Constructions of principal patches of Dupin
cyclides defined by constraints: four vertices on a
given circle and two perpendicular tangents at a
vertex
L. Garnier and L. Druoton
University of Burgundy
Abstract
Dupin cyclides are algebraic surfaces introduced for the first time
in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin
cyclide can be defined as the envelope of an one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who
thought to use these surfaces in CAD/CAM and geometric modeling.
Some authors gave algoritms to convert a Dupin cyclide patch into a rational biquadratic Bézier surface. Then, the control points of these Bézier
surfaces have some properties. From these constraints, i.e. the four vertices are on a circle and two tangent vectors at a vertex are perpendicular,
there are some algorithms which permit to construct and convert such a
Bézier surface into a Dupin cyclide patch. In this paper, we use the space
of spheres to simplify the construction of the Dupin cyclide patch. We
do not determine the control points of the underlying Bézier surface.
1 Introduction
Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by
the French mathematician Pierre-Charles Dupin (see [28]) and were introduced
in CAD by R. Martin in 1982 (see [17]). They have a low algebraic degree : at
most 4. These Dupin cyclides have a parametric equation and two equivalent
implicit equations (see [29, 23]) and they have been studied by a lot of mathematicians (see [22, 23, 2]). Since a score of years, much of authors used them
in Computer Aided Geometric Design. In [18, 16, 19, 24], Dupin cyclides are
represented by Rational Biquadratic Bézier Surfaces (RBBS). Then, the conditions of the control points of a RBBS which can represent a Dupin cyclide
patch have been given. From the previous conditions, some constructions of
RBBS which can be converted into Dupin cyclide patches have been proposed
237
238
Constructions of Dupin cyclides
(see [11, 14, 8, 26]). To obtain an unique such RBBS, two initial conditions are
necessary and sufficient (see [7, 15]): the four vertices belong to a circle; the
tangent vectors at one of the four vertices, which define two circular edges,
must be perpendicular.
The space of spheres was introduced in various ways. For example, M.
Berger (see [2]) works in the projective space of the quadratic forms on the
affine Euclidean space, M. Paluszny (see [31]) works in a four dimensional
projective space using the hypersphere of Moebius while (see [30]), T. Cecil (see [27]), R. Langevin and P. Walczak (see [13]) use a four dimensional
quadric Λ4 in the five dimensional Lorentz space which is endowed with a
non-degenerate indefinite quadratic form of signature (4, 1).
Some definitions of Dupin cyclides exist (see [20, 23, 29, 14]). The most
appropriate in our context is: a Dupin cyclide can be defined, in two different
ways, as the envelope of an one-parameter family of oriented spheres. The contribution of a sphere to the Dupin cyclide is a circle in the usual Euclidean 3D
affine space E3 called characteristic circle. At each point belonging to this circle, the Dupin cyclide and the aforementioned sphere are tangent. In the space
of spheres, a Dupin cyclide is represented by two conics in two perpendicular
2-planes. Using the transition formulae between these oriented spheres of E3
and the points of Λ4 , we can draw characteristic circles or arcs of characteristic
circles of a given Dupin cyclide. Moreover, we can obtain a mesh of a Dupin
cyclide patch using the conics in Λ4 and we need not know its parameters nor
the affine transformations to place the Dupin cyclide in the correct location of
the scene (see [4, 9]).
In this paper, we compute a Dupin cyclide principal patch using the space
of spheres. The initial conditions are given in [7, 15]: the four vertices belong
to a circle and the tangents (in the 3D space) to two of the four edges, at one
of the four vertices, are perpendicular. In comparison with [11, 8, 26], we do
not need to: construct the RBBS, determine the type of the Dupin cyclide, the
Dupin cyclide parameters, the transformation matrix. In fact, the visualization
of the principal patch can be done directly from the space of spheres.
A problem exists with the iterative method which permits the constructions
of Dupin cyclide principal patches: after four or five iterations, the control of
the Dupin cyclide patches is not satisfactory (see [3]). Thus, unlike previous
works (see [7, 15, 3, 11, 8, 26]), the use of the space of spheres to represent
Dupin cyclide patches will permit to construct different meshes: we will be
able to mix quadrilateral meshes and 3D triangular meshes with circular edges
(see [11]). Moreover, we hope to use 3D triangular meshes without three circular edges: using the space of spheres, from three contact conditions, we can
determine a one-parameter family of Dupin cyclides tangent along a common
curve (see [6, 5]).
The paper is organised as follow: after some background about Dupin cy-
L. Garnier and L. Druoton
239
clides, the space of spheres and pencils of spheres, we give, in section 3, a
theorem and an algorithm to compute a Dupin cyclide principal patch. After the conclusion and the perspectives, Appendix 5 contains the proof of the
aforementioned theorem.
2 Background
2.1 Non-degenerate Dupin cyclides in the usual Euclidean
3D spaces E3
→
→
→
Let (O3 , −
e1 , −
e2 , −
e3 ) be the orthonormal basis frame of the usual Euclidean
space E3 . In the Euclidean affine space E3 , each sphere S, with a non-negative
radius, defines two oriented spheres S+ and S− . In fact, we distinguish the
inside space and the outside space of S. To define an oriented sphere with centre C and radius ρ , we use an algebraic radius which is positive (respectively
−
→
negative) if the unit normal vector N at the point M is in the same direction as
−→
(respectively opposite direction to) than the vector CM and we have
−→
−
→
CM = ρ N
(2.1)
In E3 , a (quartic) Dupin cyclide (we do not consider degenerate Dupin cyclides which are tori, circular cones and circular cylinders) is the envelope of
two families of one-parameter oriented spheres, Figure 1, the locus of the centres are an ellipse E, equation (2.2) and Figure 1(a), and a hyperbola H, equation (2.3) and Figure 1(b), in two perpendicular planes. Moreover, the vertices
of one are the foci of the other (see [29, 21, 23]) and then we have the relation
b2 = a2 − c2
Using Table 1, a parametric equation of a Dupin cyclide is


µ (c − a cos θ cos ψ ) + b2 cos θ


a − c cos θ cos ψ




b sin θ × (a − µ cos ψ )


