Blending of Surfaces of Revolution and Planes by ... - Le2i - CNRS

Blending of Surfaces of Revolution and Planes by Dupin cyclides Lionel Garnier, Sebti Foufou and Marc Neveu

Abstract. This paper focuses on the blending of a plane with surfaces of revolution relying on Dupin cyclides, which are algebraic surfaces of degree 4 discovered by the French mathematician Pierre-Charles Dupin early in the 19th century. A general algorithm is presented for the construction of two kinds of blends: pillar and recipient. This algorithm uses Rational Quadric Bézier Curves (RQBCs) to model the relevant arcs of the principal circles of the cyclides and allows the construction of a new blending primitive: the spindle Dupin cyclide. Our algorithm can also be used for the blending of the plane with particular surfaces of revolution such as the torus, the catenary and the pseudosphere.

§1. Introduction Blending two surfaces is a traditional operation in geometric modeling. It consists in building a third surface that fills in the gap between two initial surfaces. It is therefore a tedious operation, due to either reasons of geometric continuity at the junction of the different surfaces or to parametrization problems when the blending surface is parametric. In this work, we are interested in the computation of Dupin cyclide patches that blend a surface of revolution with a plane. Dupin cyclides have an essential property: all their curvature lines are circular. This property facilitates the use of these surfaces in the blending of quadrics. We investigate the construction of two possible kinds of blends, referred to as pillar (Figure 4, left) and recipient (Figure 4, right). The proposed algorithm is general and can be used for the blending of a plane and any surface of revolution. This paper is organized as follows: Section 2 defines Dupin cyclides and gives their interesting properties for blending operations. Section 3 XXX xxx and xxx (eds.), pp. 1–4. Copyright 200x by Nashboro Press, Brentwood, TN. ISBN 0-9728482-x-x All rights of reproduction in any form reserved.

c °

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L . Garnier, S . Foufou and M . Neveu

reviews surfaces of revolutions and gives the equations of those used in our examples. Section 4 reviews existing methods for the computation of Dupin cyclide blends. Our blending algorithm is detailed in section 5, together with a set of various blending examples. Section 6 studies a complete shape design example using a combination of surfaces of revolution blended by Dupin cyclides. Section 7 draws conclusions and suggests some perspectives for future extensions of this work. In this paper, the terms Dupin cyclides and cyclides will be used interchangeably, and will always refer to Dupin cyclides. §2. Dupin Cyclides Dupin cyclides are non-spherical algebraic surfaces discovered by French mathematician Pierre-Charles Dupin at the beginning of the 19th century [10]. Pierre-Charles Dupin defined a Dupin cyclide as the envelope surface of a variable sphere that touches three given spheres in a continuous manner [5]. Other mathematicians have worked on these surfaces and have defined them in various manners [12, 8, 13].

Fig. 1. Dupin cyclides: ring (left), horned (middle) and spindle (right) Dupin cyclides depend on three parameters a, c and µ with a ≥ c and have the following two equivalent implicit equations: ¡ ¢2 2 Fy (x; y; z) = x2 + y 2 + z 2 − µ2 + b2 − 4 (ax − cµ) − 4b2 y 2 = 0 (1) ¡ ¢2 2 Fz (x; y; z) = x2 + y 2 + z 2 − µ2 − b2 − 4 (cx − aµ) + 4b2 z 2 = 0 (2) √ where b = a2 − c2 . Dupin cyclides can also be defined by the following parametric equation:  µ(c − a cos θ cos ψ) + b2 cos θ   x (θ; ψ) =   a − c cos θ cos ψ   b sin θ × (a − µ cos ψ) Γ (θ; ψ) = (3) y (θ; ψ) =  a − c cos θ cos ψ      z (θ; ψ) = b sin ψ × (c cos θ − µ) a − c cos θ cos ψ 2

where (θ; ψ) ∈ [0; 2π] . According to the values of the parameters, there are three kinds of Dupin cyclides: the ring cyclide where 0 ≤ c < µ ≤ a,

Blending by Dupin Cyclides

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the horned cyclide where 0 < µ ≤ c < a and the spindle cyclide where 0 ≤ c ≤ a < µ. Figure 1 shows examples of half Dupin cyclides with (θ; ψ) ∈ [0; 2π] × [0; π].

