Dupin cyclides passing through a fixed circle

We derive algebraic equations on the coefficients of the implicit equation to characterize all Dupin cyclides passing through a fixed circle. The results are applied to solve the basic problems in CAGD about blending of Dupin cyclides along circles.


Introduction
Dupin cyclides are remarkable algebraic surfaces that have applications in Computer Aided Geometric Design (CAGD).The prototypical example of a Dupin cyclide is a torus of revolution with major radius R and minor radius r.A canonical implicit equation of a torus is (1.1) We must have r < R for a smooth torus surface.Any torus contains two orthogonal circles through each point.These circles are curvature lines of the torus, and are called principal circles.A smooth torus has two additional circles through each point on a bitangent plane to the torus; see Figure 1(a).They are called Villarceau circles [V48].A Dupin cyclide is the image of a torus under a Möbius transformation, in particular an orthogonal transformation or an inversion with respect to a sphere.These transformations preserve the angles and the set of circles and lines on the surfaces [MV22,Ott12].Accordingly, smooth Dupin cyclides inherit the property of having 2 principal circles and 2 Villarceau circles through each point; see Figure 1 The implicit equation for a Dupin cyclide is of degree 4 or 3 and has the form (1.2) where a 0 , b 1 , . . ., f 0 ∈ R. For general values of the coefficients, this implicit equation defines a more general surface called Darboux cyclide [PSS12].The practical problem of distinguishing Dupin cyclides among Darboux cyclides is considered in [MV22].
The basic problem considered in this paper is smooth blending of two Dupin cyclides along a fixed circle.Our approach is to match implicit equations (1.2) for the two Dupin cyclides we blend.To solve the basic problem algebraically, we first consider the general linear family of Darboux cyclides passing through a fixed circle.Then we use the results in [MV22] to characterize the smaller family of Dupin cyclides in terms of the algebraic relations for the free coefficients of the general family of Darboux cyclides.This is considered in Section 3 together with the formulation of the main results of the paper.We prove them separately for quartic and cubic equations in Sections 4 and 5.The smooth blending between two implicit equations of Dupin cyclides along the fixed circle is investigated in Section 6.In the last section, we express the Möbius invariant of a Dupin cyclide equation, introduced in [MV22], to our particular smaller families of Dupin cyclides.

Preliminaries
First off, let us recall the salient results in [MV22] on distinguishing Dupin cyclides among Darboux cyclides.They are formulated using the following abbreviations of algebraic expressions in the coefficients in (1.2): Let σ 12 , σ 13 denote the permutations of the variables b and e 1 , e 2 , e 3 that permute the indices 1, 2 or 1, 3, respectively.
To recognize quartic Dupin cyclides among the form (1.2), we can assume a 0 = 1 by dividing all coefficients by a 0 .Then we apply the shift to remove the cubic terms and reduce the equation to an intermediate Darboux form Theorem 2.1.The hypersurface in R 3 defined by (2.2) is a Dupin cyclide only if the following 12 polynomials vanish: and Proof.This result is covered in [MV22, Proposition 3.6].We consider the necessity only because in this paper we start from the 12 polynomial equations to find a Dupin cyclide.In addition, the 12 polynomial equations define the following degenerate Dupin cyclides: two touching spheres, a sphere and a point on it, a double sphere, a circle, a cyclide with exactly 2, 1 or no real points; see [MV22, Section 6.3].We need to discard them from the algebraic equations.
Theorem 2.2.The hypersurface in R 3 defined by (1.2) is a cubic Dupin cyclide only if the following equations are satisfied: where Proof.This is covered in [

