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Branching blend of natural quadrics based on surfaces with rational offsets
Computer Aided Geometric Design 25 (2008) 332–341 www.elsevier.com/locate/cagd
Branching blend of natural quadrics based on surfaces with rational offsets Rimvydas Krasauskas Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24, LT-03225 Vilnius, Lithuania Received 5 July 2007; received in revised form 1 November 2007; accepted 4 November 2007 Available online 14 December 2007
Abstract A new branching blend between two natural quadrics (circular cylinders/cones or spheres) in many positions is proposed. The blend is a ring shaped patch of a PN surface (surface with rational offset) parametrized by rational bivariant functions of degree (6, 3). The general theory of PN surfaces is developed using Laguerre geometry and a universal rational parametrization of the Blaschke cylinder. The construction is extended via inversion to a PN branching blend of degree (8, 4) between Dupin cyclide and a natural quadric. © 2007 Elsevier B.V. All rights reserved. Keywords: Rational offset; PN surface; Natural quadrics; Dupin cyclides
1. Introduction
In current CAD systems, curves and surfaces are represented in a standard NURBS form, i.e. they are parametrized by rational B-splines. Unfortunately, offsets of rational surfaces arising in practical applications are in general not rational and need to be approximated. On the other hand traditional 3d modeling primitives like natural quadrics (sphere, circular cylinder/cone) or torus surfaces admit rational offsets. According to Rossignac (1987), about 99% of mechanical parts can be represented by combinations of planes and natural quadrics with the possibility of representing blends between them. Since rolling ball blends have rather intuitive shape, they are the most popular in practice, but in general they are irrational or admit rational parametrizations of high degree (see details in Section 2). On the other hand in many cases the smoothness of the blend is more important than its exact shape. This situation in CAD industry motivates us to search for more general rational surfaces with rational offsets that can serve as blending surfaces between natural quadrics. We choose to work with Pythagorean normal (PN) surfaces which were introduced by Pottmann (1993) and are direct generalizations of well-known planar rational
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Pythagorean hodograph curves (Farouki and Pottmann, 1996). A surface in R3 with a fixed rational parametrization f (t, u) = (f1 (t, u), f2 (t, u), f3 (t, u)) is called a PN surface, if its unit normal ∂f1 ∂f2 ∂f3 ∂f1 ∂f2 ∂f3 ft × fu , ft = , , , fu = , , , N (t, u) = ft × fu ∂t ∂t ∂t ∂u ∂u ∂u is rational. It is clear that the offset of such PN surface at distance d has a rational parametrization f (t, u) + dN (t, u). The paper is organized as follows: Section 2 surveys previous PN surface blending constructions. Section 3 outlines necessary facts about the Blaschke model of Laguerre geometry. Section 4 describes all steps of the main branching blend construction and derives its properties. The construction is generalized to Dupin cyclides using inversions in Section 5. Conclusions are drawn in Section 6. 2. Survey of PN surface blending constructions The majority of known PN surface blending constructions between natural quadrics are pieces of rational canal surfaces. A canal surface is an envelope of a 1-parameter family of spheres in R3 , defined by a spine curve s(t) = (s1 (t), s2 (t), s3 (t)) and a radius function r(t). For the envelope to be real the condition on derivatives ˙s (t)2 − r˙ 2 (t) 0 is necessary. Let us collect this data in a curve γ (t) = (s(t), r(t)) ∈ R4 and denote the canal surface by C(γ ). Peternell and Pottmann (1997) proved that a canal surface C(γ ) defined by a rational curve γ can be rationally parametrized. If γ has degree k then Cγ can be parametrized with degree at least (3k − 2, 2), under a mild condition ˙s (t)2 − r˙ 2 (t) > 0 (Krasauskas, 2007). This degree bound is the minimal possible in general. The first non-trivial blending application of canal surfaces was proposed by Pratt (1990) and Boehm (1990). They used Dupin cyclides, which are quartic canal surfaces defined by special conics γ ∈ R4 . For