Γd (θ , ψ ) = 



a − c cos θ cos ψ




b sin ψ × (c cos θ − µ )
(2.6)
a − c cos θ cos ψ
where the pair (θ , ψ ) belongs to [0, 2π ]2 .
We can note that a Dupin cyclide is characterized by a third independent
parameter µ (see [18, 12]). We can distinguish five kinds of Dupin cyclides
: ring, Figure 3(a); spindle, Figure 3(e); horned, Figure 3(d); one-singularity
240
Constructions of Dupin cyclides
(a)
(b)
F IGURE 1. A Dupin cyclide, envelope of two families of one parameter
spheres where the locus of the centres are an ellipse and a hyperbola in two
orthogonal planes: (a) the centres of the spheres belong to an ellipse, (b) the
centres of the spheres belong to a hyperbola (two planes belong to the family)
Object
Spheres
Centres
Radii
Planes
About the ellipse E
S1 ( θ )
About the hyperbola H
S2 ( ψ )
Ωψ
Ωθ (a cos (θ ) , b sin (θ ) , 0)
(2.2)
r1 (θ ) = µ − c cos (θ ) (2.4)
c
, 0, −b tan (ψ )
cos(ψ )
(2.3)
a
cos (ψ )
c x − µ a = ε bz
r2 (ψ ) = µ −
(2.5)
TABLE 1. Properties of the Dupin cyclides:spheres
and planes belonging to
−π π
each family, θ ∈ [0, 2π ], ψ ∈ [0, 2π ] \ 2 , 2 and ε ∈ {−1, 1}
L. Garnier and L. Druoton
241
−
→
n1
b
−
→A
n2
b
C1
µ + c = r1
b
a=
O1 O2
2
−
→
nE2
CD4A: µ − c = r2
b
O2 CD4E:
O1
c − µ = r2
C2
F IGURE 2. Orientations of a ring Dupin cyclide (CD4A) and a horned
Dupin cyclide (CD4E) which are tangent along two principal circles C1 and
C2 in the plane Py of equation y = 0. The centre of C1 (respectively C2 ) is O1
(respectively O2 ) whereas the radius of C1 (respectively C2 ) is r1 (respectively
r2 ).
(a)
(b)
(d)
(c)
(e)
F IGURE 3. Five types of Dupin cyclides in E3 : (a) ring, (b) horned, (c)
singly horned, (d) spindle, (e) one-singularity spindle
242
Constructions of Dupin cyclides
spindle, Figure 3(e) and singly horned, Figure 3(d). Note that one can find
some other definitions of Dupin cyclides (see [23, 29]).
Figure 2 shows two principal circles (Principal circles are characteristic
circles belonging to the planes of equations y = 0 or z = 0 and two coplanar
principal circles permit the computation of the Dupin cyclide parameters.) C1
and C2 in the plane of equation y = 0 for a ring Dupin cyclide, called CD4A,
and a horned Dupin cyclide, called CD4E. The centre of C1 (respectively C2 )
is O1 (respectively O2 ) whereas the radius of C1 (respectively C2 ) is r1 (respectively r2 ). The same formula permits the computation of the parameter a
by
−−−→
O1 O2 a=
2
whereas the parameters µ and c are solutions of the following system
µ + c = r1
(2.7)
|µ − c| = r2
We can note that
• along the circle C1 , the tangent spheres which give the characteristic circles of each Dupin cyclides have the same orientation. This is schema→
tized by the vector −
n1 ;
• along the circle C2 , the orientations of the spheres which give a characteristic circle of each Dupin cyclides, are opposite. This is schematized
−
→
−
→
by the vectors nA2 for the CD4A and nE2 for the CD4E, see Formula (2.1).
Using the notion of oriented spheres, we have ρ1 = r1 and ρ2 = −r2 . Thus,
the system of the Formula (2.7) becomes
µ + c = ρ1
(2.8)
µ − c = ρ2
and we do not need absolute values.
We note that, in the horned Dupin cyclide case, the change of orientation of
the spheres leads to two singular points (i.e. two spheres having a radius equal
to zero). To obtain the relation between the types of the Dupin cyclides and the
parameter values, one can see [14].
L. Garnier and L. Druoton
243
Definition
n
−−→
o
Cl = M ∈ L4,1 | Q4,1 O5 M = 0
Nature in L4,1
Sphere
Euclidean view
with a null radius
Light cone
(2.11)
Hyperboloid
n
−−→
o
Λ4 = M ∈ L4,1 | Q4,1 O5 M = 1
Unit sphere
(2.12)
of one sheet
TABLE 2. Fundamental quadrics of L4,1
2.2 Construction of the space Λ4 of oriented spheres and
planes of E3
2.2.1 The Lorentz space
−−→
→
Let L4,1 be the 5 dimensional vector space of vector basis (−
ei )i∈[[0,4]] , equipped
with the non-degenerate indefinite symmetric bilinear form L4,1 of signature
(4, 1) defined by
−−→ −−→
L4,1 : L4,1 × L4,1 −→
R
4
(2.9)
−
→
−
→
( u , v ) 7−→ −x0 y0 + ∑ xi yi
i=1
−−→
→
→
where −
u (x0 , . . . , x4 ) and −
v (y0 , . . ., y4 ) lie in L4,1 . Let Q4,1 be the non-degenerate
indefinite quadratic form associated to L4,1 i.e.
→
−
→
Q4,1 (−
u ) = L4,1 (→
u ,−
u)
(2.10)
−−→
Let L4,1 be the affine space associated to the vector space L4,1 with the
origin O5 (0, 0, 0, 0, 0). In the space L4,1 , two quadrics play an important role:
Λ4 and Cl , Table 2. The set Cl , called light cone, is the quadric of the isotropic
position vectors (for the Lorentz form) while the set Λ4 represents the space of
oriented spheres and planes of E3 . To explain the construction of this space,
we have to define the different types of vectors and planes, Tables 3 and 4.
2.2.2 Embedding of E3 on the light cone
We have to build a model of the ambient Euclidean affine space E3 embedded
as a sub-manifold of L4,1 in order to manipulate simultaneously the spheres
244
Constructions of Dupin cyclides
→
Vector types −
v
Space
Time
Light
→
−
→
Q4,1 (−
v ) > 0 Q4,1 (→
v ) < 0 Q4,1 (−
v) =0
−−→
→
TABLE 3. The three different types of vectors −
v of the vector space L4,1
Type
2 dimensional plane called 2-plane
Space
All the vectors are space-like vectors
Time Contain at least a time-like vector or exactly two non-collinear light ones
Light
Planes parallel to a tangent plane to Cl
−−→
TABLE 4. Three types of 2-planes of L4,1 and L4,1
and planes of E3 and the points of E3 . To do that, we have to build an isometry
between E3 and a paraboloid P, section of the cone Cl by a particular affine
hyperplane of L4,1 and we obtain a one-to-one relation between the oriented
spheres of E3 and the points of Λ4 .
→
→
For example, choose two vectors −
n1 (1, 0, 0, 0, 1) and −
n2 ( 12 , 0, 0, 0, − 21 ) and
→
the hyperplane H , parallel to −
n1 ⊥ (i.e. the space of vectors L4,1 -orthogonal
→
– in the rest of this paper, the term L4,1 is implied) to −
n1 ) passing through the
point m2 (1, 0, 0, 1, 0), Figure 4. The hyperplane H is defined by the point m2 ,
→
→
→
→
→
→
the three space-like vectors −
e1 , −
e2 and −
e3 and the light-like vector −
n1 = −
e0 + −
e4
and then, the equation of H is x0 − x4 = 1. The equation of the paraboloid P
is

x0 − x4 − 1 = 0

(2.13)
 x0 − 1 x2 + x2 + x2 + 1 = 0
2
3
2 1
The formulae of the one-to-one correspondence Π between E3 and P are
given in Table 5, (see [12]). So, we identify the point M (x, y, z) of E3 with the
−−−→ →
point Π (M) of P. Then, the origin O3 of E3 in L4,1 is defined by O5 O3 = −
n2 .
−
→
−
→
Moreover, each light vector m , which is not parallel to n1 , defines a point
m on the paraboloid P (m is the intersection between P and the line, passing
→
through O5 , generated by −
m ) and from this point m, we obtain a point M in E3 .
2.2.3 The space of spheres Λ4
Let n be in the set [[1, 4]]. A n-submanifold V in L4,1 is a n-surface without
singular point. One can generalize Table 4 to V replacing the term 2-plane by
tangent space, called Tσ V , to V at σ . If, for all σ in V , the type of Tσ V is the
same, one can distinguish submanifolds of L4,1 . As it had been understood by
G. Darboux, the space of the oriented spheres (the planes are particular spheres)
is represented by the quadric Λ4 , subset of L4,1 , Formula (2.12). This quadric
L. Garnier and L. Druoton
245
F IGURE 4. Construction of the 3 dimensional paraboloid P isometric to E3 ,
−
→
→
the hyperplane H is tangent to Cl : Each light-like vector −
m , non-parallel to
−
→
n1 , corresponds to a point of E3 via the paraboloid P: this point is the
→
intersection between the line (O, −
m ) and P
Point of E3
Direction
 