Fig. 2. Cut of a ring Dupin cyclide by its symmetry plane Py : (y = 0) A Dupin cyclide admits two symmetry planes Py : (y = 0) and Pz : (z = 0). The section of a Dupin cyclide by one of these two planes is the union of two circles called principal circles. Two of these principal circles are sufficient to determine the three parameters a, c and µ, and thus to define the Dupin cyclide. Figure 2 shows the two principal circles of a Dupin cyclide in the plane Py . Only a part of the principal circles is useful for the blend: the useful arc will be represented as a RQBC in standard form. The blending principal circles are C1 and C2 . For i ∈ {1; 2}, Oi is the center of Ci , ρi is the radius of Ci . We can suppose w.l.o.g. that ρ1 is greater than ρ2 . From these principal circles C1 and C2 of the Dupin cyclide, it is easy to calculate parameters a, c et µ to obtain a ring or a spindle cyclide: a=

O1 O2 2

µ=

ρ1 + ρ2 2

c=

ρ1 − ρ2 2

(4)

The lines of curvature of Dupin cyclides are circles, obtained with the θ or ψ constant [16]. The blends with Dupin cyclides are made along lines of curvature [11]: each blended surface meets the blending Dupin cyclide along lines of curvature. The cyclide trimming curves thus have a simple parametric equation: Γd (θ; ψ0 ) , θ ∈ [0; 2π] from ¡the surface of revolution ¢ side (we need determine the value of ψ0 ) and Γd θ; π2 , θ ∈ [0, ; 2π] from the plane side. For more details regarding the properties of Dupin cyclides, see [5, 6, 7, 15, 17, 18, 20]. Through each curvature line of a cyclide, there is a constant angle between the cyclide normal and the principal normal of the curvature line. This property makes it possible to construct G 1 continuous connections between the surfaces of revolution and the cyclides. §3. Surfaces of Revolution A surface of revolution is a surface generated by rotating a two-dimensional contour curve about an axis (called the axis of revolution). The resulting

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surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include: cone, cylinder, hyperboloid, paraboloid and sphere. Table 1 gives the implicit and parametric equations of the subset of quadric surfaces of revolution needed in this paper. All surfaces considered here have axis (Oz) as the axis of revolution. Name

Implicit equation

Cone

x2 + y 2 z2 − =0 a2 c2

Ellipsoid

x2 + y 2 z2 + −1=0 a2 c2

Hyperboloid of one sheet

x2 + y 2 z2 − −1=0 a2 c2

Hyperboloid of one sheet

x2 + y 2 z2 − +1=0 a2 c2

Cylinder

x2 + y 2 −1=0 a2

Paraboloid of revolution

x2 + y 2 − 2pz = 0 a2

Parametric equation at cos (θ) at sin (θ) ct a cos (θ) cos (ψ) a sin (θ) cos (ψ) c sin (ψ) a cos (θ) ch (t) a sin (θ) ch (t) c sh (t) a cos (θ) sh (t) a sin (θ) sh (t) εc ch (t) a cos (θ) a sin (θ) t at cos (θ) at sin (θ) t2 2p

Tab. 1. Equations of quadrics with (Oz) as the axis of revolution Besides the primitives given in table 1, three others, the torus of revolution, the catenoid and the pseudosphere, will be used in our illustrative examples. The torus is a Dupin cyclide having c = 0. The catenoid is the surface of revolution generated by a meridian, called the catenary, defined by the equation (ach( at ), t). The ³ meridian of ´the pseudosphere, called 1 tractrix, is defined by equation t − th (t) ; ch(t) , with t ∈ R. Equations (5) and (6) are those of the catenoid and the pseudosphere:

¡ ¢  a cos (θ) ch ¡ at ¢  a sin (θ) ch t  a t 

 (5)

 cos (θ) (t − th (t))  sin (θ) (t − th (t))  1 ch(t)