The main results
Without loss of generality, we assume that a fixed circle Γ ⊂ R 3 with radius r > 0, is given by the equations Computing the variety of Dupin cyclides passing through the circle Γ turns out to be non-trivial.The defining equations are obtained by restricting the coefficients of (1.2) to cyclides passing through Γ and by considering the effects on the equations in Theorems 2.1 and 2.2.The Darboux cyclides passing through the circle Γ form a linear subspace of the space of coefficients in (1.2).
Proof.The equation of a Darboux cyclide passing through the circle Γ is an algebraic combination of x and x 2 + y 2 + z 2 − r 2 .Besides, the terms of degree four and degree three should match to the Darboux form (1.2).We expand the quartic and cubic terms to so that they would be contained in the ideal (x, x 2 +y 2 +z 2 −r 2 ) of the polynomial ring R(r)[x, y, z] over R(r) -the fraction field of R[r].The remaining terms of degree ⩽ 2 should be in the same ideal, hence they have the shape The ambient space Darboux cyclides passing through the circle Γ is then identified as P 8 , with the coordinates [u 0 , . . ., u 4 ; v 1 , . . ., v 4 ].The Dupin cyclides with real points will form an algebraic variety D Γ in this projective space.If we would consider the radius r as a variable, the variety D Γ would be invariant under the scaling of (x, y, z) ∈ R 3 , and its equations would also be weighted-homogeneous, with the weight 1 for r and the respective weights 0, 1, 1, 1, 2, 2, 2, 2, 3 of the coordinates of P 8 .We rather consider r as a parameter, and assume r ̸ = 0.
As it turns out, the variety D Γ is reducible, reflecting the fact that the circle Γ could be either a principal or Villarceau circle on a Dupin cyclide.Accordingly, we split the main result into two theorems as follows.
Theorem 3.2.The hypersurface in R 3 defined by (3.2) is a non-degenerate Dupin cyclide containing Γ as a Villarceau circle if and only if the equations and the inequality are satisfied.
Theorem 3.3.The hypersurface in R 3 defined by (3.2) is a non-degenerate Dupin cyclide containing Γ as a principal circle only if the ranks of the following two matrices are equal to 1: Remark 3.4.The rank conditions mean vanishing of the 2 × 2 minors of the matrices N and M. The 2 × 2 minors from the first 3 rows of M differ from the minors of N by the common factor v 4 − 2r 2 u 1 .Incidentally, this factor appears as an equation for the Villarceau case.Localizing with (v 4 − 2r 2 u 1 ) −1 leads to the ideal for the principal circle case.But the Villarceau case equations of Theorem 3.2 do not imply a lesser rank of M, as the second column does not necessarily vanish fully, particularly in the fourth row.Rather similarly, the 2 × 2 minors from the last 3 rows differ of M from the minors of N by the common factor −8r 4 u 1 , as the terms −4r 2 v i (v 1 + u 4 − 2r 2 u 0 ) are proportional to the first column.Therefore, the 2 × 2 minors formed only by the first 3 rows or only by the last 3 rows of M can be ignored.
Remark 3.5.Using Maple or Singular, the Hilbert series of the ideal principal circle component is given by H p (t)/(1 − t) 4 , where Hence, the dimension of the variety equals 4 and the degree equals H p (1) = 8.The Hilbert series of the ideal of Villarceau circle component is computed as , where It follows that the dimension of the variety equals 4 and the degree equals 4.
The Zariski closure of the Villarceau circle component is a complete intersection.