example, any two circular cones with a common inscribed sphere can be blended by a part of a Dupin cyclide bounded by two circles (see Fig. 1, left). All possible blend surfaces of this type were studied in details by Shene (1998). Canal surfaces defined by general conics γ ∈ R4 can blend natural quadrics in more general positions (Fig. 1, middle) as was shown in Krasauskas and Zub˙e (2007). Results on rational parametrizations (Peternell and Pottmann, 1997; Peternell, 1997; Krasauskas, 2007) enabled Kazakeviˇci¯ut˙e (2005a) to develop a theory of rational variable radius rolling ball blends between natural quadrics in arbitrary positions. Here we will consider just one important case. Example 1. Let Qa and Qb be two cylinders in R3 defined by equations x12 + x22 = ra2 and x22 + x32 = rb2 , where 0 < ra < rb (Fig. 1, right). The condition that a sphere touches both cylinders Qa and Qb defines a quartic surface V ⊂ R4 . Any curve on V generates a canal surface touching both cylinders, i.e. a rolling ball blend. Unfortunately a fixed radius case corresponds to an irrational curve on V . Nevertheless, a certain rational quartic curve γ ⊂ V can be found (Krasauskas and Kazakeviˇci¯ut˙e, 2005). This construction generates a canal surface Cγ of degree (10, 2) which is minimal possible (Krasauskas, 2007). It is impossible to construct such a blending with a boundary circle on the cylinder Qa , since the corresponding curve on V and the associated canal surface are irrational. We are going to search for a more general PN surface blending construction as one piece of an algebraic surface. Other available methods related to surfaces with rational offsets, like patching with cubic cyclides (Pottmann and Peternell, 1998) or linear normal constructions (Jüttler and Sampoli, 2000), are not acceptable for us because they generate a 1–1 Gaussian map, which is too restrictive for our purposes.
Fig. 1. Blending constructions with canal surfaces.
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3. Blaschke model of Laguerre geometry The main idea of Laguerre geometry is to consider oriented spheres in R3 as points in R4 (see Pottmann and Peternell (1998) for details). A sphere c with a center (x1 , x2 , x3 ) and radius x4 is mapped to a point ζ (c) = (x1 , x2 , x3 , x4 ) ∈ R4 (the orientation is encoded by a sign of x4 ). We use also the projective extension RP 4 of R4 with additional coordinate x0 . Each oriented plane e: e0 + e1 x1 + e2 x2 + e3 x3 = 0, e12 + e22 + e32 = 1, in R3 is mapped to the hyperplane ζ ∗ (e) in (RP 4 )∗ , defined as e0 + e1 x1 + e2 x2 + e3 x3 + x4 = 0, i.e. with homogeneous coordinates (e0 , . . . , e3 , 1). A Laguerre transformation of R4 is an affine transformation f (x) = λA(x) + b, where λ = 0 and A is a linear transformation that preserves the Minkowski metric v, wM = v1 w1 + v2 w2 + v3 w3 − v4 w4 . From this projective point of view Laguerre transformations are exactly those projective transformations of RP 4 that preserve the absolute quadric Ω: x0 = 0, x12 + x22 + x32 − x42 . A line in R4 is called isotropic line if v, v = 0 for its directional vector v. All the spheres tangent to a surface X in R3 define an isotropic hypersurface Iso(X) in R4 . It is a union of isotropic lines. Iso(X) is also a union of all offsets of X in the form of hyperplane sections Iso(X) ∩ {x4 = d}. For example, the zero section (d = 0) can be identified with X. It is natural to apply Laguerre transformation L to the surface X in R3 using the zero section L(X) = L(Iso(X)) ∩ {x4 = 0}. In particular, offsetting corresponds to the simple Laguerre transformation—translation in the x4 -axis direction. Pottmann and Peternell (1998) proposed to use the Blaschke model of Laguerre geometry for PN surface modeling. The Blaschke map δ : (RP 4 )∗ → RP 4 maps hyperplanes to points. For an appropriate affine coordinate system δ(ζ ∗ (e)) = (1, e1 , e2 , e3 , e0 ). Therefore the image is lying in a quadric x12 + x22 + x32 = 1 in R4 which is called the Blaschke cylinder B ⊂ R4 . The main observation of Pottmann and Peternell (1998) is the following 1–1 correspondence: {PN surfaces in R3 } ↔ {rational surfaces in the Blaschke cylinder B}. x12
+ x22
+ x32
= x02
RP 3
in can be parametrized by homogeneous coordinates of 2 PS (z0 , z1 ) = |z0 | + |z1 |2 , 2 Re(z0 z¯ 1 ), 2 Im(z0 z¯ 1 ), |z0 |2 − |z1 |2 .
