G 1 -blend between a differentiable superquadric of

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1 G 1 -blend between a differentiable suerquadric of revolution and a lane or a shere using Duin cyclides Lionel GANIE and Sebti Foufou and Yohan Fougerolle Laboratoire LEI, UM CNS 5158, Université de Bourgogne, BP. 4787, 178 Dijon, France {firstname.lastname}@u-bourgogne.fr Abstract In this article, we resent a method to erform G 1 -continuous blends between a differentiable suerquadric of revolution and a lane or a shere using Duin cyclides. These blends are atches delimited by four lines of curvature. They allow to avoid arameteriation roblems that may occur when arametric surfaces are used. ational quadratic Béier curves are used to aroximate the rincial circles of the Duin cyclide blends and thus a comlex 3D roblem is now reduced to a simler D roblem. We resent the necessary conditions to be satisfied to create the blending atches and illustrate our aroach by a number of suerellisoid/lane and suerellisoid/shere blending examles. I. INTODUCTION Duin cyclides have been introduced in 18 by the mathematician Pierre-Charles Duin [6]. Although these surfaces have many interesting geometric roerties [6], [4], [9], [1], this aer mainly uses the following ones: Duin cyclides lines of curvature are circles. Duin cyclides have two imlicit equivalent equations and a canonical arametric formulation. A Duin cyclide has two airs of two colanar rincial circles from which the surface arameters can be determined. In the 198s,. Martin revived the study of Duin cyclides and alied them to geometric modeling [16]. Since then, Duin cyclides have been used in this domain to make blends between circular quadrics and/or lanes [1], [], [17], [19], []. In this aer we consider the use of Duin cyclides as G 1 -continuous blends between a lan or a shere and a differentiable suerquadric of revolution which we define as a regular suerquadric verifying certain roerties that make it differentiable everywhere. An examle of such blending is illustrated in Figure 1. Two different kinds of blends are ossible, namely the illar and container configurations. ational Quadratic Béier Curves QBCs are used to comute the rincial circles of the blending Duin cyclide. The whole Duin cyclide is not necessary and has to be trimmed: only a arametric quarter in the angular arameter domain [ ], π or [ π, π] of the whole surface is used to define the blending atch. Fig. 1. A Duin cyclide blends a differentiable suerquadric of revolution and a lane. To: Pillar construction. Bottom: Container construction. Notations and definitions: for convenience and without loss of generality, we assume = O is axis of revolution of the rimitive to be blended with a given lane. We also further resent extension to erform rimitive/rimitive blends. The rimitive or suerquadric circle, that is the intersection of the considered rimitive and the lane orthogonal to the revolution axis, is considered lying on the lane P : = const. We use for our examles P : =, and we consider the uer art of the rimitive, i.e. > const. Figure 1 illustrates the roblem and the main geometric notions. Princial circles arcs γ 3 and γ 4, in red, are the intersection of the Duin cyclide S 1 and the construction lane P y : y =. P or P y are the lanes of symmetry for both rimitives. In blue is, the axis of the suerquadric of revolution S. In