(6)


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§4. Existing Methods for Blending with Cyclides The literature concerning the use of Dupin cyclides for the blending of quadric primitives is relatively rich as many authors have addressed this problem. The circular curvature line property of Dupin cyclides facilitates the construction of the cyclide (or the portion of a cyclide) for the blending of two circular quadric primitives. Chandru et al. [6] suggest using Dupin cyclides to compute variable radius blends and define an approximation strategy that can be applied in case of complex explicit construction of blends. Degen [9] defines generalized cyclides as a new class of cyclides having more degrees of freedom than traditional quadrics, gives a universal representation of these surfaces and discusses the general conditions for the blending of a given surface with a generalized cyclide. Pratt [16, 18] shows how to make piecewise cyclide blending surfaces, and extends the symmetric cyclide blends to include asymmetric problems. The particular case of the computation of double cyclide blends between two cones is given as an example of possible blending applications. Aumann [4] investigates curvature continuous blending of cones and cylinders using normal ringed surfaces which are generated, like cyclides, by a circle of variable radius sweeping through space. Allen and Dutta [3] present a method to generate nonsingular transition surfaces between natural quadrics using either Dupin cyclides or parabolic cyclides. Necessary and sufficient conditions for the existence of the cyclide transition surface are given. Also, Allen and Dutta [1, 2] study a special type of blend called pure blends. Paluszny and Boehm [14] study the joining of quadrics and propose working outside the Moebius hypersphere in the 4-space. The joining of two cones 3-space is thus reduced to the construction of a piecewise conic in 4-space, that satisfies two specific conditions. Shene [19] shows how one can use cyclides to define the blending surface of two cones. Later on, the author makes use of offset surfaces to improve and generalize this work in order to compute the blending surface of two randomly positioned half cones [20]. M. Pratt has developed an algorithm to blend a cylinder of revolution and a plane P by a Dupin cyclide [16]. To determine the parameters of the cyclide blend, the resolution is made in the construction plane Py which contains two principal circles of the cyclide. Py contains ∆, the cylinder axis, and the normal of the plane P. With this construction, only the pillar type of blend is possible (Figure 4, left). Let ρ0 be the cylinder radius. Let {Ω1 } = ∆ ∩ Pz , {Ω2 } = ∆ ∩ P, h = Ω1 Ω2 and β be the angle between ∆ and P. The radii of the principal circles of the Dupin cyclide are: ³ ´ ρ1 = (h + ρ0 cot β) cot

β 2

³ ´ ρ2 = (h − ρ0 cot β) tan

β 2

(7)

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and the Dupin cyclide parameters are: a = (h + ρ0 cosec β) cosec (β) c = (h + ρ0 cosec β) cot β µ = h cosec β + ρ0 cot2 β

(8)

We present a geometric method to effect the blending of the plane and surfaces of revolution. Instead of using trigonometric functions, our method uses RBQCs to represent the principal circles of the cyclide blends. §5. The Proposed Blending Algorithm To join a surface of revolution and a plane by a Dupin cyclide, the following steps are necessary: - Find RQBC representations of the principal circles of the blending Dupin cyclide, - Compute the cyclide parameters a, c and µ using formulae (4), - Trim the Dupin cyclide (ie. find the interval of θ and ψ) to retain only the useful part for the blending. As depicted in Figure 4, two kinds of blends (called pillar and recipient) can be constructed using the proposed blending method. 5.1. Representing Circular Arcs Using RQBCs Let P0 and P2 be two points. Let P1 be a point on the perpendicular bisector of the segment [P0 P2 ] such that P1 does not belong to this segment, Figure 3. There is a single circle passing through P0 and P2 , tangent to (P0 P1 ) at P0 and tangent to (P2 P1 ) at P2 . The center of this circle is given by: −−−→ −−→ P0 P12 P1 O0 = t0 P1 I1 , t0 = −−→ − (9) −−→ I1 P1 • P0 P1