Distinguishing principal and Villarceau circles
As we will discuss in Section 5, the variety D Γ turns out to be reducible.We discard some of the components because they either represent only degenerate reducible cases of cyclides, or the cases with complex (rather than real) coefficients in (3.2).Theorems 3.2 and 3.Under Euclidean transformations, we move the torus equation (1.1) so that the circle Γ is a principal circle (with radius r) or a Villarceau circle (with radius R).The principal circles on the vertical plane x = 0 are given by (y ± R) 2 + z 2 = r 2 .Identifying one of those circles with Γ by the shift y → y + R, we obtain an equation of the form (3.2) with for the representative (under the Möbius transformations) tori with Γ as a principal circle.It is straightforward to check that the representative tori (4.1) do not satisfy the second the fourth equations of Theorem 3.2 generically, while the second column of the matrix M in Theorem 3.3 consists of zeroes for them.Now consider a Villarceau circle on z = αx + βy where α = r/ϱ, β = 0, ϱ = √ R 2 − r 2 is moved onto Γ by the Euclidean transformation Then the torus equation becomes as an implicit equation for the representative tori with Γ as a Villarceau circle.The representative tori (4.4) satisfy the equations of Theorem 3.2, while the rows with u 2 and u 0 in the first column form a lower-triangular matrix with non-zero determinant generically.Therefore, Theorem 3.2 describes cases with Γ as a Villarceau circle, and Theorem 3.3 describes cases with Γ as a principal circle, as claimed.
Remark 4.1.The variety D Γ contains a non-interesting big component consisting of touching spheres and touching sphere/plane cases.This component is defined by the 2 × 2 minors of the matrix and the equation In addition, the degeneration to the circle Γ satisfy rank(L) = 0 and with the two more equations: u 1 = 0 and v 1 = 2r 2 u 0 .Furthermore, the principal component restricted to rank(L) = 0 only has double sphere cases.Therefore, the matrix L has to be of full rank 2 in order to obtain a non-degenerate Dupin cyclide in the principal circle component.
(5.1) and let us denote the 2 × 2 minors N by (5.4) Let us also denote (5.5) We split the proofs into two cases for quartic and cubic Dupin cyclides by the use of Theorems 2.1 and 2.2 in a parallel way.We arrive at parallel options to simplify the reducible variety D Γ from the full consideration of equations in those theorems.Most of the considered particular equations or factors appear naturally by considering the mentioned eliminations and localizations.
Let I Γ ⊂ R Γ denote the ideal generated by the ρ-images of the 12 polynomials in Theorem 2.1.The polynomials in this ideal have to vanish when (3.2) is a Dupin cyclide.Our goal is to describe real points representing non-degenerate Dupin cyclides on this variety.The polynomial ρ(K 1 ) factors in R Γ , namely ρ(K 1 ) = − 1 4 T 4 V 0 , where This shows that the variety defined by I Γ is reducible.To investigate real points of the variety, we consider the possible three options: (5.7) If V 2 = 0, then U 0 − 2u 4 = 0, u 1 = 0 as we look only for real components.
The augmented ideal contains this sum of squares: , hence an empty variety.With V 3 = 0 we obtain the equations of Theorem 3.2 in the homogenized form with u 0 .The points on the corresponding variety describe cases when Γ is a Villarceau circle, as analyzed in the previous section.
Secondly, assume that V 0 ̸ = 0. Localization of I Γ in the ring R Γ [V −1 0 ] gives an ideal generated by the 2×2 minors of the matrix L in (4.5) and the additional equation (4.6) with u 0 = 1.Hence we only obtain degenerate Dupin cyclides according to Remark 4.1.
The last option is T 4 = V 0 = 0. We notice polynomial multiples of T 2 2 + T 2 3 in the Gröbner basis of (I Γ , T 4 , V 0 ).Localization at T 2 2 + T 2 3 ̸ = 0 gives an ideal that contains the 4 polynomials of Theorem 3.2.Hence, it describes some points in the Villarceau circle component (of the option T 4 ̸ = 0).We assume further that T 2 = T 3 = 0. Consideration of the following polynomial allows further progress: (5.8) The localization V 4 ̸ = 0 leads to a subcase (describing touching spheres) of the option V 0 ̸ = 0. Hence, we assume that V 4 = 0. Elimination of v 2 , v 3 , v 4 in the ideal (I Γ , T 2 , T 3 , T 4 , V 0 , V 4 ) leads to some generators that factor with (5.9) The further localization V 5 ̸ = 0 leads to the principal circle component in Theorem 3.3.The remaining case V 5 = 0 splits into these two subcases, as we are interested in the real points only: (i) u 1 ̸ = 0, so that u 2 = u 3 = u 4 = 0.The obtained ideal is reducible, with the prominent factor V 6 = u 2 1 (v 2 2 + v 2 3 ) + 4v 2 4 after elimination of v 1 .The localization V 7 ̸ = 0 belongs to the principal circle component.The case 1 = 0, and the cyclide degenerates to a double sphere case.
Elimination of the variables u 1 , u 2 , u 3 , u 4 gives us a principal ideal, and the generator factors with (5.10) The localization V 7 ̸ = 0 belongs to the principal circle component.With V 7 = 0 we get v 2 = v 3 = v 4 = 0, and the resulting ideal contains the product Either of the factors leads to points on the principal circle component.

Proof for cubic cyclides
We use Theorem 2.2 to recognize cubic Dupin cyclides in the form (3.2) with u 0 = 0.The equation is first transformed to the form (1.2) be the ring homomorphism defined by the coefficients identification.Since ρ 0 (B 0 ) = U 0 , all remaining computations will be considered over the localized ring R Γ [U −1 0 ].Let us denote by I * Γ the ideal generated by the numerators of the ρ 0 -images of the 4 equations in Theorem 2.2.This ideal contains the product T 4 V * 0 , where (5.11) Similar to the simplification method in the quartic case, we consider the three options T 4 ̸ = 0; V * 0 ̸ = 0; and T 4 = V * 0 = 0.The localization T 4 ̸ = 0 gives us directly the u 0 = 0 part of the Villarceau circle component in Theorem 3.2.
The localizing V * 0 ̸ = 0 gives an ideal containing the 2 × 2 minors of the matrix L. Hence, this case describes only degenerate cyclides, see Remark 4.1.Lemma 6.2.The function F(y, z) ≡ λ on the circle Γ for some constant λ if and only if the envelope surface of tangent planes of the cyclide (3.2) along Γ is given by the equation It is a circular cone if λ ̸ = 0, or a cylinder if λ = 0.
Remark 6.3.The envelope of tangent planes degenerates to the plane of the circle Γ when λ = ∞.If the circle is a Villarceau circle, then the envelope of tangent planes is a much more complicated surface of degree 4.