2 black is the section of the trimming Duin cyclide. The Duin cyclide S 1 connects to the suerquadric S res. the lane P along the curve γ res. γ 1. When the Duin cyclide kind is known, two rincial circles determine one unique Duin cyclide: thus, the roblem of comuting a Duin cyclide in 3D is reduced to the simler D roblem of determining two of its colanar rincial circles. This aer is organied as follows: Section resents a short background on rational quadratic Béier curves and Duin cyclides. In Section 3, we resent the differentiable suerquadrics of revolution. Section 4 is dedicated to the conditions to obtain a G 1 blend between a Duin cyclide and a differentiable suerquadrique of revolution. Section 5 shows how it is ossible to join a differentiable suerquadric of revolution and a lane using a Duin cyclide and an algorithm is given. Blends between a shere and a suerquadric are resented in Section 6. A comlete examle is given in section 7. The last section resents our conclusions and future work. II. BACKGOUND In this section, we recall the main roerties of ational Quadratic Béier Curves QBC [8], [14], [5] and Duin cyclides. A background on suerquadrics suerellisoids and suerhyerboloids of one sheet and their differential roerties are resented in more detail in the next section. A. ational Quadratic Béier Curves The ational Quadric Béier Curves QBC are definite arametric curves starting from the Bernstein s olynomials of degree, defined by B t = 1 t, B 1 t = t 1 t and B t = t. The symmetry of the Berstein s olynomials is essential for the construction of an arc of circle since it ensures the symmetry of geometrical constructions. A QBC is the set of oints Mt defined by OMt = wbt OP + w 1B 1t OP 1 + w B t OP, 1 w B t + w 1B 1t + w B t where t [, 1]. The definition of a QBC does not deand on the oint O. The lines P P 1 and P P 1 are tangent to the curve at the oints P and P resectively. This tye of curve, noted QBC {P ; P 1 ; P }, makes ossible to model conics arcs using three control oints P i i and three weights w i i. This definition can be simlified, and is known as standard form, by setting w = w = 1 and w 1 = w >. The weight w determines the nature of the curve, which is an arc of ellise if and only if < w < 1, an arc of arabola if and only if w = 1, and an arc of hyerbole if and only if w > 1. B. Duin cyclides There are various ways to define Duin cyclides. A Duin cyclide is the image of a circular cone by an inversion. A Duin cyclide is the image of a torus by an inversion. A Duin cyclide is the enveloe of sheres centered on a given conic, called deferent, and orthogonal to a given fixed shere, called shere of inversion: the latter is centered on the focal axis of the cyclide [4]. A Duin cyclide is the enveloe surface of a set of sheres, the centers of which lie on a given conic with focus F, and the radius of which is such that the distance F M + is a given constant this definition is due to Maxwell. A Duin cyclide is the enveloe surface of the sheres tangent to three given fixed sheres. Duin cyclides of degree 4 deend on three indeendent arameters a, c and µ with a c. It is convenient to define b = a c. The arametric equation of a Duin cyclide is defined by Γ d θ; ψ = µc a cosθ cosψ + b cosθ a c cosθ cosψ b sinθ a µ cosψ a c cosθ cosψ b sinψ c cosθ µ a c cosθ cosψ where θ; ψ [, π]. A Duin cyclide has two equivalent imlicit equations [4], [9] x + y + µ + b = 4 ax cµ + 4b y 3 and x + y + µ b = 4 cx aµ 4b 4 According to the arameter values, there are three kinds of Duin cyclides: ring cyclide, horned cyclide and sindle cyclide, as illustrated in Figure. Fig.. Three kinds of Duin cyclides. Left: ring with c µ a. Middle: horned with < µ c < a. ight: half sindle with c a < µ. Fig. 3. Cross section of a Duin cyclide along the symmetry lane P y : y = Duin cyclides admit two symmetry lanes P y : y = and P : =. The section of a Duin cyclide by one of these two lanes is the union of two circles called rincial circles. From these two rincial circles, the three arameters a, c and µ can be determined to define a Duin cyclide. The Figure 3 shows the two rincial circles of a Duin cyclide in the lane P y and arameters are determined as follows: Theorem 1: : Determination of Duin cyclide s arameters in P y y = Let C 1 and C be the rincial circles of a Duin cyclide in P y y =. The centers of the circle C 1 and C are O 1 and O, and their radii are ρ 1 and ρ, resectively, with ρ 1 ρ.