Fig. 3. Modeling an arc of a circle with a RQBC where I0 is the midpoint of the segment [P0 P2 ]. Points P0 , P1 , P2 with their weights w0 = 1, w and w2 = 1 are the control points of the RQBC


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representing the circle. When w > 0 the curve represents the short arc of the circle, when w < 0 it represents the long arc. 5.2. Principal Circles of Cyclide Blends We notice here that we are working in the plane Py . Let D be a line defined by z = mx + p, and C (xC ; 0; zC ) ∈ D. Let B be a point in Py . It is possible to define a point A ∈ D in such a way that BC = AC: ¶ µ l lm , ε ∈ {−1; 1} (10) A xC + ε √ ; 0; zC + ε √ 1 + m2 1 + m2 Figure 4 shows the construction used for the computation of the principal circles of the Dupin cyclide blends between a cylinder and a plane. This construction is effected in the plane Py and is valid for the blending of any other surface of revolution and a plane. Computations of the control points of RQBCs representing these circles are detailed in algorithm 1. ∆

O1

B1

1 0 0 1

I1 A1 11 00 11 00

1 0 0 1

∆

Ω

1 0 0 1

1 0 0 1

B2

1 0 1 0

C1

1 0 0 1

00 11 00 11 00 11

C2

B1

1 0 0 1

O2

1 0 1 0

O1

1 0 1 0

I2

1 0 1 0

1 0 1 0

Ω

O2

1 0 1 0

I1

C1 plane

surface of revolution

1 0 1 0

B2

I2

1 0 1 0 1 0 1 0

A2

1 0 1 0 1 0 1 0

A1 11 00 11 00

1 0 1 0

A2

1 0 1 0

C2

plane surface of revolution

Fig. 4. Computing principal circles of a cyclide blend between a surface of revolution and a plane: Left, Pillar blend. Right, Recipient blend Algorithm 1. Given: A plane P and a surface of revolution S with axis of symmetry ∆. 1. Choose two points B1 and B2 on S such that B1 and B2 are symmetrical in relation to ∆, 2. Compute lines T1 and T2 tangent to S at B1 and B2 respectively, 3. Compute points C1 = T1 ∩ P and C2 = T2 ∩ P 4. Compute A1 = P ∩ C1 and A2 = P ∩ C2 where C1 (C1 , l1 ) is the circle centered at C1 with radius l1 = C1 B1 and C2 (C2 , l2 ) is the circle centered at C2 with radius l2 = C2 B2 . 5. Compute, using formula (9), centers O1 , O2 and radii ρ1 , ρ2 of the two circles represented by the RQBCs defined by control points B1 , C1 , A1 and B2 , C2 , A2 , Result: Principal circles of Dupin cyclide blends between surfaces of revolution and planes.

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Fig. 5. Cylinder-plane blending with ring (left) and spindle (right) Dupin cyclides The computation of points A1 and A2 can be carried out in the same way as point A of equation (10). As the diameters of circles are axis of symmetry, we have C1 A1 = C1 B1 and C2 A2 = C2 B2 . We notice that in the case of a pillar blend (Figure 4 left), A1 and A2 do not belong to the segment [C1 C2 ], while in the case of a recipient blend, Figure (4 right), A1 belongs to the half-line [C1 C2 ) and A2 belongs to the half-line [C2 C1 ). The RQBCs defined by control points A1 , C1 , B1 and A2 , C2 , B2 represent the principal circles of the blending cyclides.