x
M=y
z
−→

x1
M =  x2 
x3

←−
Point of P ⊂ L4,1


1 2
2
2
 2 x +y +z +1 




x





y
m=






z


1

2
2
2
x +y +z −1
2
 
x0
 x1 
 

m=
 x2 
 x3 
x4
Condition


 x0 = x4 + 1

 x0 = 1 x21 + x22 + x23 + 1
2
TABLE 5. Correspondence between the points of E3 and the points of the
paraboloid P
246
Constructions of Dupin cyclides
is an unitary sphere with centre O5 (for the Lorentz form) and a submanifold
of type (3, 1). Indeed, at each point σ of Λ4 , the tangent hyperplane Tσ Λ4 to
Λ4 is the set of points M verifying
−−→ −−→
L4,1 O5 σ , σ M = 0
(2.14)
We can note this hyperplane is time-like: the signature of the restriction of Q4,1
to this hyperplane is (3, 1). Formula (2.14), which is equivalent to
−−→ −−→
(2.15)
L4,1 O5 σ , O5 M = 1
−−→
shows that the vector O5 σ and the gradient to Tσ Λ4 at a point σ are colinear
(see Formula (2.1), for the particular case of E3 ). The deduction is easy using
the Chasles relation and the formula
−−→ −−→
L4,1 O5 σ , σ O5 = −1
In the same way, a curve on Λ4 can have a type. Our interest is the spacelike curve i.e. at each point, the tangent vector is space-like. One can note too
that Λ4 contains a lot of affine lines which are light-like. Indeed, for each point
σ on Λ4 , the intersection between Λ4 and Tσ Λ4 is a light cone of dimension 3
(all the generatrices are light-like).
The relation between the points on Λ4 and the oriented spheres in E3 is
obtained using the L4,1 -orthogonality: if σ belongs to Λ4 , then the intersection
between the hyperplane parallel to Tσ Λ4 passing through O5 and the paraboloid
P is a sphere or a plane. These calculations are left to the readers.
It is possible to prove (see [13]) that two spheres S1 and S2 , with the same
orientation at their contact point M, are tangent if and only if we have
→
Q4,1 −
σ−
1 σ2 = 0
where σ1 and σ2 are the representations, in the space of spheres, of the spheres
S1 and S2 respectively. Moreover, the point M is defined by the intersection
between the paraboloid P and the light-like line generated by the point O5 and
−−σ
→
the vector σ
1 2.
Let Hx0 =x4 be the hyperplane of equation x0 = x4 . This hyperplane permits
to distinguish the spheres and the planes. Let Ω (a, b, c) (respectively r) be
the centre (respectively the algebraic radius) of an oriented sphere in E3 . The
representation of this sphere on Λ4 is
a2 + b2 + c2 − r2 − 1
1 a2 + b2 + c2 − r2 + 1
(2.16)
, a, b, c,
σ=
r
2
2
L. Garnier and L. Druoton
247
and, it is very easy to remark the two different oriented spheres, defined from
the same geometrical sphere, are symmetric with respect to O5 on the one
hand, and σ does not belong to the hyperplane Hx0 =x4 on the other hand. The
representation of an oriented 2-plane of equation
p
a x + b y + c z = d, with a2 + b2 + c2 = 1
(2.17)
is
σ = (d, a, b, c, d)
(2.18)
and
• the point σ belongs to the hyperplane Hx0 =x4 ;
• the condition given in Formula (2.17) ensures the point σ belongs to Λ4
i.e. we have
−−→
Q4,1 O5 σ = 1
Reciprocally, each point σ (x0 , x1 , x2 , x3 , x4 ) on Λ4 defines an oriented plane
or an oriented sphere. If we have x0 = x4 , σ represents the affine plane (in E3 )
of equation
x1 x + x2 y + x3 z = x0
(2.19)
whereas if we have x0 6= x4 , σ represents the oriented sphere (in E3 ), its centre
is
x1
x2
x3
Ω
,
,
(2.20)
x0 − x4 x0 − x4 x0 − x4
and its algebraic radius is
1
(2.21)
x0 − x4
2.2.4 Representation of Dupin cyclides in the space of spheres
First, let us recall a Dupin cyclide is the envelope of two families of oneparameter oriented spheres. The representation of a Dupin cyclide in the space
of spheres Λ4 is the union of two circles or the union of a circle and a line for
the quadratic form Q4,1 (see [12]).
Using an Euclidean construction, we can see these conics as:
• two ellipses, figure 5(a), for a ring Dupin cyclide;
• an ellipse and a hyperbola, figure 5(b), for a spindle or horned Dupin
cyclide;
• an ellipse and a parabola, figure 5(c), for a singly horned Dupin cyclide
or one-singularity spindle Dupin cyclide.
248
Constructions of Dupin cyclides
(a)
(b)
(c)
F IGURE 5. Representation of Dupin cyclides on Λ4 : (a) a ring Dupin
cyclide is two connected circles, (b) a Dupin cyclide with two singular points
is the union of two circles, one is connected, the other has two connected
components, (c) a Dupin cyclide with one only singular point is the union of a
circle and a line
We note in the space of spheres, the tangency directions to these conics are
space-like directions in the hyperplane tangent to the quadric Λ4 . Obviously,
→
if the kind of tangency directions, generated by a vector −
v1 , was time-like, the
→
intersection between the half-line [O5 , −
v1 ) and Λ4 would be the empty set.
The characteristic circles of a Dupin cyclide can be determined by the computation of the intersection of two orthogonal spheres (see [4, 9]). Let t 7→ σ1 (t) be
the parametrisation of the conic representing one of the two families of spheres.
The representation of the first sphere S (t0 ) in E3 is the point σ0 = σ1 (t0 ) in Λ4 .
•
−−−−→
The second sphere σ (t0 ) is given using the tangent vector ddtσ (t0 ) to the curve
at the point σ (t0) and we have
• "
−−−−→!
dσ
σ (t0 ) = O5 ,
(t0 ) ∩ Λ4
(2.22)
dt
2.2.5 Linear pencils of spheres and their correspondence in Λ4
In [13], the authors recall that any linear pencil of spheres of E3 is represented
in L4,1 by the section of the quadric Λ4 by an affine 2-plane P passing through
−
→
O5 . Given the type of the plane P , we have different types of pencil of spheres
Proposition 1 :
• The section of the quadric Λ4 by a space-like plane P corresponds to a
pencil of spheres with a base circle i.e. all the spheres of the pencil have
a common circle, Figures 6(a) and 7(a).
L. Garnier and L. Druoton
249
(a)
(b)
(c)
F IGURE 6. Three pencils of spheres in E3 : (a) a pencil of spheres with a
base circle, (b) a Poncelet pencil, (c): A a pencil of tangent spheres at a point
• The section of the quadric Λ4 by a time-like plane P corresponds to a
Poncelet pencil, i.e. the balls, defined by the spheres, are contained in
each other and tend to two limit points, Figures 6(b) and 7(b).
• The section of the quadric Λ4 by a light-like plane P corresponds to a
pencil of tangent spheres in a point is the union of two light-like lines,
Figures 6(c) and 7(c).
3 Algorithm to construct a Dupin cyclide principal patch
3.1 The constraints in E3
Let us recall the definition of Rational Biquadratic Bézier Surfaces (RBBS).
2
A point M (u, v),
(u, v) ∈ [0, 1] , belongs to the RBBS defined by the weighted
points Pi j , ωi j (i, j)∈[[0,2]]2 , if we have the relation
−−−−−−→
OM (u, v) =
1
2
2
∑ ∑ wi j Bi (u) B j (v)
i=0 j=0
2
2
−−→
∑ ∑ ωi j Bi (u) B j (v) OPi j
i=0 j=0
(3.1)
250
Constructions of Dupin cyclides
(a)
(b)
(c)
F IGURE 7. Representation of three pencils of spheres in Λ4 , each pencil is
the section of Λ4 by a 2-plane: (a) a pencil of spheres with a base circle and
the type of the 2-plane is space-like, (b) a Poncelet pencil and the type of the
2-plane is time-like, (c) a pencil of tangent spheres at a point and the type of
the 2-plane is light-like
Name
Properties
(PG1)
P00 , P02 , P22 and P20 are cocyclic (i.e. form a cyclic quadrilateral)
−−−→ −−−→ −−−→ −−−→ −−−→ −−−→ −−−→ −−−→
(PG2) P00 P01 = P01 P02 , P02 P12 = P12 P22 , P22 P21 = P21 P20 , P00 P10 = P10 P20 (PG3)
−−−→ −−−→
−−−→ −−−→
−−−→ −−−→
−−−→ −−−→
P00 P10 • P00 P01 = 0, P02 P01 • P02 P12 = 0, P22 P12 • P22 P21 = 0, P20 P21 • P20 P10 = 0