3 To obtain a ring or a sindle Duin cyclide, the values of arameters are a = O 1O, µ = ρ 1 + ρ, c = ρ 1 ρ. Similar results can be obtained with the lane P. Lines of curvature of Duin cyclides are circles, obtained with θ or ψ constant [17]. The blends with Duin cyclide are made along lines of curvature [7]: each blended surface meets the blending Duin cyclide along lines of curvature. Only a art of rincial circles is useful and is reresented as a standard QBC [1], [1]. Thus, trimming curves have a simle arametric equation, Γ d θ; concerning the suerquadric side and Γ d θ; π concerning the lane side. For other roerties of Duin cyclides, see [4], [6], [17], [18], [19], [1]. III. DIFFEENTIABLE SUPEQUADICS Suerellises have been introduced by G. Lam [15] and M. Gardiner [11] as a generaliation of ellises. In the same way, in 1981, Alan Barr used owers of cosines and sines to introduce the suerquadrics [3], [1]. There are four kinds of suerquadrics suerellisoids, suerhyerboloids of one sheet, suerhyerboloids of two sheets, and suertoroids. In this aer, we illustrate our aroach using suerellisoids and suerhyerboloids of one sheet. Suerquadrics have both imlicit and arametric formulations. Unfortunately, in the original formulation, a non continuous function was used, so we cannot use the usual comutation of differential geometry. We roose to restrict the owers to a subset of rational numbers. We call the resulting surfaces differentiable suerquadrics. We show further that any suerquadric is a limit of a sequence of differentiable suerquadrics. The suerquadrics have been defined with real ower and a function sgn, which is not continuous and therefore not differentiable. To retrieve the differential roerties of these surfaces, we roose to use articular rational owers where the numerator and the denominator q are odd ositive integers. Under this condition, t t q is a differentiable one-to-one function from into almost everywhere. The suerquadrics defined using these owers are called differentiable suerquadrics. Two examles of these surfaces suerellisoids and suerhyerboids of one sheet are given in sections III-A and III-B and their differentiability is shown through the comutation of artial derivatives. A. Suerellisoids The arametric equation of a suerellisoid is Γ sb u, v = a Cos ε1 u Cos ε v a 1 Sin ε1 u Cos ε v, 5 a Sin ε v where Cos ε t = sgncost cost ε, Sin ε t = sgnsin t sint ε and u, v [, π] [ π, ] π. The imlicit equation of a suerellisoid is ε 1 x ε 1 y ε ε ε 1 =. 6 a a 1 When ε 1 = ε = 1, the suerellisoid is an ellisoid. When ε 1 and ε tend to, the suerellisoid tends to a rectangle aralleleied. 1 Differentiable suerellisoid: First, we will define a subset Q 1 which is dense for the canonical toology in + : { } Q 1 =, ; q N N 7 q It is easy to rove that Q 1 = +. We will remlace the real owers ε 1 and ε in equation 5 by two rationals, belonging to Q 1, 1 q 1 and. So, the arametric equation of a differentiable suerellisoid is Γ sd : u, v a a cos 1 q 1 u cos v a 1 sin 1 q 1 u cos v a sin v, 8 where u, v [, π] [ π, ] π and 1 q 1, Q 1. The imlicit equation of a differentiable suerellisoid is x a q 1 1 y q 1 1 q =. 9 a 1 a Proerties of differentiable suerquadrics: Firstly, let Γ sb the ma of a suerquadrics, equation 5. One can find a sequence 1,n res.,m,m of Q 1 such that 1,n,m lim = ε 1 and lim = ε n+ m+,m For each ower 1,n and,m,m, let Γ 1,n q,,m the ma of a 1,n,m differentiable suerquadric. It is easy to see that : and From the sequence lim n+ Γ 1,n,,m,m = Γ ε1,,m,m 1 lim m+ Γ 1,n,,m,m = Γ 1,n Γ 1,n extract a sub-sequence of ma Γ 1,k q 1,k,,k,k q,,m 1,n,m n N,m N,ε 11, it is ossible to such that k N lim k+ Γ 1,k q 1,k,,k,k = Γ ε1,ε 1 Finally, we can state Theorem : For each suerquadric of revolution, there is a sequence of differentiable suerquadrics tending to it. Secondly, the function f : x sgnx x ε has two exressions : x < imlies that fx = e εln x x > imlies that fx = e εlnx

