ : Y → X be the smooth model of X and recall that we defined the class of a complex curve U ⊂ X as the divisor class of the strict transform of U in the smooth surface Y . Let C ⊂ X and C ⊂ X be complex lines and/or circles such that C = C . We denote their strict transforms via ϕ by D ⊂ Y and D ⊂ Y , respectively. If W ⊂ B(X) is a component, then we denote by C W the union of curves in Y whose divisor class is in W .
[C] · W 0, and thus claim 5 is a direct consequence of claim 3. In this article, N (X) ∼ = 0 , 1 , ε 1 , ε 2 , ε 3 , ε 4 Z , where the nonzero intersections between the generators are 0 · 1 = 1 and ε 2 1 = ε 2 2 = ε 2 3 = ε 2 4 = −1. We use the following shorthand notation for elements in B(X), G(X) and E(X): We consider the following unimodular involutions σ * : N (X) → N (X) that are induced by the real structure σ : X → X: The singular locus of X is denoted by sng X. A component W ⊂ B(X) defines by Proposition 3a a Dynkin graph of type either A 1 , A 2 or A 3 and corresponds to an isolated double point that is a node, cusp or tacnode, respectively. We underline if the isolated singularity is real (for example A 1 ) and take formal sums to denote the disjoint union of singularities (for example A 1 + 2A 1 ).
A Darboux cyclide X ⊂ S 3 is called a S1 cyclide or S2 cyclide, if X(R) is homeomorphic to a sphere or the disjoint union of two spheres, respectively.
Theorem A. If X ⊂ S 3 is a λ-circled Darboux cyclide such that λ ≥ 2, then σ * , sng X, B(X), E(X), G(X) and λ are up to Aut N (X) defined by a row in Table 2 and Table 3.  if underlined, and otherwise to its left or right neighbor in the listing, if its position is even or odd, respectively. The dashed row dividers indicate when σ * is defined by {g 0 , g 3 , g 12 , g 34 , g 13 , g 24 , g 14 , g 23 , g 1 , g 2 } Example 1. The diagrams in Figure 3, Figure 4 and Figure 7 show the incidences between complex lines and isolated singularities in EY-, CY-, Perseus-, CH1-, ring-, CO-, and EO-cyclides. First suppose that X ⊂ S 3 is an EY cyclide. We apply Theorem A and find that sng X = A 3 , B(X) = {b 1 , b 2 , b 12 } and E(X) = {e 11 , e 12 , e 3 , e 4 } (see Table 2 and Table 3    b) If X is covered by at least two pencils of circles with complex conjugate base points, then X is a ring cyclide.
c) If X is a CY cyclide or CO cyclide, then all complex lines in X meet some real isolated singularity.
Proof. a) We apply Theorem A and consider for each of the 14 cases in Table 2 and Table 3 for the triple sng X, B(X), G(X) . We know from Proposition 3[a,c] that complex conjugate base points correspond to complex conjugate isolated singularities. It follows from the sng X column in Table 2 that only the Perseus cyclide, ring cyclide, CH1 cyclide, CY cyclide and CO cyclide contain two complex conjugate isolated singularities. First suppose that X is a Perseus cyclide. It follows from Propo-sition 3c that a pencil of circles corresponds to a class in {g 0 , g 1 , g 12 , g 2 , g 3 } ⊂ G(X) and a base point corresponds to a component of B(X) = {b 1 , b 2 }. Since g 12 ·{b 1 } 0,  Proof. It follows from Bézouts theorem that a line in X that meets a hyperplane in P 4 in more than one point, must be contained in this hyperplane. Assertions a) and b) are thus a direct consequence of the incidences of complex lines and isolated singularities in X as depicted in the corresponding diagram of Figure 4 (see Example 1).