(PG4)
P11 ∈ Aff {P00 , P01 , P10 } ∩ Aff {P02 , P12, P01 } ∩ Aff {P20 , P21 , P10 } ∩ Aff {P22 , P21 , P12 }
TABLE 6. Geometrical properties of a rational biquadratic Bézier surface
→
→
obtained by conversion of a Dupin cyclide, the notation −
u •−
v designates the
−
→
−
→
Euclidean dot product between u and v in E3 and Aff {A, B,C} designates
the affine space generated by the points A, B and C
where
B0 (t) = (1 − t)2 , B1 (t) = 2t (1 − t) and B2 (t) = t 2
(3.2)
are the quadratic Bernstein polynomials (see [18]).
Several authors gave algorithms to convert a Dupin cyclide patch into a
rational biquadratic Bézier surface (see [18, 16, 25, 14]) and we can cite some
of the properties of the control points, Table 6. These conditions are necessary
and sufficient.
We can note that a Dupin cyclide principal patch is defined by six points
P00 , P20 , P22 , P02 , P10 and one of the two following points P11 or P01 (see [1, 8])
and the constraints :
• the points P00 , P20 , P22 , P02 form a cyclic quadrilateral (see [17] and
L. Garnier and L. Druoton
251
Figure 8);
• the tangent lines (P00 P10 ) and (P00 P01 ) to the RBBS at P00 are perpendicular.
3.2 Transfer of the constraints in E3 onto Λ4
From the previous section, we start with four points M00 , M01 , M11 and M10 on
−−→
→
a circle and two perpendicular tangents vectors −
v−
Cθ0 and vCψ0 at M00 (see [7,
−−→
→
15]), Figure 8. The vectors −
v−
Cθ0 and vCψ0 replace the previous tangent lines.
→
−−→
The points M00 , M01 (respectively M10 ) and the vector −
v−
Cθ0 (respectively vCψ0 )
define the circle Cθ0 (respectively Cψ0 ).
Each circle of E3 permits to construct a pencil of spheres with a base circle
C (Cθ0 or Cψ0 ) and two elements of this family are particular: one is an oriented
sphere having C as great circle and the second is an oriented plane passing
through C. We can remark this sphere and this plane are perpendicular.
First, in the space of spheres, each pencil of spheres is represented by a
circle obtained as the section of Λ4 by a space-like plane. The point O5 belongs
to this plane and is the centre of the circle. Then, using Theorems 1 and 2, we
compute a point σθ0 on one circle and a point σψ0 on the other circle such as
their representations in E3 are to spheres tangent at M00 .
Second, we build the light-like line defined by the point σψ0 and the vector
−
→
m10 which is the representation of the point M10 . Third, we compute the point
σθ2 on this light-like line such that its representation in E3 is a sphere containing
the point M11 .
In [15], the data are four points on a circle, a tangent vector to a characteristic circle Cθ0 at M00 and a normal vector to the surface at M00 . The authors
use some iterations to build the other characteristic circles.
The sphere Sθ0 which contains the circle Cθ0 is computed. The vector tangent
which determines the second characteristic circle Cψ0 is the cross product between the two previous vectors.
Then, the sphere Sψ0 , containing the circle Cψ0 is computed. The two other
circles Cθ2 and Cψ2 , the two spheres Sθ2 and Sψ2 are built in the same way.
In the space of spheres, our method permits to determine the adequate
spheres and we do not calculate intersections of straight lines. We directly compute the sphere Sθ0 (respectively Sψ0 ) from the pencils of spheres with a base
circle Cθ0 (respectively Cψ0 ). Then, we determine the sphere Sθ2 which generates the characteristic circle Cθ2 . To summarize, we just need three spheres to
determine the Dupin cyclide.
Some preliminary results must be given before the presentation of Algorithm 1: Theorem 1 characterises the relative positions of two spheres using
252
Constructions of Dupin cyclides
Mr 10
C
Cθ2 ⊂ Sθ2
Cψ0 ⊂ Sψ0
−
→
v−
Cψ0
M11
r
Cψ2 ⊂ Sψ2
−
→
v−
Cθ
0
r
M00
Cθ0 ⊂ Sθ0
r
M01
F IGURE 8. Conditions, in E3 , to obtain a Dupin cyclide principal patch, the
points M00 , M01 , M11 and M10 belong to a circle (i.e. form a cyclic
→
quadrilateral): The points M00 , M01 (respectively M10 ) and the vector −
v−
Cθ0
→
(respectively −
v−
Cψ0 ) define a circular arc Cθ0 (respectivelyCψ0 ): The vectors
−−→
−
→
v−
Cθ0 and vCψ0 are perpendicular
L. Garnier and L. Druoton
253
the Lorentz form and their representations in Λ4 , Theorem 2, proved in Appendix I, permits the computation of the spheres Sθ0 and Sψ0 .
3.3 Relative positions of two spheres
Theorem 1 : Relative positions of two spheres
Let S and Sx be two oriented spheres in E3 . Let σ and σx be their representations on Λ4 .
Then, we have three cases:
−−→ −−−→
• S ∩ Sx is a circle if and only if L4,1 O5 σ , O5 σx < 1
−−→ −−−→
• S and Sx are tangent if and only if L4,1 O5 σ , O5 σx = 1
−−→ −−−→
• S ∩ Sx = 0/ if and only if L4,1 O5 σ , O5 σx > 1
Proof: Let P be the affine 2-plane generated by the points O5 , σ and σx .
−
→
−−−→
−−→
→
Let P be the vector plane defined by the vectors O5 σx and O5 σ . Let −
u be a
−
→
vector belonging to P such as
−−−→
→
• −
u and O5 σx are perpendicular i.e.
−−−→
→
(3.3)
L4,1 −
u , O5 σx = 0
→
→
• if −
u is not a light-like vector, we have Q4,1 (−
u ) = 1.
Thus, we have the following property
−−−→
−−→
→
∃(α , β ) ∈ R2 | O5 σ = α O5 σx + β −
u
(3.4)
−−−→
−−−→
→
u and O5 σx are perpenSince O5 σx is a space-like vector on the one hand, −
→
dicular on the other hand, the type of the vector −
u determine the type of the
−
→
plane P
−
→
→
• Q4,1 (−
u ) = 1 implies that the type of P is space-like and then the pencil
of spheres is a pencil of spheres with a base circle,
−
→
→
• Q4,1 (−
u ) = 0 implies that the type of P is light-like and then the pencil
of spheres is a pencil of tangent spheres,
−
→
→
• Q4,1 (−
u ) = −1 implies that the type of P is time-like and then the
pencil of spheres is a Poncelet pencil.
254
Constructions of Dupin cyclides
→
−
Type of −
u Q4,1 (→
u ) Formula (3.5) Conclusion
Time-like
Light-like
Space-like
−1
0
1
1 = α2 − β 2
1 = α2
1 = α2 + β 2
|α | > 1
|α | = 1
|α | < 1
TABLE 7. Comparison of |α | to 1 from the formula (3.5)
Morover, as σ and σx belong to Λ4 , from formulae (3.4) and (3.3), we
obtain
−−→
→
1 = Q4,1 O5 σ = α 2 + β 2 Q4,1 (−
u)
(3.5)
→
The Formula (3.5) permits to compare |α | to 1 according to the type of −
u,
Table 7. Since
−−→ −−−→
L4,1 O5 σ , O5 σx = α
(3.6)
we have the expected result.
We can note for a Dupin cyclide, if σE (respectively σH ) represents an
oriented sphere centered on the ellipse E (respectively hyperbola H), then the
type of the line (σE σH ) is light-like and we have
−−−→ −−−→
L4,1 O5 σE , O5 σH = 1
(3.7)
3.4 Theorem and algorithm
We can remark that if C is a great circle on the oriented sphere S, and if P
is an oriented plane containing the circle C, then, an expression (on Λ4 ) of a
sphere which belongs to the pencil of spheres with C as base circle is
−−−→
−−−−−→
−−−→
O5 γθ (θ ) = cos (θ ) O5 σθI + sin (θ ) O5 σθ⊥I
(3.8)
where θ ∈ [0, 2π ] and σθI (respectively σθ⊥I ) is the representation of the oriented
sphere S (respectively the oriented plane P). Moreover, we have
−−−→ −−−→
⊥
L4,1 O5 σθI , O5 σθI = 0
→
Let us recall that the points M00 and M01 and the tangent vector −
v−
Cθ0 at M00
define a characteristic circle of the future Dupin cyclide. We have the same
→
−−→
remark if we replace M01 by M10 and −
v−
Cθ by vCψ0 .
0
L. Garnier and L. Druoton
255
Theorem 2 permits to compute, from two pencils of spheres with a base
circle, a sphere in each pencil, such that these spheres are tangent. In this
theorem, the four spheres σθI , σψI , σθ⊥I and σψ⊥I verify