4 and lnx is not defined for x = then there is discontinuity at. Moreover, to derive the fonction f, we must consider three cases: x < ; x > ; the connection of the two exression at x =. If we want to calculate the nth derivative of f, this roblem subsist. The fact that the function x 7 x q, q Q1 is a nth derivable one-to-one almost everywhere simlifies the roblem: for all real x, the function has just one exression and the roblem of connection at x = disaears. The class of this function is C almost everywhere. Only a roblem at can arrive but, in our case, the determination of the tangents to the curve, the fact that the derived function admits an infinite limit is not a roblem. So, from formula 8, it is easy to calculate the exression of the tangent vector of the suerellisod. The exression of the first vector is 1 q1 a sin q1 u cos q v 1 Γsd q u, v = a1 cos 1q1 1 u cos q v, 13 u q1 whereas the exression of the second vector is 1 q a cos q1 u sin q v 1 q Γsd u, v = a1 sin q1 u sin q v v q q a cos q v, 14 with the convention 1+ = + and 1 =. 3 Differentiable suerellisoid of revolution: As the curvature lines of Duin cyclides are circles, the differentiable suerquadrics to be blended must be revolution surfaces. So, in equation 8, we set 1 = q1 to obtain a differentiable suerellisoid of revolution, illustrated on Figure 4. Its arametric equation can be given by: cos u cos q v Γse : u, v 7 sin u cos q v, 15 sin q v where u, v [, π] π, π and q Q1. B. Suerhyerboloid of one sheet Figure 5 resents different suerhyerboloids of one sheet. The arametric equation of a differentiable suerhyerboloid of one sheet of revolution is defined by ch q v cos u Γsh : u, v 7 ch q v sin u, 17 sh q v where u, v [, π] and q Q1. The imlicit equation of a differentiable suerhyerboloid of one sheet of revolution is q q x + y 1 =. 18 Fig. 5. Three differentiable suerhyerboloids of one sheet of revolution From left to right: q = 1, q = 3, q = 11. C. Suerhyerboloid of two sheet Figure 6 resents different suerhyerboloids of two sheets. The arametric equation of a differentiable suerhyerboloid of two sheets of revolution is defined by sh q v cos u Γsh : u, v 7 sh q v sin u, 19 ε ch q v where ε { 1, 1}, u, v [, π] and q Q1. The imlicit equation of a differentiable suerhyerboloid of two sheets of revolution is x + y q q + 1 =. Fig. 4. Three differentiable suerellisoids of revolution. From left to right: 3 = 1, q = 17, q = 95. q The imlicit equation of a differentiable suerellisoid of revolution is x + y q + q 1 =. 16 Fig. 6. Three differentiable suerhyerboloids of two sheets of revolution. 7, q = 91. From left to right: q = 1, q = 13

5 IV. G 1 BLEND BETWEEN A DIFFEENTIABLE SUPEQUADIC OF EVOLUTION AND A DUPIN CYCLIDE Along the symmetry curve of the differentiable suerquadric of revolution γ, Figure 1, it is ossible to comute a Duin cyclide atch as a blending surface between and two of the three differentiable suerquadrics of revolution discussed in section III: a suerellisoid and a suerhyerboloid of one sheet. Indeed, the curve γ does not exist on a suerhyerboloid of two sheets, and this case will be considered in q forthcoming study. Figure 6. A. Suerellisoid The trimming circle between the Duin cyclide and the suerellisoid is γ = {Γ se u,, u [, π]}. It is well known that along this curve, the Duin cyclide second tangent vectors are vertical. So the continuity of the connection between the Duin cyclide and the differentiable suerquadric of revolution is G 1 if the suerellisoid second tangent vectors are vertical too. As this suerellisoid is a surface of revolution, it is sufficient to calculate the tangent to the surface along this circle for only one value of u, for examle u =. So, we have γ se v = Γ se, v = cos v sin v, 1 and M = γ se is a oint of the trimming circle. Let M = γ se v. The tangent to the curve at M is the limit osition of the straight line M M and is characteried by lim v γ se v if M is a regular oint. If M is a singular oint, this limit osition is characteried by a Taylor exansion of γ se v at and then the blend is G1 if the limit osition of the straight line M M is vertical: Theorem 3: Let γ se be the curve defined by equation 1 and M = γ se. The limit osition of M M is 1 Vertical iff <. Horiontal iff >. Proof: In this roof, we consider that we are in the lane P y y =. We have : γ se v = sin v cos q v. cosv sin q v Three cases arise: 1 < 1 We have q < and then lim v γ se v = lim v q sin v and so the tangent is vertical. = 1 We have q = and then: = + and then: γ se v = sin v cosv γ se v = and so the tangent is vertical. 