Smooth blending along principal circles
In this section, we focus on smooth blending between Dupin cyclide equations in the principal circle component.The main case to investigate is by fixing a tangent cone along the circle Γ and find Dupin cyclides in the principal circle component that fit the blending conditions along the circle, see Figure 2(a).
Proposition 6.4.Let us fix the parameter λ ̸ = 0 and the cone (6.2).The family of Dupin cyclides in the principal circle component touching the cone along the circle Γ is defined by the 5 equations Proof.From Lemmas 6.1-6.2, the tangency conditions along the circle are given by v i = λu i for i ∈ {2, 3, 4}.We specialize u 0 , v 2 , v 3 , v 4 in the ideal generated by the 2 × 2 minors of N and M, and obtain an ideal We notice many multiples of u 2 , u 3 , u 4 in a Gröbner basis of I λ .If u 2 u 3 u 4 ̸ = 0, we obtain an ideal I * λ ⊂ R λ [(u 2 u 3 u 4 ) −1 ] generated by the 5 equations of the proposition.The points with u 2 u 3 u 4 = 0 satisfy the equations of I * λ ∪ R λ , by checking the cases u 2 = u 3 = u 4 = 0, Remark 6.5.The five equations of Proposition 6.4 are linear in the five variables u 4 , v 1 , v 2 , v 3 , v 4 .Hence, we can easily solve the equations in those variables and obtain a parametrization of the family of Dupin cyclides touching the cone along the circle Γ. Apart from the first 3 equations, the variables u 2 , u 3 appear only within the expression u 2 2 + u 2 3 , representing a rotational degree of freedom: rotating the two Dupin cyclide patches independently around the xaxis preserves the smooth blending along the circle Γ.
The limit cases λ = 0 and λ = ∞ of principal circle component also contain interesting families of Dupin cyclides.The family in the case λ = 0 allows us in particular to blend two torus equations or a torus and a Dupin cyclide, see Figure 2(b),(c),(d).The family in the case λ = ∞ allows us in particular to blend a Dupin cyclide with a plane, see Figure 2(e).Proposition 6.6.Dupin cyclides touching the cylinder along the circle Γ are defined by the equations Dupin cyclides here are symmetric with respect to plane of the circle Γ.
Proof.The equations v 2 = v 3 = v 4 = 0 follow from the condition λ = 0.With those constraints, the ideal of principal circle component reduces to the other two equations u 1 = 0 and (6.8).The symmetry property with the plane x = 0 follows from equations (6.7).
Proposition 6.7.Dupin cyclides touching the plane of the circle Γ along the same circle are defined by u 2 = u 3 = u 4 = 0, v 4 = 2r 2 u 1 , (6.9) Proof.The proof is similar to the proof of Proposition 6.6.The equations u 2 = u 3 = u 4 = 0 follow from the tangency condition λ = ∞ and the ideal of principal circles reduces to the other two equations.
Remark 6.8.A cubic cyclide equation among the family in Proposition 6.7 degenerates to a reducible surface (touching sphere/plane).It is interesting to distinguish toruses in the principal circle component.We get two cases, depending on the position of the circle Γ (wrapping around the torus hole or around the torus tube).Figure 2(c) and (d) illustrate two different configuration of torus blending using those two kinds of principal circles.The circle wraps around the torus tube of both toruses in Figure 2(c).The circle wraps around the torus tube for one torus and around the torus hole for the other torus in Figure 2(d).
Proposition 6.9.The equation (3.2) defines a torus in the principal circle component if and only if one of the following applies: Proof.Assume that the circle Γ is wrapping around the torus tube.Then we have a tangent cylinder along the circle, encoded by v 2 = v 3 = v 4 = 0 as in Proposition 6.6.The cross section of (3.2) with the plane x = 0 is a pair of circles of the same radius (Γ, Γ ′ ) Besides the equality between radiuses, we have u 2 2 + u 2 3 = 2u 0 u 4 .Hence the equation (6.8) factors into (v 1 + u 4 )(v 1 + u 4 − 2r 2 u 0 ).To recognize a torus equation, we can assume that u 3 = 0 and u 2 = √ 2u 0 u 4 by applying rotation preserving the circle Γ.The rotated cyclide equation is This is a torus only if u 0 = 1 and v 1 = −u 4 .This proves (i).
Assume now that the circle Γ is wrapping around the torus hole.Then we have a tangent cone along the circle, i.e. v 2 = λu 2 , v 3 = λu 3 , v 4 = λu 4 as in Proposition 6.4.The section with x = 0 should be a pair of concentric circles.Hence, u 2 = u 3 = 0. Again with u 0 = 1 and the parametrization in Proposition 6.4, the cyclide equation reduces to This is a torus equation, see (1.1).