−−−→ −−−→
−−−→ −−−→

⊥


 L4,1 O5 σψI , O5 σθI × L4,1 O5 σψI , O5 σθI ≤ 0
(3.9)
−−−→
−−−→ −−−→

−
−
−
→

⊥
⊥
⊥

 L4,1 O5 σθI , O5 σψI × L4,1 O5 σψI , O5 σθI ≤ 0
Theorem 2 Let Cθ and Cψ be two circles passing through the point M00 such
that the tangent vectors to these circles at M00 are perpendicular.
Let σθI (respectively σψI ) be the point on Λ4 corresponding to an oriented
sphere having Cθ (respectively Cψ ) as great circle.
Let σθ⊥I (respectively σψ⊥I ) be the point on Λ4 corresponding to an oriented
plane containing the circle Cθ (respectively Cψ ).
On Λ4 , the parameterizations θ0 7→ γθ (θ0 ) and ψ0 7→ γψ (ψ0 ) of the circles
which represent these two pencils of spheres, are given by the Formula (3.8).
Let A and B be
A=
s
B=
s
2
−−−→ −−−→2
−−−→ −−−→
⊥
L4,1 O5 σψI , O5 σθI + L4,1 O5 σθI , O5 σψI
(3.10)
2
−−−→ −−−→ 2
−−−→ −−−→
⊥
⊥
⊥
L4,1 O5 σψI , O5 σθI + L4,1 O5 σθI , O5 σψI
Let θs be the solution of the system
(
cos (θs ) = A2 − B2
sin (θs ) = 2 A B
Let θ0 = −
θs
θs
or θ0 = − + π .
2
2
Let aθ0 and bθ0 be
−−−→ −−−→
−−−→ −−−→
⊥
aθ0 = cos (θ0 ) L4,1 O5 σθI , O5 σψI + sin (θ0 ) L4,1 O5 σψI , O5 σθI
−−−→ −−−→
−−−→ −−−→
⊥
⊥
⊥
bθ0 = cos (θ0 ) L4,1 O5 σθI , O5 σψI + sin (θ0 ) L4,1 O5 σψI , O5 σθI
Let ψ0 be the solution of the system
(
cos (ψ0 ) = aθ0
sin (ψ0 ) = bθ0
(3.11)
(3.12)
256
Constructions of Dupin cyclides
Then, in E3 , the spheres corresponding to the points γθ (θ0 ) and γψ (ψ0 ) are
tangent at M00 .
Proof: Appendix 5.
We can remark that the spheres γθ (θ0 ) and γθ (θ0 + π ) represent the same
geometric sphere in E3 .
From a pencil of tangent spheres at a point, Lemma 3.1 permits to determine the sphere of this pencil passing through another point. This lemma will
be useful in the steps (12) to (14) of Algorithm 1.
Lemma 3.1 :
Let σ0 (respectively −
m→
00 ) be the representation of an oriented sphere (respectively a point M00 ) of E3 .
4
Let l00 : t 7→ σ0 + t −
m→
00 be a parameterization of a light-like line (on Λ )
which represents the spheres tangent to σ0 at M00 .
Let −
m→
10 be a light-like vector which represents the point M10 of E3 .
The point M10 belongs to the sphere defined by the point l00 (t0 ) if and only
if
−−−→
−
→
L4,1 O5 σ0 , m10
−→
m→
t0 = f (σ0 , −
(3.13)
00 , m10 ) = −
L (−
m→, −
m→)
4,1
00
10
Proof:
By construction of the space of spheres, the point M10 belongs to the sphere
defined by the point l00 (t0 ) if and only if (an analytic proof is easy) :
−−−−−→
−
→
L4,1 O5 l00 (t0), m10 = 0
(3.14)
and we have
−−−−−→
−−−→
−
→
−
→
−
→
L4,1 O5 l00 (t0 ), m10 = 0 ⇐⇒ L4,1 O5 σ0 + t0 m00 , m10 = 0
−−−→
−→ −→
⇐⇒ L4,1 O5 σ0 , −
m→
10 + t0 L4,1 (m00 , m10 ) = 0
−−−→
m→
L4,1 O5 σ0 , −
10
⇐⇒ t0 = −
−
→
−
→
L (m , m )
4,1
00
10
L. Garnier and L. Druoton
257
Algorithm 1 Computation of a Dupin cyclide principal patch in the space of
spheres
Input: Four points Mi j (i, j)∈[0,1]2 on a circle and two perpendicular vectors
−
→
−−→
v−
Cθ0 and vCψ0 .
1. For i from 0 to 1 do
For j from 0 to 1 do
Computation of −
m→
i j which represents the point Mi j on the
paraboloid P
End do
End do.
2. Construction of the circle Cθ passing through the points M00 and M01
→
such that the tangent vector at M00 is −
v−
Cθ0 .
3. Construction of the circle Cψ passing through the points M00 and M10
→
such that the tangent vector at M00 is −
v−
C ψ0 .
4. Construction of σθI which is the representation of an oriented sphere in
E3 having Cθ as great circle.
5. Construction of σθ⊥I which is the representation of an oriented plane in
E3 containing Cθ .
6. Construction of σψI which is the representation an oriented sphere in E3
having Cψ as great circle.
7. Construction of σψ⊥I which is the representation of an oriented plane in
E3 containing Cψ .
8. Determination of the circle γθ (θ0 ) representing the pencil of spheres
with base circle Cθ , Formula (3.8).
9. Determination of the circle γψ (ψ0 ) representing the pencil of spheres
with base circle Cψ , Formula (3.8).
10. Computation of θ0 and ψ0 using Theorem 2.
11. Computation of the point σθ0 = γθ (θ0 ) and σψ0 = γψ (ψ0 ).
12. Determination of a parameterization of the light-like line l01 : t 7→ σψ0 +
t−
m→
10 .
−→
m→
13. Computation of t1 = f σψ0 , −
10 , m11 using Formula (3.13).
14. Computation of the point σθ2 = l01 (t1).
•
15. Computation of the point σ θ0 , Formula (2.22).
•
Output: Three points σθ0 , σθ2 and σ θ0 on Λ4 which generate one of the two
planes defining a Dupin cyclide.
258
Constructions of Dupin cyclides
r
σψ2
Λ4
γθ ( θ )
σψr 0 = γψ (ψ0 )
O5 b
γψ ( ψ )
l10 (t)
l11 (t)r σ
r
σθ0 = γθ (θ0 )
θ2
F IGURE 9. Construction of a Dupin
cyclide
of
principal
patch in the space
4
spheres Λ , the lines σθ0 σψ0 , σθ2 σψ0 , σθ2 σψ2 , and σθ0 σψ2 , are
−→ −→
−→
light-like lines and have −
m→
00 , m10 , m11 , and m01 as direction vectors
respectively: The pink (respectively green) 2-plane is the plane containing the
pencil of spheres with base circle Cθ0 (respectively Cψ0 ): We use Theorem 2
to compute the spheres σθ0 and σψ0 whereas we use Lemma 3.1 to compute
the spheres σθ2 and σψ2 : Each pencil of spheres is represented by a circle (for
the Lorentz form) which is the intersection between Λ4 and an affine
space-like 2-plane passing through O5
L. Garnier and L. Druoton
259
r
σψ2
Λ4
γθ ( θ )
r
σψ0
Ob 5
•
σθ0
b
γψ ( ψ )
−−
−−→
γθ0 (θ0 )
r σ
θ2
r
r
Ωθ
σθ0
F IGURE 10. Construction of a Dupin cyclide principal patch in the space of
spheres Λ4 , the pink 2-plane contains the circle representing the pencil of
spheres with a base the circle Cθ : The green 2-plane contains the circle
representing a family of spheres which defines the Dupin cyclide: This plane
•
σθ2 and
the characteristic circle σθ0 ∩ σθ0 which is
is generated by the sphere
−−−−→
defined by the pair σθ0 , γθ0 (θ0 ) , these three elements are computed using