Fig. 6. Ellipsoid-plane blending: Left, a ring pillar blend. Right, a spindle recipient blend. 5.3. Parameters of Cyclide Blends We now use equation (4) to compute parameters a, c and µ of the cyclide from its principal circles. The last operation one needs to perform is the trimming of the resulting cyclide surface to retain only the useful part that constitutes the blend. This amounts to determining ψ0 as the trimming value for the surface of revolution side and ψ1 as the trimming value for the plane side. In order to compute ψ0 and ψ1 we need to express equations of the cyclide, of the plane and of the surface of revolution in the same gen-


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eral reference system. In such a system, ψ0 is the parameter value such that {Γ (θ, ψ0 ) , θ ∈ [0; 2π]} is the intersection of the Dupin cyclide and the surface of revolution. So, to find ψ0 we have to solve the system of equations resulting from the substitution of x(θ, ψ), y(θ, ψ) and z(θ, ψ) of equation (3) for coordinates of point B1 or B2 with θ = 0 or θ = π. In the same way, ψ1 is the parameter value such that {Γ (θ, ψ1 ) , θ ∈ [0; 2π]} is the intersection of the Dupin cyclide and the plane, which implies that ψ1 = ± π2 . 5.4. Illustrative Blending Examples Figures 5, 6 and 7 show a set of blending results of the proposed algorithm. Pillar as well as recipient blends defined either as ring or as spindle Dupin cyclides are illustrated. The cases considered are plane with cylinder, ellipsoid and cone. Figures 8 and 9 give examples of blends between a plane and two other particular surfaces of revolution: the torus which is a four degree surface and the catenary, which is a non algebraic surface.

Fig. 7. Cone-plane blending: Left and Middle, Pillar blends. Right, Spindle recipient blend

Fig. 8. Torus of revolution-plane blending: Left and middle, Ring pillar blends. Right, Spindle recipient blend.

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Fig. 9. Catenary of revolution-plane blending: Left, Ring pillar blend. Middle and right, Spindle recipient blends §6. Combining Surfaces of Revolution Using Dupin Cyclides for Shape Modeling: A Complete Example In this section, we present the modeling of a complete satellite antenna as an example of shape modeling by means of a combination of surfaces of revolution using Dupin cyclide (Figure 10). The complete satellite antenna is an assembly of three parts: the antenna, the decoding device and the adjustment lever that allows orientation of the antenna to better capture waves:

Fig. 10. Two different views of the satellite antenna • The antenna is made up of a paraboloid of revolution S1 and a parallelepipedal base S2 blended by a Dupin cyclide patch S3 . • The decoding device is made up of a head represented by an ellipsoid of revolution S4 and supported by a metal rod represented by a cylinder of revolution S5 . The two surfaces are blended together by a Dupin cyclide patch S6 . • The antenna and the decoding device are blended using a Dupin cyclide S7 .


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• The adjustment lever is made up of a sphere S8 and a cylinder of revolution S9 linked together by Dupin cyclide S10 . All mentioned cyclide blends between the components of this satellite antenna are G 1 continuous. Table 2 gives more details on the primitives used. The first column indicates the name of the primitive. The second column notes the primitive in relation to Figure 10. Column three gives the parameters defining the primitive. For cyclides, in addition to values of parameters a, µ and c, the visualization interval of parameter ψ is shown, given that parameter θ varies in [0; 2π]. The last column indicates the part of the antenna in which the primitive is used. Blends constructed in this example are not limited to the cases of planesurfaces of revolution but include more complex blends such as cylinderparaboloid, cylinder-ellipsoid and cylinder-sphere, whose construction algorithms are not covered in this paper. Primitive Parallelepiped Paraboloid Dupin Cyclide

Ref. S2 S1 S3

Ellipsoid Cylinder Dupin cyclide

S4 S5 S6

Dupin cyclide

S7

Sphere Cylinder Dupin cyclide

S8 S9 S10

Parameters m = 0.35, p = −1.5 a = 1, p = 2 a £ ' 1.05, ¤ µ ' 0.83, c ' 0.182, −1.2; π2 a = 0.35, c = 0.0875 a = 0.1, h = 0.74923 a ' 0.29, µ ' 0.2, c ' 0, [−1.33; 0] a ' 0.23, µ ' 0.13, c = 0, [0; 1.7] a = c = 0.125 a = 0.05, h = 0.29 a ' 1.26, µ ' 1.21, c = 0, [0; 0.34]

Where base antenna antenna-base blend head head rod head-rod blend antenna-rod blend lever top lever rod lever rod-top blend

Tab. 2. Breakdown of the satellite antenna components §7. Conclusion This work is another proof of the importance and the usefulness of Dupin cyclide surfaces in the geometric modeling. This paper dealt with the blending of planes and surfaces of revolution using Dupin cyclides. This is a small application of these surfaces in a long list of possible and very interesting applications. The proposed algorithm uses RQBCs to define the principal circles of two kinds of cyclide blends. A set of illustrative figures are given to show the results of this algorithm.