The unit sphere S:
(1) CP 1 : (2)
The map PS : C2 → R4 is homogeneous in the sense that PS (λz0 , λz1 ) = |λ|2 PS (z0 , z1 ). Bézier curves and patches on S can be lifted to C2 uniquely up to a constant factor. Therefore PS is called a universal rational parametrization of S (see Cox et al. (2003), Krasauskas (2006) for details). From the projective point of view the Blaschke cylinder is a cone over a sphere, i.e. it is a toric threefold. Its universal rational parametrization is slightly more complicated (cf. Cox et al., 2003): (3) PB (d0 , d1 , z0 , z1 ) = d0 , 2d1 Re(z0 z¯ 1 ), 2d1 Im(z0 z¯ 1 ), d1 |z0 |2 − |z1 |2 , d1 |z0 |2 + |z1 |2 . The map PB : R2 × C2 → R5 is homogeneous, PB (ρ|λ|2 d0 , ρd1 , λz0 , λz1 ) = ρ|λ|2 PB (d0 , d1 , z0 , z1 ), and can be useful for studying rational surfaces in the Blaschke cylinder. According to (1), this is equivalent to studying PN surfaces in R3 . 4. Branching blends based on PN surfaces The general scheme of the proposed method consists of three steps: (1) construction of a Gaussian map; (2) definition of a support function; (3) conversion from dual to point representation. Define a branching blend of two circular cylinders/cones as a ring shaped C1 blend between them, such that one of its boundary curve girds Qa and the other lies on the side of Qb . Here we illustrate the new method in the case of branching blend of two cylinders considered in Example 1. Construction 2. Our goal is to generate a branching blend of cylinders Qa and Qb defined in Example 1, which is a PN surface bounded by a circle Ca , x3 = h, on the vertical cylinder Qa and by a rational curve Cb on the upper side of the horizontal cylinder Qb (see Fig. 2, left). Step 1. Normals along Ca and Cb define the following curves on the unit sphere: a circle na on the equator and a circular arc nb on the plane section x1 = 0, see Fig. 2(right). We do not fix endpoints of nb yet: we keep them
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Fig. 2. Boundary curves Ca (in black), Cb (in grey) and their Gaussian images na (in black), nb (in grey) on the unit sphere.
Fig. 3. Blending two cylinders in three Laguerre-different positions.
symmetric with respect to the plane x2 = 0. In order to build a symmetric Gaussian map it remains to find a Bézier representation of the spherical quarter, x1 , x3 0. Consider symmetric quadratic parametrizations of these circles in homogeneous form: na (t) = (1 + t 2 , 2t, 1 − t 2 , 0),
n b (t) = (1 + t 2 , 2t, 0, 1 − t 2 ).