Algorithm 1
260
Constructions of Dupin cyclides
For simplicity , sometimes, we use the same notation for a sphere in E3 and
its representation on Λ4 .
In Algorithm 1, we propose a method to compute the Dupin cyclide. To
obtain the required principal patch, it is enough to trim the previous Dupin
cyclide and to keep the useful part. The steps (1) to (7) permit to transfer, on
Λ4 , the constraints enforced in E3 . This algorithm uses Theorem 2, the spheres
σθI , σθ⊥I , σψI and σψ⊥I verify the conditions given by the Formula (3.9).
The steps (8) and (9) permit to obtain the usual parameterization of a circle in
a affine 2-plane equipped with a dot product. The steps (10) and (11) permit
to determine the two spheres tangent at the point M00 , each sphere belongs to
each pencil.
The steps (12) to (14) permit to compute the second sphere σθ2 . Using Lemma
3.1, we compute the sphere σθ2 , tangent to Sψ0 at M10 , passing through the
point M11 .
•
In the step (15), we compute the last sphere σθ0 defined by the sphere σθ0 and
−−−−−→
the tangent vector γθ 0 (θ0 ), Formula (2.22).
We can remark that iterating the part of the algorithm corresponding to the
steps (12) to (14) computes the sphere σψ2 of the second family : the second
circle on Λ4 representing the same Dupin cyclide would be defined by the
•
intersection between Λ4 and the affine 2-plane generated by σψ2 , σψ0 and σψ0 .
Continuing the process, we would compute the point σθ3 which represents the
oriented sphere Sθ3 in E3 . Since we use Theorem 2 and Algorithm 1, we would
have σθ3 = σθ0 , generally, this result is wrong and σθ3 belongs to the curve γθ
but is distinct to σθ0 .
Figure 9 shows the two pencils of spheres, each pencil is represented by
a circle with centre O5 and belongs to a space-like 2-plane passing through
O5 . The points σθ0 and σψ0 which belong to the two pencils, are determined
using Theorem 2. The points σθ0 (respectively σψ0 ) is the representation of the
oriented sphere Sθ0 (respectively Sψ0 ) in E3 . The computation of the point σθ2
is made using Lemma 3.1: σθ2 belongs to a light-like line and we know −
m→
11
which represents, in E3 , the point M11 belonging to the sphere Sθ2 represented
by σθ2 . The spheres Sθ0 and Sψ0 are tangent at M00 . The spheres Sθ2 and Sψ0
are tangent at M10 .
The spheres Sθ2 and Sψ2 would be tangent at M11 . Finally, The spheres Sψ2
and Sθ0 would be tangent at M01 .
Figure 10 shows the point σθ0 which belongs
the circle γθ . The charac to−
−−0 −−→
−−−−−→
teristic circle on σθ0 is determined by the pair σθ0 , γθ (θ0 ) where γθ 0 (θ0 ) is
the tangent vector to the curve γθ at σθ0 . Thus, the Dupin cyclide is completely
defined : it is the intersection between Λ4 and the affine 2-plane generated by
−−−−−→
the points σθ0 and σθ2 and the vector γθ 0 (θ0 ). This vector determines the point
L. Garnier and L. Druoton
261
Name
In E3
In Λ4
σθI
σθ⊥I
((0.565, 4.294, −0.556), 2.753)
(2.268, 0.205, 1.560, −0.202, 1.905)
0.057 x − 0.429 y + 0.901 z − 1.375 = 0
(1.375, 0.057, 0.429, 0.901, 1.375)
σψI
((2.737, 0.360, 1.886), 3.108)
(0.405, 0.881, 0.116, 0.607, 0.083)
σψ⊥I
−0.968 x − 0.128 y − 0.214 z + 3.100 = 0 (−3.100, −0.968, −0.128, −0.214, −3.100)
TABLE 8. Spheres defined by the principal patch
•
•
σθ0 which represents, in E3 , the sphere Sθ0 orthogonal to the sphere Sθ0 .
3.5 Example
In this section, we introduce the following notation: C (Ω, r) (respectively
S (Ω, r)) is a circle (respectively sphere) with centre Ω and
with radiusr.
√
√
The four vertices of the future principal patch M00 = 2 2, 2 2, 0 , M10 =
√
√ √
√ 2 3, −2, 0 , M01 = −2, 2 3, 0 and M11 = −2 2, −2 2, 0 belong to
the circle C (O3 , 4) (in black) in the plane Pz : z = 0, Figure 11(a).
The characteristic circles are modelized by two couples of rational quadratic
Bézier curves defined by the weighted control points (M00 , 1), (P01 , ±ω01 ),
(M01 , 1) and (M00 , 1), (P10 , ±ω10 ), (M10 , 1) with
(P01 , ω01 ) ' ((−0.129, −0.980, 2), 0.466)
and
1
3.798, , −3 , 0.621
(P10 , ω10 ) '
2
The initial spheres are characterized by the centres and the radii, noted
(Ω, r), and their representations on Λ4 are given in the Table 8.
We obtain
(θs , θ0 , ψ0 ) ' (0.7470776248, −0.3735388124, 2.164840770)
and the spheres which define the Dupin cyclide are given in the Table 9. Since
we have
−−−→
−−−→
−−−→
O5 σθ0 = cos (θ0 ) O5 σθI + sin (θ0 ) O5 σθ⊥I
•
the sphere σθ0 is computed using the following formula
−−−→
−−−→
•
−−−→
O5 σθ0 = − sin (θ0 ) O5 σθI + cos (θ0 ) O5 σθ⊥I
262
Constructions of Dupin cyclides
(a)
(b)
F IGURE 11. Example: (a) the vertices M00 , M01 , M11 and M10 belong to the
black circle in the plane Pz : z = 0, the other circle are the characteristic circle
to the Dupin cylide to compute, (b) The Dupin cyclide is characterized, on Λ4 ,
•
by the three points σθ0 , σθ2 and σθ0 which represent, in E3 the three spheres
•
•
Sθ0 , Sθ2 and Sθ0 : We can note the two spheres Sθ0 and Sθ0 are orthogonal
L. Garnier and L. Druoton
263
Name
In Λ4
In E3
σθ0
(1.610, 0.171, 1.295, −0.517, 1.272)
((0.504, 3.830, −1.529), 2.957)
(2.107, 0.128, 0.969, 0.765, 1.975)
((0.962, 7.310, 5.774), 7.544)
σθ2
(0.704, 0.131, −0.994, −0.517, 0.473)
((0.561, −4.291, −2.232), 4.317)
σψ0
(−2.796, −1.296, −0.171, −0.517, −2.616)
((7.192, 0.947, 2.871), −5.552)
•
σθ0
σψ2
(−1.890, 0.994, −0.131, −0.517, −1.816) ((−13.516, 1.779, 7.030), −13.597)
TABLE 9. Spheres which define the Dupin cyclide
and one of two affine 2-planes which characterize the Dupin cyclide is gener−−−→
•
ated by the points σθ0 and σθ2 and the vector O5 σθ0 .
We obtain the Dupin cyclide parameters
(a, c, µ ) ' (4.934, 1.803, 2.849)
and using the Formula (2.6) and the following affine matrix (see [14]),