Cliffordian Darboux cyclides
We develop a necessary condition for a Darboux cyclide X to be Cliffordian in terms of the sets B(X), E(X) and G(X) in Table 3.
Suppose that X ⊂ S 3 is a Darboux cyclide.  If neither F nor F has base points in E, then it follows from Lemma 2 and Bézout's theorem that X ∩ E consist of two left generators and two right generators. By Proposition 3b, the classes of these generators form a Clifford quartet and thus the proof is concluded for this case.
In the remainder of the proof we assume without loss of generality the F has complex conjugate base points in E. Recall from Proposition 3[a,c] that each base point corresponds to a complex isolated singularity of X.
First suppose that F has base points in E as well. These base points must be complex conjugate and thus X is a ring cyclide by Lemma 3. It follows from Lemma 4a that the hyperplane section E ∩ X consists of four complex lines. Recall from Example 3 that E(X) defines a Clifford quartet and thus we concluded the proof.
Finally, suppose that F does not have base points in E. In this case, the hyperplane section X∩E contains two right generators R and R by Lemma 2. We apply Lemma 3 and find that X is either a Perseus cyclide, ring cyclide or CH1 cyclide. The main assertion now follows from Lemma 4b.
We call the Clifford data (A, a, g, U ) a certificate if g · u = 0 for all u ∈ U . We say that X satisfies the Clifford criterion if there exists at least one certificate. Blum cyclide, is not a certificate, since g 12 · e 11 = 0 (see Figure 5). In fact, we shall see that a Blum cyclide does not satisfy the Clifford criterion.   If F or F has base points on E, then these base points must be complex conjugate as E R = ∅. Thus it follows from Lemma 3 and Lemma 5 that X is either a Perseus cyclide, ring cyclide or CH1 cyclide. In this case we conclude the proof as the main assertion holds for these cases by Example 4.
In the remainder of the proof we assume that neither F nor F has base points on E.
Notice that the member F i is a complex possibly reducible conic for all i ∈ P 1 .
Moreover, for all complex p ∈ X there exists a complex i ∈ P 1 such that p ∈ F i .
By Bézout's theorem, the intersection P(A) ∩ E consists of two complex conjugate points {a, a}. The Hamiltonian product for the unit quaternions S 3 extends via γ : S 3 R → S 3 to a productˆ on projectivized unit quaternions S 3 \ E with complex coefficients. See forward Example 5 for an explicit coordinate representation of such a product. Thus for all complex α ∈ P(A)\{a, a} we obtain the complex left Clifford translation ϕ α : S 3 → S 3 such that ϕ α (x) = αˆ x for all x ∈ S 3 \ E. It follows from Lemma 5 that X ∩E consists of two left generators L, L ⊂ E and two right generators R, R ⊂ E whose incidences with P(A) and ϕ α (P(B)) are illustrated in Figure 6. Example 5. In order to construct parametrizations of Cliffordian surfaces we consider the 5 × 5 matrices M i :=Mi in Table 4. Notice that M 0 is the identity matrix.
Let J be the diagonal matrix with (−1, 1, 1, 1, 1) on its diagonal. We verify that and thus each M i defines an automorphism with parameters 0 ≤ α, β < 2π. We can implicitize the parametric representation of X and compute deg X and sng X. If deg X = 4, then X is a Darboux cyclide and sng X consists of complex double points. To assign X to one of the 14 cases of Theorem A we use the following observations from Table 2. If | sng X| = 2 and | sng X R | = 0, then X must be a Perseus cyclide.
If | sng X| = 4, then X must be a ring cyclide. If | sng X| = 3 and the stereographic projection from the real isolated singularity of X(R) is a CH1 quadric (or not a CY quadric), then X must be a CH1 cyclide. We deduce that if (A, B) is defined by (M0,M1), (M2,M3), or (M4,M5), then deg X = 4 and X is a ring cyclide, Perseus cyclide and CH1 cyclide, respectively (see Figure 1). If (A, B) is defined by (M0,M6) or (M7,M8), then deg X = 8 and X is in Figure 2 middle and right, respectively. See [14, orbital] for an implementation of these methods.

Remark 2.
Recall from Example 4 that the Perseus cyclide, ring cyclide and CH1 cyclide satisfy the Clifford criterion and each of these three cyclides is realized as a Cliffordian Darboux cyclide in Example 5.
Proof. The Darboux cyclide X satisfies the Clifford criterion by Lemma 6. We apply Theorem A and consider 11 of the 14 triples B(X), E(X), G(X) in Table 3. For each such triple we go through all possible Clifford quartets in E(X). For each such Clifford quartet A we consider all possible Clifford data (A, a, g, U ). We verify that each such Clifford data (A, a, g, U ) is not a certificate. We used [16, cyclides] to do the verification automatically. In particular, we find that a Clifford quartet exists only if X is a Blum cyclide, EH1 cyclide, HP cyclide or S1 cyclide.

Bohemian Darboux cyclides
We show that Bohemian Darboux cyclides are the pointwise sum of a line and a circle in R 3 , namely a CY or EY as in Figure 1.