3 > 1 We have q > and then: γ se = A Taylor exansion is needed to obtain 1 γ se v v 1 v 1 v v 1 v 1 v 1 v v v = γ 1 v = So, two cases are ossible: a < i.e. <. We have lim v γ 1 v = 1 lim v = + v so we conclude that the tangent is vertical. b > i.e. >. We have lim v γ 1 v = 1 v = 1 v v so we conclude that the tangent is horiontal. Figure 7 shows the G 1 blend between a Duin cyclide and a differentiable suellisoid of revolution. The section of the differentiable suellisoid of revolution by the construction lane is the suerellise γ. The section of the Duin cyclide by the construction lane is the union of two circles C 1 and C. At the oint B 1 res. B, one can see that the tangent T 1 res. T to the cercle C 1 res. C is also the tangent to the suerellise γ, which roves the G 1 continuity of this connection. B. Suerhyerboloid of one sheet The method here is similar to the one of the revious section, only the exression of the curve γ differs, i.e. ch v γ sh v = Γ sh, v =. sh v So we can write the following theorem:

6 Fig. 7. G 1 continuous blend between a Duin cyclide and a differentiable suellisoid of revolution. Theorem 4: Let γ sh be the curve defined by equation and M = γ sh. The limit osition of M M is 1 vertical iff <. horiontal iff >. Proof: This roof can be made using Taylor exansion and is similar to the roof of theorem 3. V. BLENDING A DIFFEENTIABLE SUPEQUADIC OF EVOLUTION AND A PLANE It is well known that it is ossible to make a G 1 blend between a Duin cyclide and a lane [17], [13] or a shere [17], [1]. To create a G 1 -blend between a suerquadric of revolution and a lane using a Duin cyclide, we need to determine a construction lane that is a symmetry lane for both surfaces. Such lane is used to comute two circular arcs γ 1 and γ that are the rincial colanar circles of the Duin cyclide. In order to define γ 1 and γ, we need two trilets of oints A 1, B 1, C 1 and A, B, C. Points A i and B i are the contact oints with one of the surface and the Duin cyclide. Points C i are used to determine the tangents in the construction lane, i.e. the lines A i C i and B i C i. To guarantee the curve γ i to be a circular arc, we must have A i C i = B i C i. Finally, oints A 1 and A must be symmetric around the revolution axis of the rimitive they belong to similarly for B 1 and B. In other words, to guarantee the curves γ i to be arcs of circle and the symmetries for coules A 1, A and B 1, B, the following equation system must be verified A 1 C 1 = B 1 C 1 A C = B C, s A A 1 = A s B B 1 = B where A and B are the revolution axes of the surfaces carrying oints A 1 and A, and B 1 and B, resectively. If one of the surfaces to be blended is a lane or a shere, these secifications are verified because: Any line that is orthogonal to a lane is a revolution axis for this lane Any line crossing the center of a shere is a revolution axis for this shere For the general case, we need to introduce the following lemma: Lemma 1: : Let C x C ; ; C be a oint belonging to a straight line 1 : = mx +, B a oint which does not belong to this line. Let l = BC. There are two oints A ε, ε { 1; 1}, belonging to 1 such that A ε C = l, given by A x ε l C + ε ; ; l m C + ε 1 + m 1 + m The roof of this lemma is evident using directrix coefficient of a straight line and Thales theorem. We recall that is the axis of the suerquadric E and 1 is the intersection of the construction lane P y and the lane P. Thus, the user intervenes only twice. First, the user chooses a oint belonging to the suerquadric E, for examle B 1 used to determine B and the oints C 1 and C. These two oints are the intersections of the straight line 1 and the tangent to the suerquadric E assing through the oints B 1 and B, as illustrated on Figure 8. When choosing B 1, we also check that the oints B 1 and B are in the same half sace delimited by P to obtain a connected blend using ring or sindle Duin cyclide. The second user intervention relates to the choice of the oint A 1 belonging to the straight line 1 such as A 1 C 1 = B 1 C 1. This oint A 1 determines the comutation of the oint A. B 1 µ+ c =ρ 1 I 1 O 1 C 1 A 1 C 1 A 1 d= a B1 O Ω B