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The use of supercyclides for the blending of elliptical surfaces of revolution primitives is now being studied. §8. Acknowledgments We particularly wish to thank Mike Pratt who introduced the Dupin cyclide surfaces to us and with whom we have had a pleasant and very instructive discussions about cyclides. §9. References 1. S. Allen and D. Dutta. Cyclides in pure blending I. Computer Aided Geometric Design, 14(1):51–75, 1997. ISSN 0167-8396. 2. S. Allen and D. Dutta. Cyclides in pure blending II. Computer Aided Geometric Design, 14(1):77–102, 1997. ISSN 0167-8396. 3. S. Allen and D. Dutta. Results on nonsingular, cyclide transition surfaces. Computer Aided Geometric Design, 15(2):127–145, 1998. ISSN 0167-8396. 4. G. Aumann. Curvature continuous connections of cones and cylinders. Computer-aided Design, 27(4):293–301, 1995. 5. V. Chandru, D. Dutta, and C. M. Hoffmann. On the geometry of Dupin cyclides. Technical Report CSD-TR-818, Purdue University, November 1988. 6. V. Chandru, D. Dutta, and C. M. Hoffmann. Variable radius blending using Dupin cyclides. Technical Report CSD–TR–851, Purdue University, January 1989. 7. G. Darboux. Thèse la faculté des sciences de Paris. Annales scientifiques de l’école normale, 1866. 8. G. Darboux. Leons sur la Théorie Générale des Surfaces, volume 1. Gauthier-Villars, 1887. 9. W. L. F. Degen. Generalized Cyclides for Use in CAGD. In A. Bowyer, editor, The Mathematics of Surfaces IV, pages 349–363, Oxford, 1994. Clarendon Press. 10. C. P. Dupin. Application de Géométrie et de Méchanique la Marine, aux Ponts et Chaussées, etc. Bachelier, Paris, 1822. 11. D. Dutta, R. R. Martin, and M. J. Pratt. Cyclides in surface and solid modeling. IEEE Computer Graphics and Applications, 13(1):53–59, January 1993. 12. J. C. Maxwell. On the cyclide. Quarterly Journal of Pure and Applied Mathematics, pages 111–126, 1868.


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13. Moutard. Annales de Mathématiques. Bulletin de la société Philomatique et compte rendu de l’Accadémie, 1804. 14. M. Paluszny and W. Boehm. General cyclides. Computer Aided Geometric Design, 15(7):699–710, 1998. 15. U. Pinkall. Dupin hypersurfaces. Math. Ann., 270(3):427–440, 1985. 16. M. J. Pratt. Cyclides in computer aided geometric design. Computer Aided Geometric Design, 7(1-4):221–242, 1990. 17. M. J. Pratt. Dupin cyclides and supercyclides. In G. Mullineux, editor, Proceedings of the 6th IMA Conference on the Mathematics of Surfaces (IMA-94), pages 43–66, Brunel University, September 1994. Oxford University Press. 18. M. J. Pratt. Cyclides in computer aided geometric design II. Computer Aided Geometric Design, 12(2):131–152, 1995. 19. C. K. Shene. Blending two cones with Dupin cyclides. Computer Aided Geometric Design, 15(7):643–673, 1998. 20. C. K. Shene. Do blending and offsetting commute for Dupin cyclides. Computer Aided Geometric Design, 17(9):891–910, 2000. Lionel Garnier, Sebti Foufou and Marc Neveu Lab. LE2I, CNRS, UFR Sciences, Université de Bourgogne, BP 47870, F-21078 Dijon Cedex, France @u-bourgogne.fr