Since the arc nb is traced twice when we go around Cb , we reparametrize the second arc by the formula t → 2kt/(1 + t 2 ),
(4)
i.e. nb (t) = (1 + t 2 )2 n b (2kt/(1 + t 2 )). The endpoints nb (±1) = n b (±k) of nb symmetric with respect to the plane x2 = 0 and depend on the parameter k > 0. Now it is a standard situation where a universal rational parametrization of a sphere can be used as described in Krasauskas (2006) in order to obtain a unique Gaussian map N (t, u) of degree (4, 2). We lift the quadratic and the quartic curves na and nb to a linear and a quadratic curves in the space C2 (i.e. PS (ca ) = na , PS (cb ) = nb ) ca = (1 − it, t − i),
cb = (1 + t 2 , 2kt).
Then the Gaussian map is defined by N (t, u) = PS (Λ(t)(1 − u)ca + ucb ), where Λ(t) = 1 + it is a path between 1 and 1 + i in C (see Krasauskas (2006) for details): N(t, u) = 8 − 12u + 16t 2 − 24ut 2 + 5u2 + 10u2 t 2 + 8t 4 − 12t 4 u + 5t 4 u2 − 16ku2 t 2 + 4k 2 u2 t 2 + 16kut 2 , 4t (1 + t 2 )(2 − u)(ku − 2u + 2), 4(1 − u)(2 − u)(1 − t 4 ), u(4 + 8t 2 − 3u − 6ut 2 + 4t 4 − 3t 4 u + 16kut 2 − 4k 2 ut 2 − 16kt 2 ) . Step 2. Combining Eq. (2), Eq. (3) and taking d1 = 1, the tangent plane formula for a PN surface x = x(t, u) with the Gaussian map N(t, u) has the following form (here we assume x4 = 0) S(t, u)x0 + N1 (t, u)x1 + N2 (t, u)x2 + N3 (t, u)x3 + N0 (t, u)x4 = 0.
(5)
Let us call S(t, u) a support function (it is d0 in Eq. (3)). In fact the usual support function is defined as a distance from the tangent plane to the origin and it is equal to S/N0 . In our case the Gaussian map is fixed, so N0 is not changing.
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Fig. 4. Blending cylinders with different angles between their generators.
A support function Sa (t, u) of a sphere touching the vertical cylinder along the circle Ca is derived from the same Eq. (5) by substituting x = (1, 0, 0, h, ra ). The other substitution x = (1, 0, 0, 0, rb ) will define a support function Sb (t) of the horizontal cylinder along the curve Cb . Finally we define a support function for our blending surface as a combination S(t, u) = Sa (t, u) + u2 (Sb (t) − Sa (t, 1)), which preserves C 1 contact with S(t, u) along the boundary u = 0. Initial data depend on three parameters ra , rb and h. Assume that h = ra + rb (see Remark 3). If we choose k = ra /rb then ra = kr and h = (k + 1)r, where we denote r = rb , and the support function S(t, u) depends only on k and r: S(t, u) = r (1 + t 2 )2 u(4 − 3u)/4 + 2k(1 − 2t 2 u + 2t 2 + t 4 )(1 − u) + k 2 t 2 u2 . Step 3. From the plane representation we obtain the point representation by solving the following linear system (where t and u indicate partial differentiation with respect to t and u, respectively) Sx0 + N1 x1 + N2 x2 + N3 x3 = 0, St x0 + (N1 )t x1 + (N2 )t x2 + (N3 )t x3 = 0, Su x0 + (N1 )u x1 + (N2 )u x2 + (N3 )u x3 = 0. If the degree of (S, N1 , N2 , N3 ) is (dt , du ) then the degree of the solution (x0 , x1 , x2 , x3 ) is (3dt − 2, 3du − 2) in general. Since (dt , du ) = (4, 2) we can expect a solution of degree (10, 4). Fortunately, the solution expressed in terms of minors of the corresponding 3 × 4 matrix has the following common factor g = gcd(x0 , x1 , x2 , x3 ) = 2(1 + t 2 )2 (1 − u) + k(u + 6t 2 u + t 4 u − 8t 2 ), that enables us to drop the degree down to (6, 3) by factoring X = x/g: X(t, u) = 2(1 + t 2 )(2 − u) (1 + t 2 )2 (8 − 12u + 5u2 ) + 16kt 2 u(1 − u) + 4k 2 t 2 u2 , 8rkt (2 − u)(3u2 − 6u + 4)(1 + t 2 )2 + 4ku(1 − u)(t 4 − t 4 u + 2t 2 + 1 − u) , 