Smooth blending along Villarceau circles
Due to different tangency conditions along the circle Γ, see Remark 6. Proof.
] be a Dupin cyclide touching the given Dupin cyclide D along the circle Γ in the Villarceau circle component.We obtain the following equations in matrix form: The first two rows of the matrix are linear equations obtained from D ′ being in the Villarceau circle component.The last 5 rows are the tangency conditions to the given Dupin cyclide D from Lemma 6.1.Note that the 7 by 8 matrix has full rank.We must have v i ̸ = 0 for some i ∈ {2, 3, 4} due to degeneracy to horn cyclides.Then by setting s = u ′ i /v i , we can solve u ′ j = su j , v ′ j = sv j , j ∈ {2, 3, 4}, (6.12) Hence, dividing the equation of D ′ by s, all coefficients are fixed except v ′ 1 = 2r 2 u 0 /s − 2u 4 and u 0 becomes u 0 /s.Hence with t = u 0 /s − u 0 , u 0 and v ′ 1 become u 0 + t and 2r 2 u 0 − 2u 4 + 2r 2 t = v 1 + 2r 2 t respectively.The obtained family is neccessary in the Villarceau circle component because only the second equation in Theorem 3.2 depends on u 0 and v 1 , and it is preserved by the perturbation.
(7.5)By further elimination of u 2 2 + u 2 3 using (6.5)-(6.6),we obtain the more compact form It is interesting that this compact form (7.6) also covers the J 0 expression of the family of cubic Dupin cyclides u 0 = 0 in Proposition 6.4.Since the majority of Dupin cyclides in the principal circle component belong to the family of Dupin cyclides in Proposition 6.4, three equivalent expressions for J 0 in the principal circle component are obtained by substituting λ = v i /u i to (7.6) for each i = 2, 3, 4.This coincidence also can be checked by reducing the numerator of the difference between the general J 0 and each of the found three expressions modulo the ideal of principal circle component.
In the two limit cases Propositions 6.6 and 6.7 of principal circle component, we use the same method and obtain the following expression: for the family of Dupin cyclides in Proposition 6.6, and for the family of Dupin cyclides in Proposition 6.7.Note that the latter formula is well-defined because the family in Proposition 6.7 does not contain nondegenerate cubic Dupin cyclides, see Remark 6.8.

Figure 1 :
Figure 1: A smooth torus (a) and a smooth Dupin cyclide (b).The solid circles are principal circles and the dashed circles are Villarceau circles.

Remark 4. 2 .
In the Villaceau circle component, we must have u 2 4 ⩽ r 2 (u 2 2 + u 2 3 ).Otherwise, there is no real solution for v 2 in (3.4).The strict inequality (3.5) throws away horn cyclides (J 0 = 0 with reference to Section 7) among Dupin cyclides defined by the main equations (3.3) − (3.4).Those horn Dupin cyclides are also in the principal circle component.Note that the intersection between the touching spheres component and the principal circle component represents touching spheres with touching point on the circle Γ.The touching spheres component intersects the Villarceau component at a sphere containing Γ and a point on Γ.The latter intersection is contained in the principal component.5 Proving Theorems 3.2 and 3.3 Let us define the ring

Figure 2 :
Figure 2: Two Dupin cyclide equations with different coefficient values [u 0 , . . ., u 4 ; v 1 , . . ., v 4 ] are smoothly blended along the circle Γ with r = 1.The two cyclides in (e) are obtained from the parameter values a = 1 and a = 1.8.The two cases at (f ) are obtained from the parameter values t = 0 and t = 0.4.