d= a I 1 C 1 C A µ c =ρ I O I O C 1 C µ+ c =ρ µ c =ρ 1 O1 d= a Ω O d= a C B A C Fig. 8. Section of the construction by the lane P y of the blend between a differentiable suerquadric of revolution E and a lane by a Duin cyclide. To: illar construction. Bottom: container construction. The suerquadric is a surface of revolution, so the G 1 -blend can be verified in the construction lane Figure 8: the fact that γ se and γ seπ are vertical, theorem 3 and theorem 4 the fact that C 1 B 1 and C B are vertical imlies that the four vectors are colinear. The stes one needs to execute to comute the G 1 blend are enumerated in algorithm 1. Figure 8 shows x x

7 Algorithm 1 Comutation of a G 1 Duin cyclide blend between a differentiable suerquadric of revolution and a lane. Inut: A differentiable suerquadric E of revolution with axis. A lane P. 1 Choice of the oint B1 belonging to E. Comutation of the oint B, symmetrical to B1 in relation to. 3 Comutation of the vector v1 res. v tangent to E at B1 res. B. 4 Comutation of the oint C1 res. C, intersection of the lane P and the straight line B1, v1 res. B, v. 5 Choice of the oint A1 belonging to the lane P such as A1 C1 = C1 B1, lemma 1. 6 Comutation of the oint A belonging to the straight line C1 C such as A C = C B and: if A1 [C1 C then A [C C1 else A / [C C1. 7 Comutation of the rincial circle centre O1 res. O and rincial circle radius ρ1 res. ρ determined by QBC {B1 ; C1 ; A1 } res. QBC {B ; C ; A } [1]. 8 Comutation of Duin cyclide arameters a, c and µ using theorem 1. 9 Comutation of the Duin cyclide trimming values. Outut: A trimmed Duin cyclide making a G 1 -blend between a differentiable suerquadric of revolution and a lane. Fig. 1. A container blend between a differentiable circular suerellisoid 3, and a lane by a Duin cyclide. From left to right: q = 1, q = =. q 17 the generatrix 1 res. by the straight line B1 ; v1 res. B ; v where B1 and B are symmetrical in relation to E axis and the vector v1 res. v is the tangent vector to E at B1 res. B. Figure 1 shows two blends between a differentiable suerellisoid of revolution and a shere by a ring Duin cyclide. On the left subfigure res. suerellisoid right subfigure, a rameters are ; ; q1 = 4; 8; 11 res. ; ; q1 = ; 8; 35, shere centre coordinates are 1 ; ; 15 res. 5 ; ; 17, shere radius is r = 8 res. r = and Duin cyclide arameters are a; c; µ 6.57; 3.3;.57 res. a; c; µ 16.; 5.; 14.. Fig. 11. Two Duin cyclide blends between a differentiable suerellisoid of revolution and a shere. Fig. 9. A illar blend between a differentiable circular suerellisoid and a 3, q = 95. lane by a Duin cyclide. From left to right: q = 1, q = 17 the influence of the osition of oint A1 ste 5 of algorithm 1 on the tye of the construction illar or container. On the left art of Figure 8, A1 does not belong to the half-line [C1 C, A does not belong to the half-line [C C1 and the blend tye is illar. On the right art of Figure 8, A1 belongs to the half-line [C1 C, A belongs to the half-line [C C1 and the blend tye is container. Figure 9 shows some illar constructions between a differentiable suerellisoid of revolution and a lane by a ring Duin cyclide. Figure 1 shows some container constructions between a differentiable suerellisoid of revolution and a lane by a sindle Duin cyclide. VI. B LENDING A DIFFEENTIABLE SUPEQUADIC OF EVOLUTION AND A SPHEE To adat the method of section V for the blending of a suerquadric of revolution and a shere one needs to relace VII. A N EXAMPLE : A STUDY DESK WITH A LAMP In this section, we show a comlete examle: a study desk with a lam, Figure 1 and 13. This object is modeled using two lanes, quadrics two sheres, two cylinders, suerquadrics two suerellisoids and a suerhyerboloid of one sheet and seven Duin cyclides which make G 1 -blends between these surfaces, Table I. These blends are built using rational quadratic Be ier curves. The arameters of the lane S are m =.95 and = 1. The radius coule of the differentiable suerellisoid of revolution S res. S4 is ; = 3; 8 res. ; = 4; and the ower is q = 17 for the two suerellisoids. The center coordinates of the shere S9 res. S13 are 8; ; 15 res. 98; ; 1 and the radius is 1. The radius of the cylinder S7 res. S11 is res. 3 and its height is 11 res. 5. The radius coule of the differentiable suerhyerboloid of one sheet of revolution S15 5. Duin cyclides is ; = 6; 7 and the ower is q = 11 arameters are given by the Table II knowing that θ describes the interval [, π]. VIII. C ONCLUSION Every suerquadric of revolution can be seen as a limit of a sequence of rational suerquadrics. The differentiable