0.059 0.039 0.998 0.698
 0.379 −0.925 0.013 −0.301 


(3.15)
 0.923 0.377 −0.069 0.471 
0
0
0
1
we can place the Dupin cyclide in the scene, Figure 12. Then, we can calculate the bounds of the Dupin cyclide principal patch, Figure 13. We obtain θ0 ' 2.521616545, θ1 ' 4.652818777, ψ0 ' −0.9430762257 and ψ1 '
1.266085740.
4 Conclusions
In this paper, we have given an algorithm to compute a principal patch Dupin
cyclide in the space of spheres. The constraints are four vertices on a circle and
two perpendicular tangents to the patch at one of the vertices. The use of the
space of spheres permits to simplify the work made by Garnier and Gentil: we
never solve any equations.
In the future, we should compute a Dupin cyclide triangle tangent to three
planes at one point on each previous planes. We expect to build some subdivision schema with Dupin cyclide patches and Dupin cyclide triangles.
264
Constructions of Dupin cyclides
(a)
(b)
F IGURE 12. Two views of a Dupin cyclide defined by four points on a circle
and two characteristic circles
Acknowledgements
The authors thank D. Michelucci, from the University of Burgundy, for their
constructive remarks about this paper.
The authors thank the "Qatar National Research Fund" because this research work has been funded in part by NPRP grant number NPRP-09-906-1137 from the Qatar National Research Fund (a member of the Qatar Foundation).
References
[1] W. Degen. Handbook of Computer Aided Geometric Design. Elsevier,
North Holland, ISBN: 9780080533407, 575-602, 2002.
[2] M. Berger, M. Cole and S. Levy. Geometry II. Springer, ISBN:
9783540170150, 2009.
[3] D. McLean. A method of Generating Surfaces as a Composite of Cyclide
Patches. The Computer Journal, 28(4), 433-438 1985.
L. Garnier and L. Druoton
265
F IGURE 13. Two views of the Dupin cyclide principal patch obtained from
Figure 8 using Algorithm 1
266
Constructions of Dupin cyclides
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L. Garnier and L. Druoton
269
5 Appendix I: proof of theorem 2:
Before the proof, we need two lemmas.
5.1 Lemmas
Lemma 1 First, we have
−−−→ −−−→
−−−→ −−−→
⊥
⊥
L4,1 O5 σψI , O5 σθI × L4,1 O5 σθI , O5 σψI =
−−−→ −−−→
−−−→ −−−→
⊥
⊥
L4,1 O5 σθI , O5 σψI × L4,1 O5 σψI , O5 σθI
and
2
−−−→ −−−→2
−−−→ −−−→
+
L4,1 O5 σψI , O5 σθI
+ L4,1 O5 σθI , O5 σψ⊥I
L4,1
2
−−−→ −−−→ 2
−−−→ −−−→
⊥
⊥
⊥
O5 σψI , O5 σθI
= 1
+ L4,1 O5 σθI , O5 σψI
(5.1)
(5.2)
Second, we have
(A cos θ0 − B sin θ0 )2 = 1 and aθ20 + b2θ0 = 1
(5.3)
where A and B are given by Formula (3.10) whereas aθ0 and bθ0 are given by
Formula (3.11).
Proof:
Without loss of generality, we can always consider the following case.
The point M00 (0, 1, 0) belongs to two circles. Let Ωθ0 (0, 0, 0) (respectively
rθ0 = 1) be the centre (respectively the radius) of the first circle Cθ0 (and of the
→
first sphere). Let −
v−
Cθ0 (−1, 0, 0) be a tangent vector to the circle at M00 . Using
Formulae (2.16) and (2.18), we have
σθI = (0, 0, 0, 0, −1)
and σθ⊥I = (0, 0, 0, −1, 0)
−−→
→
Let −
v−
Cψ0 (0, α , β ) be an unit vector orthogonal to vCθ0 . We have to distinguish two cases.
5.1.1 First case: α = 1 =⇒ β = 0.
The centre of the second circle Cψ0 is
Ωψ0 (x0 , 1, z0 )
270
Constructions of Dupin cyclides
and the radius of Cψ0 is
rψ0 =
p
x0 2 + z0 2
Thus, the sphere S Ωψ0 , rψ0 is represented on Λ4 by


x0
1
z0
1
σψI =  q
,q
,q
,q
, 0
2
2
2
2
2
2
2
2
x0 + z0
x0 + z0
x0 + z0
x0 + z0
whereas the plane containing the circle Cθ0 , orthogonal to the previous sphere,
is represented on Λ4 by


z0
x0
, 0, − q
, 0
σψ⊥I = 0, q
2
2
2
2
x0 + z0
x0 + z0
and
We have
−−−→ −−−→
z0
−−−→ −−−→
⊥
×0 ≤ 0
L4,1 O5 σθI , O5 σψI × L4,1 O5 σψI , O5 σθI = − q
x20 + z20
−−−→ −−−→
−−−→ −−−→
x0
⊥
L4,1 O5 σθI , O5 σψI × L4,1 O5 σψ⊥I , O5 σθ⊥I = q
×0 ≤ 0
x20 + z20
and the conditions given in Formula (3.9) are true.
Then, the Formulae (5.1) and (5.2) are true.
As we have
−−−→ −−−→
−−−→ −−−→
⊥
L4,1 O5 σθI , O5 σψI = L4,1 O5 σθI , O5 σψI
the condition given in Formula (3.9) is true.
Moreover, we have A = 0 and B = 1. Thus, we obtain θs = π . Choosing
θ0 = π2 and using Formula (3.11), we obtain

z0

a θ0 = − q




x20 + z20
(5.4)
x0


q
=
b

θ

 0
x20 + z20
and we have aθ20 + b2θ0 = 1. We have too
π
π 2
(A cos θ0 − B sin θ0 )2 = 0 × cos − 1 × sin
=1
2
2
(5.5)
L. Garnier and L. Druoton
271
5.1.2 Second case: 0 ≤ α < 1 and β =
√
1 − α 2.
Without loss of generality, we can always consider the following case.
The centre of the second circle Cψ0 is
α (y0 − 1)
Ωψ0 x0 , y0 , − √
1 − α2
whereas its radius is
rψ0 =
s
x0 2 (1 − α 2 ) + (y0 − 1)2
1 − α2
Thus, we have on the one hand
where
σψI = q
1
(1 − α 2 ) x20 + (y0 − 1)2
ΣψI
p
p
p
p
ΣψI = y0 1 − α 2 ; x0 1 − α 2 ; y0 1 − α 2 ; (1 − y0 ) α ; (y0 − 1) 1 − α 2
and we have on the other hand
where
σψ⊥I = q
1
(1 − α 2 ) x20 + (y0 − 1)2
Σ⊥
ψI
p
2
2
2 , x 1 − α2
,
1
−
y
,
x
,
−x
1
−
α
α
α
1
−
α
=
x
1
−
Σ⊥
0
0
0
0
0
ψI
We can remark the condition given in Formula (3.9) is true
√
−−−→ −−−→
−−−→ −−−→
1 − α 2 α (y0 − 1)2
⊥
L4,1 O5 σψI , O5 σθI ×L4,1 O5 σψI , O5 σθI = −
≤0
(1 − α 2 ) x20 + (y0 − 1)2
and
L4,1
√
−−−→ −−−→
x20 α 2 − 1 α 1 − α 2
−−−→ −−−→
⊥
⊥
⊥
O5 σθI , O5 σψI × L4,1 O5 σψI , O5 σθI =
≤0
(1 − α 2 ) x20 + (y0 − 1)2
Moreover
−−−→ −−−→
−−−→ −−−→
(1 − y0 ) x0 α 1 − α 2
⊥
⊥
=
L4,1 O5 σθI , O5 σψI × L4,1 O5 σψI , O5 σθI =
(1 − α 2 ) x20 + (y0 − 1)2
−−−→ −−−→
−−−→ −−−→
⊥
⊥
L4,1 O5 σθI , O5 σψI × L4,1 O5 σψI , O5 σθI
and the Formula (5.1) is true. Thus
272
A2
Constructions of Dupin cyclides
=
=
=
B2
2
−−−→ −−−→2
−−−→ −−−→
⊥
L4,1 O5 σθI , O5 σψI + L4,1 O5 σθI , O5 σψI
2
(1 − y0 )2 1 − α 2 + x20 α 2 − 1
(1 − α 2 ) x20 + (y0 − 1)2
1 − α2
(1 − y0 )2 + x20 1 − α 2
(1 − α 2 ) x20 + (y0 − 1)2
=
1 − α2
=
2
−−−→ −−−→ 2
−−−→ −−−→
⊥
⊥
⊥
L4,1 O5 σψI , O5 σθI + L4,1 O5 σψI , O5 σθI
=
α 2 (y0 − 1)2 + x20 α 2 1 − α 2
=
α
=
α2
(1 − α 2 ) x20 + (y0 − 1)2
2
2 1 − α2
(y
−
1)
+
x
0
0
2
(1 − α 2 ) x20 + (y0 − 1)2
Thus, A2 + B2 = 1 and the Formula (5.2) is true. Moreover, θs = π is
solution of the system
(
cos (θs ) = 1 − 2 α 2
√
sin (θs ) = 2 α 1 − α 2
and using the relation 2 cos2 (θ0 ) = 1 + cos (2 θ0 ) = 1 + cos (−θs ), θ0 verifies
(
cos2 (θ0 ) = 1 − α 2
sin2 (θ0 ) =
α2
and we obtain
(
√
cos (θ0 ) = εc 1 − α 2
sin (θ0 ) =
εs α
,
(εc , εs) ∈ {−1, 1}
From the following formula
2 tan (θ0 )
= tan (2θ0 ) = − tan (θs )
1 − tan2 (θ0 )
L. Garnier and L. Druoton
we have
2ε α
√ s
×
εc 1 − α 2
which can be simplified into
and then, we obtain
273
√
2 α 1 − α2
1
=−
1 − 2 α2
α2
1−
1 − α2
√
εs
1 − α2
1 − α2
√
=
−
×
1 − 2 α2
ε c 1 − α 2 1 − 2α 2
εs = −εc
We can remark we have the following relation
p
2
p
2
1 − α 2 εc 1 − α 2 − α εs α = εc 1 − α 2 + α 2
(A cos θ0 − B sin θ0 )2 =
=1
(5.6)
and then, we obtain