Lemma 7.
Suppose that X ⊂ S 3 is a Darboux cyclide that contains two circles through a general point that do not meet in two points. If complex conjugate lines in X intersect, then these lines meet at an isolated singularity.
Proof. It follows from Proposition 3b that there exists f, g ∈ G(X) such that σ * (f ) = f , σ * (g) = g and f ·g = 1. We apply Theorem A and find using Table 3 that σ * = 2A 1 in Table 2. For each of the 10 cases we verify that e · σ * (e) = 0 for all e ∈ E(X).
Thus by Proposition 3b there exists e ∈ E(X) and a component W ⊂ B(X) such that e · W 0 and σ * (e) · W 0. We concluded the proof as this component W corresponds to an isolated singularity of X.
Proposition 5. If A, B ⊂ R 3 are generalized circles such that S(A+B) is a Darboux cyclide, then A + B is either a CY or EY.
Proof. Let X := S(A + B). If deg π(X(R)) = 2, then we go through the well-known classification of quadric surfaces up to Euclidean similarity (see [15,Proposition 4]) and conclude that A + B is either a CY or EY as in Figure 1. First suppose that F has no base points. We know from Lemma 1 that X contains complex conjugate lines meet at the vertex of U. A general member of F and F meet in one point and thus X has an isolated singularity at this vertex by Lemma 7.
Hence, deg π(X(R)) = 2 so that A + B is as asserted.
Next we suppose that F has a real base point in U R . In this case deg π(X(R)) = 2, since U R ⊂ sng X by Proposition 3[a,c] and thus A + B is as asserted.
Finally, suppose by contradiction that both F and F have non-real base points in U.
Then X must be a ring cyclide by Lemma 3 so that | sng X| = 4 by Proposition 3[a,c].
It follows from Lemma 4a that the hyperplane section X ∩ U consists of four lines.
We arrived at a contradiction with the diagram in Figure 4 (right), as all lines in U should be concurrent.
We considered all possible cases and thus concluded the proof.

Great Darboux cyclides
In this section we develop necessary conditions for great celestial Darboux cyclides.
Our main tool is the central projection τ : S 3 → P 3 which sends (x 0 : x 1 : to (x 1 : x 2 : x 3 : x 4 ). Thus τ is a 2:1 projection with ramification locus E and branching locus τ (E). If U p is a local neighborhood of a complex point p ∈ P 3 \ τ (E), then τ restricted to one of the two sheets of τ −1 (U p ) defines a complex analytic isomorphism. Notice that a real fiber of τ corresponds to antipodal points in S 3 = S 3 (R) and that great circles are send to lines. Example 6 (great CO cyclide). The surface , is a great CO cyclide, since π(X(R)) = {z ∈ R 3 | z 2 1 + z 2 2 − z 2 3 = 0} is a CO and τ (X) is ruled. Like in Example 1, we use Proposition 3 to encode the corresponding row in Table 3 in terms of the diagram in Figure 7 (left), where G(X) = {g 0 , g 1 , g 14 , g 23 }.
By Proposition 3, the components {b 34 } and {b 12 } correspond to real antipodal singularities of X and are centrally projected to the vertex of the quadric cone τ (X).
The great circles in X have class g 1 and form a pencil with real antipodal base points in sng X. Two complex antipodal little circles in X have class g 14 and g 23 , respectively and are send to real conic sections of τ (X). The components {b 13 } and {b 24 } correspond to the complex isolated singularities that lie in the ramification locus E. The central projection of these complex singular points are smooth complex branching points in τ (X).
Example 7 (great EO cyclide). The surface X := τ −1 ({y ∈ P 3 | y 2 0 + 2y 2 1 − y 2 2 }) = {x ∈ S 3 | x 2 1 + 2x 2 2 − x 2 3 }, is a great EO cyclide, since π(X(R)) = {z ∈ R 3 | z 2 1 + 2z 2 2 − z 2 3 = 0} is a EO and τ (X) is ruled. Like in Example 1, we use Proposition 3 to encode the corresponding row in Table 3 in terms of the diagram in Figure 7 (right).  Example 8 (great ring cyclide). We consider the following surface and claim that X ⊂ S 3 is a great ring cyclide. Since τ (X) is a doubly ruled quadric it follows that X is great. The Jacobian matrix of the generators of its ideal is up to scaling of the rows as follows: This matrix has rank one at the four complex conjugate isolated singularities (0 : 1 : 0 : 0 : ±i) and (0 : 0 : 1 : ±i : 0) in X ∩ E. We use Theorem A (see the sng X column of Table 2) to verify that X is a ring cyclide (see also Example 1).
Example 9 (great Blum cyclide). We consider the following surface , and claim that X ⊂ S 3 is a great Blum cyclide. Since τ (X) is a doubly ruled quadric it follows that X is great. The Jacobian matrix of the generators of its ideal is up to scaling of the rows as follows: We observe that this matrix has rank two at all points in X and thus X must be smooth. Hence, X is a Blum cyclide by Theorem A (see the sng X column in Table 2). In Figure 8 we depicted its stereographic projection π(X(R)) = {z ∈ R 3 | x 4 + 2x 2 y 2 + 2x 2 z 2 + y 4 + 2y 2 z 2 + z 4 − 6x 2 − 4y 2 + 1}.
Let us analyze the geometry of X in more detail using our methods. By Theorem A, we may assume up to Aut N (X) that B(X), E(X) and G(X) are as in Table 3.
Since τ (X) is a doubly ruled quadric, there exists great circles C, C ⊂ X such that |C∩C | = 2 and, by Proposition 3[b,c], we may assume without loss of generality that  Lemma 8. Suppose that X ⊂ S 3 is a great celestial Darboux cyclide.
a) The number of components W ⊂ B(X) such that σ * (W ) = W is even. Equivalently, the number of real isolated singularities of X is even.
b) If B(X) = ∅, then e · σ * (e) = 0 for all e ∈ E(X) and X is covered by no more and no less than two pencils of great circles.
Proof. a) Suppose that there exists a component W ⊂ B(X) such that σ * (W ) = W . Recall that E is the ramification locus of τ and that E(R) = ∅. It follows from Proposition 3 that X(R) is not smooth so that sng X E. Hence, the real singularities of X consist of two antipodal points that are send via τ to the vertex of the quadric cone τ (X). This concludes the proof for assertion a). b) We know from Proposition 3a that X is smooth. It follows that τ (X) is a doubly ruled quadric and thus X is covered by exactly two pencils of great circles. Now suppose by contradiction that there exists e ∈ E(X) such that e · σ * (e) = 0. In this case, there are complex conjugate lines L, L ⊂ X that intersect at some point p.
Complex conjugate lines in τ (X) do not intersect and thus τ (L) = τ (L) so that p ∈ E. We arrived at a contradiction since p ∈ X R and E R = ∅. This concludes the proof for assertion b).