8 Primitve Name Where Plane S Suort Plane S 1 Edge Suerellisoid S Small leg Duin cyclide S 3 S - S blend Suerellisoid S 4 Great leg Duin cyclide S 5 S - S 4 blend Duin cyclide S 6 S - S 7 blend Cylinder S 7 Fixed stem Duin cyclide S 8 S 7 - S 9 blend Shere S 9 First articulation Duin cyclide S 1 S 9 - S 11 blend Cylinder S 11 Stem Duin cyclide S 1 S 11 - S 13 blend Shere S 13 Second articulation Duin cyclide S 14 S 13 - S 15 blend Suerhyerboloid S 15 Lamshade TABLE I STUDY DESK SUFACES. Fig. 13. A study desk with a lam. EFEENCES Name a µ c ψ min ψ max S π/ S π/ S π/ S S S S TABLE II DUPIN CYCLIDES PAAMETES. Fig. 1. A study desk with a lam. suerquadrics make ossible the use of differential geometric concets. Thus, using Taylor exansion, we determine surface tangents where the function sgn is not continuous. Then, we determine a G 1 Duin cyclide atch between the differentiable suerquadric of revolution and a lane or a shere. Possible exansions of this work may be the generaliation of this construction to the blending of two differentiable suerquadrics and the introduction of suercyclide blends. [1] S. Allen and D. Dutta. Cyclides in ure blending I. Comuter Aided Geometric Design, 141:51 75, ISSN [] S. Allen and D. Dutta. Cyclides in ure blending II. Comuter Aided Geometric Design, 141:77 1, ISSN [3] A. H. Barr. Suerquadric and angle reserving transformations. IEEE Comuter Grahics and Alications, 11:11 3, January [4] G. Darboux. Princies de géométrie analytique. Gauthier-Villars, [5] G. Demengel and J. P. Pouget. Mathématiques des Courbes et des Surfaces. Modèles de Béier, des B-Slines et des NUBS, volume 1. Ellise, [6] C. P. Duin. Alication de Géométrie et de Méchanique la Marine, aux Ponts et Chaussées, etc. Bachelier, Paris, 18. [7] D. Dutta,.. Martin, and M. J. Pratt. Cyclides in surface and solid modeling. IEEE Comuter Grahics and Alications, 131:53 59, January [8] Gerald E. Farin. NUBS: From Projective Geometry to Practical Use. A. K. Peters, Ltd., Natick, MA, USA, [9] A.. Forsyth. Lecture on Differential Geometry of Curves and Surfaces. Cambridge University Press, 191. [1] Sebti Foufou and Lionel Garnier. Duin cyclide blends between quadric surfaces for shae modeling. Comut. Grah. Forum, 33:31 33, 4. [11] M. Gardiner. The suerellise: A curve that lies between the ellise and the rectangle. Sci. Amer., 133: 34, Setember [1] L. Garnier. Mathématiques our la modélisation géométrique, la rerésentation 3D et la synthèse d images. Ellises, 1st edition, july 7. ISBN : [13] L. Garnier, S. Foufou, and M. Neveu. Blending of surfaces of revolution and lanes by duin cyclides. In book of the 8th SIAM Conference on Geometric Design and Comuting, Seattle. Nashboro Press, Mike Neamtu editor, Nashville, TN, USA, 4, August-Setember 3. [14] J. Hoschek and D. Lasser. Fundamentals of Comuter Aided Geometric Design. A.K.Peters, Wellesley, Massachussets, [15] G. Lamé. Examen des différentes méthodes emloyées our résoudre les roblèmes de géométrie. Courcier, Paris, [16].. Martin. Princial atches for comutational geometry. PhD thesis, Engineering Deartment, Cambridge University, 198. [17] M. J. Pratt. Cyclides in comuter aided geometric design. Comuter Aided Geometric Design, 71-4:1 4, 199. [18] M. J. Pratt. Duin cyclides and suercyclides. In G. Mullineux, editor, Proceedings of the 6th IMA Conference on the Mathematics of Surfaces IMA-94, ages 43 66, Brunel University, Setember Oxford University Press. [19] M. J. Pratt. Cyclides in comuter aided geometric design II. Comuter Aided Geometric Design, 1:131 15, [] C. K. Shene. Blending two cones with Duin cyclides. Comuter Aided Geometric Design, 157: , [1] D. Terooulos and D. Metaxas. Dynamic 3d models with local and global déformations: Deformable suerquadrics. IEEE Transactions on Pattern Analysis and Machine Intelligence, 137:73 714, July 1991.