√
√

εc 1 − α 2 (1 − y0 ) 1 − α 2 − εc α 2 (y0 − 1)


q
a θ0 =




(1 − α 2 ) x20 + (y0 − 1)2
√
√
2x α 2 − 1 − ε α x α 1 − α 2

ε
1
−
α

c
c
0
0

 b θ0 =
q



(1 − α 2 ) x2 + (y − 1)2
0
0
which is equivalent to

2 − ε α2

−
ε
α
1
−
c
c


aθ0 = (y0 − 1) q




(1 − α 2 ) x20 + (y0 − 1)2

√
εc α 2 − 1 − εc α 2


2

bθ0 = x0 1 − α q



(1 − α 2 ) x2 + (y0 − 1)2
0
which is equivalent to

−εc (y0 − 1)


a θ0 = q




(1 − α 2 ) x20 + (y0 − 1)2
√

−
ε
x
1 − α2

c
0
 bθ = q


 0
(1 − α 2 ) x20 + (y0 − 1)2
(5.7)
and, finally, we obtain
a2θ0 + b2θ0 = 1
(5.8)
274
Constructions of Dupin cyclides
Lemma 2 : We have
s
2
−−−→ −−−→2
−−−→ −−−→
⊥
L4,1 O5 σψI , O5 σθI + L4,1 O5 σθI , O5 σψI
×
s
2
−−−→ −−−→ 2
−−−→ −−−→
⊥
⊥
⊥
L4,1 O5 σψI , O5 σθI + L4,1 O5 σθI , O5 σψI
+
(5.9)
−−−→ −−−→
−−−→ −−−→
⊥
+
L4,1 O5 σψI , O5 σθI L4,1 O5 σψI , O5 σθI
−−−→ −−−→
−−−→ −−−→
⊥
⊥
⊥
L4,1 O5 σθI , O5 σψI L4,1 O5 σθI , O5 σψI = 0
Proof:
Let us recall the conditions given in Formula (3.9). Let us suppose Formula
(5.9) is true, then, we have
2 !
−−−→ −−−→2
−−−→ −−−→
×
L4,1 O5 σψI , O5 σθI + L4,1 O5 σθI , O5 σψ⊥I
2
!
−−−→ −−−→ 2
−−−→ −−−→
L4,1 O5 σψI , O5 σθ⊥I + L4,1 O5 σθ⊥I , O5 σψ⊥I
=
−−−→ −−−→
−−−→ −−−→ 2
−−−→ −−−→
−−−→ −−−→
⊥
⊥
⊥
⊥
L4,1 O5 σψI , O5 σθI L4,1 O5 σψI , O5 σθI + L4,1 O5 σθI , O5 σψI L4,1 O5 σθI , O5 σψI
which is equivalent (after some computations) to
2
−−−→ −−−→
−−−→ −−−→
−−−→ −−−→
−−−→ −−−→
⊥
⊥
⊥
⊥
=0
L4,1 O5 σψI , O5 σθI L4,1 O5 σθI , O5 σψI − L4,1 O5 σθI , O5 σψI L4,1 O5 σψI , O5 σθI
and this relation is true from Formula (5.1).
5.2 Proof of Theorem 2 :
From Formula (3.8), we have
−−−→
−−−−−−→
−−−→
O5 γθ (θ0 ) = cos (θ0 ) O5 σθI + sin (θ0 ) O5 σθ⊥I
and
−−−→
−−−−−−→
−−−→
O5 γψ (ψ0 ) = cos (ψ0 ) O5 σψI + sin (ψ0 ) O5 σψ⊥I
L. Garnier and L. Druoton
275
Using Theorem 1 and Formula (3.7), the spheres represented by γθ (θ0 ) and
γψ (ψ0 ) are tangent if and only if
−−−−−−→ −−−−−−→
(5.10)
L4,1 O5 γθ (θ0 ), O5 γψ (ψ0 ) = 1
Then, we can simplify aθ0 into
−−−→ −−−→
−−−→ −−−→
⊥
aθ0 = cos (θ0 ) L4,1 O5 σθI , O5 σψI + sin (θ0 ) L4,1 O5 σψI , O5 σθI
and bθ0 into
bθ0 = cos (θ0 ) L4,1
−−−→ −−−→
−−−→ −−−→
⊥
O5 σθI , O5 σψI + sin (θ0 ) L4,1 O5 σψ⊥I , O5 σθ⊥I
Thus, Equation (5.10) is equivalent to
aθ0 cos (ψ0 ) + bθ0 sin (ψ0 ) = 1
(5.11)
which admits solution is and only if aθ20 + bθ20 ≥ 1. Using Lemma 1, we have
a2θ0 + b2θ0 = 1
which is equivalent to
cos (2θ0 + θs ) = 2
1 − B2
− cos (θs )
A2 + B2
where θs is the solution of the system

A2 − B2


 cos (θs ) =
A2 + B2

AB

 sin (θs ) = 2
2
A + B2
The relation A2 + B2 = 1 implies
cos (2θ0 + θs ) = 2 − 2B2 − A2 − B2 = 2 − A2 + B2 = 1
and we obtain two solutions θ0 = −
θs
θs
and θ0 = − + π .
2
2
The relation aθ20 + b2θ0 = 1 implies the Formula (5.11) is equivalent to
cos (ψs ) cos (ψ0 ) + sin (ψs ) sin (ψ0 ) = 1
(5.12)
276
Constructions of Dupin cyclides
where ψs is the solution of the system
(
cos (ψs ) = aθ0
sin (ψs ) = bθ0
and, finally, from Formula (5.12), we obtain ψs = ψ0 .
The Mathematics of Surface XIV
Edited by Robert J. Cripps, G. Mullineux and M. A. Sabin
Published by The Institute of Mathematics and its Applications
ISBN 978–0–905091–30–3