Lemma 9.
A Perseus cyclide X ⊂ S 3 is not great. Suppose by contradiction that X is great and that p, p ∈ E. Since τ is a local complex analytic isomorphism outside E, it follows that τ (X) is a smooth doubly ruled quadric. We observe that L, L, R, R E otherwise τ (X) contains three complex lines τ (R), τ (L) and τ (T ) = τ (M ) through the complex point τ (p) instead of two.

Proof. By Theorem
It follows that τ (L) = τ (R) and τ (L) = τ (R). This is a contradiction, since τ (L) intersects its complex conjugate line τ (L) and thus τ (X) must be an ellipsoid instead of being doubly ruled.
Now suppose by contradiction that X is great and that p, p / ∈ E. Thus τ (X) is quadric cone with real vertex τ (p) = τ (p). The line in P 4 that passes through the complex conjugate points p and p is of the form P(V ) for some line V ⊂ R 4 . Since P(V ) passes through the center of τ , we find that V is a radial line that passes through the center of S 3 . We arrived at a contradiction as V intersect S 3 in two antipodal points and thus p and p must be real.
We concluded the proof, since either p, p ∈ E or p, p / ∈ E and we arrived at a contradiction for each of these two cases.
Proposition 6. If X ⊂ S 3 is a great celestial Darboux cyclide, then X is either a ring cyclide, CO cyclide, EO cyclide or Blum cyclide.
Proof. It follows from Theorem A (see Table 2) and Lemma 8a that X is either a EO cyclide or CO cyclide, or that X(R) is smooth. If X(R) is smooth, then X is either a Blum cyclide or ring cyclide, by Lemma 8b and Lemma 9.  for almost all i, j ∈ P 1 , then there exists a linear projection ρ : P 4 P 3 such that deg ρ(X) = 2. If X is not Möbius equivalent to a great surface, then the center of ρ must lie "outside" of S 3 .

Combining the results
In order to prove Theorem 1 we use the following theorem from [ c) Direct consequence of Proposition 6.
Corollary 1 and Corollary 2 are direct consequences of Theorem 1.
Proof of Corollary 3. The central projection of a surface Z ⊆ S n that is covered by two pencils of great circles is a doubly ruled quadric. Therefore, n ≤ 3 and Z has no real singularities. In particular, the stereographic projection π(Z) is not a CO cyclide or EO cyclide. Thus the proof is concluded by Theorem 1c.

Acknowledgements
I would like to thank M. Skopenkov for interesting discussions. The surface figures were generated using [26,Sage]. This work was supported by the Austrian Science Fund (FWF) project P33003.