4kr(1 − t 2 ) (1 + t 2 )2 (2 − u)(3u2 − 6u + 4) + 8kt 2 u2 (1 − u) + 4k 2 t 2 u3 , 2r(1 + t 2 )(2 − u) (8 − 12u + 5u2 )(1 + t 2 )2 + 8k(1 − u + 2t 2 − t 4 u + t 4 )(1 − u) − 4k 2 t 2 u2 . Remark 3. Offsetting in a distance d means the transformation of the support function S → S + dN0 . This adds the linear variable d to the first column of the system of equations (6), and therefore, also to the solution x(t, u). The common factor g will be the same and the solution X(t, u, d) will be of degree (6, 3, 1) in variables (t, u, d). It follows that offsetting can be used to extend the construction to other cases with arbitrary parameters ra , rb , h using simple expressions r = (h − ra + rb )/2, k = h/r − 1, d = (h − ra − rb ). Any circular cone/cylinder defines a line with a positive signature in the Minkowski space R4 . Pairs of such lines in R4 are classified from the point of view of Laguerre geometry in Kazakeviˇci¯ut˙e (2005b). Any class can be characterized by three invariants (σ2 , σ3 , α):
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• σ2 = (+, +), (+, 0), (+, −) is a signature of a 2-plane generated by directional vectors of the lines; • σ3 = (+, +, +), (+, +, 0), (+, +, −) is a signature of the 3-dimensional affine span of both lines in R4 ; • α is a Euclidean (resp. hyperbolic) angle between lines in case σ2 = (+, +) (resp. σ2 = (+, −)). A condition σ2 = (+, +) in the theorem below defines almost half of general positions. Indeed, σ2 = (+, −) defines the other half, and σ2 = (+, 0) is a lower dimensional subset of all possible configurations. In particular, for any pair of two cylinders σ2 = (+, +). Theorem 4. A PN surface construction of a branching blend of two circular cones/cylinders of degree (6, 3) is possible in all positions with σ2 = (+, +). These parametrization degrees are stable under Laguerre transformations and are the minimal possible ones with this property. Proof. A key point of Construction 2 is degree reduction in Step 3 that depends on the Gaussian map, i.e. Step 1. Our plan is to generalize this particular blend to all other positions. At first we check that Construction 2 can be adapted to branching blends for all cases of two cylinders. It is enough to modify the Gaussian map. Let us consider just two examples (others can be defined as their compositions): (1) If the cylinder Qa is translated in distance l in direction of the x2 -axis (see Fig. 3) then we change the quadratic transformation Eq. (4) by the following one ra − rb + h + l ra − rb + h − l (k1 + k2 )t k1 − k2 , k1 = , k2 = , + 2 ra − rb − h + l ra − rb − h − l 1 + t2 combined with the linear projective transformation t + w − 1 + wt 2−l , w= . t → 1 + w − t + wt 2+l (2) In case when the cylinder Qb is rotated about x2 -axis by the angle α (see Fig. 4) we denote e = tan(α/2), then rotate n b (t) to n b (t) = (1 + e2 )(1 + t 2 ), 2(1 + e2 )t, −2e(1 − t 2 ), (1 − e2 )(1 − t 2 ) , t →
and then define nb (t) by the following parameter transformation in n b (t) k − e2 Kt , K = . 1 + t2 1 − ke2 Finally Λ(t) is also modified to Λ(t) = (1 − e) + (1 + e)ti. t →
A case of two cones with σ2 = (+, +) can be reduced to the previous case of cylinders using Laguerre transformation, that is a composition of suitable Lorenz rotations (see details in Kazakeviˇci¯ut˙e, 2005b). For example, a Lorenz rotation with a fixed (x2 , x3 )-plane has the following simple form H (x0 , . . . , x4 ) = (1 − v 2 )x0 , (1 + v 2 )x1 + 2vx4 , x2 , x3 , 2vx1 + (1 + v 2 )x4 , where v = tanh(α/2), α is a hyperbolic angle (see Fig. 5). It remains to prove that degree (dt , du ) = (6, 3) is the minimal one. Stable degree du = 2 means that X is a canal surface, and Example 1 shows that this is impossible in general case. From topology of the curve Cb position on the side of Qb follows that dt 4. Unit normals to the cylinder along Cb define a ruled surface R in R4 with two directrices in general position of degree 2 (the axis of the cylinder traced twice) and 4 (a conic at infinity traced twice). Therefore deg R = 6 and Laguerre stable degree dt = 6 as it is a general hyperplanar section. 2 Let X be the branching blend surface from Construction 2. Then a parametrization X(t, u, d) (see Remark 3) is in fact rational parametrization of the associated isotropic hypersurface Iso(X) in R4 . We decompose the 3d variety Iso(X) as a union of 1-parameter family of surfaces in two ways: Rt = X(t, u, d) for fixed t and Ru = X(t, u, d) for fixed u. They are ruled surfaces of degrees deg Rt = 3 and deg Ru = 6, since by Theorem 4 their general hyperplane sections are of the same degrees.
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Fig. 5. A result of hyperbolic rotation.
Fig. 6. Four tangent spheres of the symmetric branching blend.
Corollary 5. The blending surface generated by Construction 2 is the envelope of a rational family of circular cones. Proof. For every t a surface Rt is a ruled cubic. Hence Rt necessarily contains a linear directrix Lt that defines a circular cone (or cylinder), which is tangent to X. It is enough to calculate Lt for every Laguerre class separately and then make use of the Laguerre invariance of the construction. 2 5. Shape of the blending surface The branching blend of the cylinders Qa and Qb generated by Construction 2 has two planes of symmetry x1 = 0 and x2 = 0. Consider the quarter of the blend parametrized by X(t, u) with a parameter domain 0 t, u 1. Straightforward calculations show that vertical boundary curves X(0, u) and X(1, u) are circles. Hence the blending surface is tangent to four spheres along circles, and these spheres are touching the both cylinders as it is illustrated in Fig. 6. Moreover, these properties are preserved under Laguerre transformations. The surface X(t, u) behaves very much like a blend generated by a rolling ball that traces a circle Ca on the cylinder Qa . Such a rolling ball blend is irrational (see Example 1). Therefore, Construction 2 and its Laguerre transformations can be treated as a “rational PN approximations” of the corresponding rolling ball blends. The symmetric Construction 2 is unique for given data ra , rb and h in the sense that the first two steps (Gaussian map and dual representation) are unique of degree (4, 2). Nevertheless one can introduce additional degree of freedom by destroying a symmetry with respect to a plane x2 = 0. Let a modified Gaussian map be defined as N (t, u) = PS (Λ (t)(1 − u)ca (t) + ucb (t)), where ca (t) = m(1 − t)ca (−1) + (1 + t)ca (1) /2, Λ (t) = m−1 (1 − t)Λ(−1) + (1 + t)Λ(1) /2. This is essentially a projective reparametrization of the circle Ca . The rest of the construction and the resulting degree (6, 3) will be the same but the symmetry with respect to x2 = 0 will not be preserved when m = 1. This additional parameter m can be useful in asymmetric situations. For example, by changing m one can improve shape of a blend in Fig. 3(right) as it is shown in Fig. 7.
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Fig. 7. Shape dependence on parameter values m = 1, 1.5, 2.
6. Generalization to Dupin cyclides The usual inversion in R3 can be naturally extended to the Minkowski space R4 by the formula: Invp (x) = p + (x − p)/x − p, x − pM ,
(6)
where p ∈ R4 is the center of the inversion. Lemma 6. Any inversion Invp , p ∈ R4 , transforms PN surfaces to PN surfaces. Proof. By translations in any R3 direction and offsetting the point p can be moved to the origin, and the problem is reduced to the classical inversion Inv(x) = x/x2 in R3 . Let f = f (t, u) be a rational parametrization of a PN surface. Hence f 2 is rational. Then partial derivatives of F = Inv(f ) = f/f 2 can be described in the form Ft = Prf (ft )/f 2 ,
Fu = Prf (fu )/f 2 ,
where Pra (x) = x − 2(a, x)/a2 )a is a reflection in R3 . It is clear that Pra preserves cross products and distances, therefore Ft × Fu = ft × fu /f 4 is rational.
2
In order to generalize Construction 2 we need two properties of Dupin cyclides (see e.g. Degen, 2002): • a Dupin cyclide is uniquely defined by its three inscribed/circumscribed spheres; • an inversion of a circular cylinder/cone is a Dupin cyclide. Construction 7. Let Q be a cylinder/cone with a fixed circle C on it and let D be a Dupin cyclide. Then one can take any sphere S inscribed in D and find the other Dupin cyclide D , which contains S (as inscribed sphere) and is tangent to Q along the circle C. If S corresponds to a point s ∈ R4 then we apply an inversion Invs to D and D and get two cylinders/cones Invs (D) and Invs (D ) which can be blended using a suitable branching blend X according to Theorem 4 (if the required conditions are satisfied). Then we apply the inversion Invs again and get a certain blending Invs (X) between Q and D. Construction 7 is illustrated in Fig. 8, where S is a point, i.e. a sphere with zero radius. In Fig. 9 an inversion with a center in a sphere with the non-zero radius (on the left) is shown, where a blend of two cylinders of the same radii are transformed to a torus/cylinder blend (on the right). Theorem 8. Construction 7 produces a blending PN surface Y = Invp X of degree (8, 4) between a cylinder/cone Q with a fixed circle on it and a Dupin cyclide D. The parametrization degrees are stable under Laguerre transformations. Proof. In general case when the inversion center p is not incident with a curve ρ there is a simple formula for degree calculation deg Invp (ρ) = 2 deg ρ − #(ρ ∩ Ω),
(7)
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Fig. 8. Blending construction between a cylinder and a Dupin cyclide.
Fig. 9. An inversion generates blending between a cylinder and a torus.
where #(ρ ∩ Ω) is a number of intersection points with the absolute quadric Ω. Cubic isoparametric curves ρt on X are hyperplanar sections of ruled surfaces Rt that intersect Ω along conics. Similarly sextic isoparametric curves ρu on X are hyperplanar sections of ruled surfaces Ru that intersect Ω by quartic curves. Hence, from Eq. (7) follows deg Invp (ρt ) = 2 · 3 − 2 = 4 and deg Invp (ρu ) = 2 · 6 − 4 = 8. 2 7. Conclusions We proposed a new PN surface construction for branching blend of two circular cones and cylinders in quite general positions relatively to each other. The construction is Laguerre invariant (in particular offset-stable) and has the minimal possible parametrization degree (6, 3). The construction can be generalized to the blending of a circular cone/cylinder and a Dupin cyclide of degree (8, 4). These results show that the dual approach to PN surfaces is a natural one which can generate blendings of reasonable degree. Future research will be devoted to Bézier representation, shape analysis and singularity structure of the introduced branching blend. Also it will be interesting to extend the proposed construction to other cylinders/cones positions that are not presented in Theorem 4. References Boehm, W., 1990. On cyclides in geometric modeling. Computer Aided Geometric Design 7, 243–255. Cox, D., Krasauskas, R., Musta¸taˇ , M., 2003. Universal rational parametrizations and toric varieties. Topics in Algebraic Geometry and Geometric Modeling, Contemporary Mathematics, vol. 334, pp. 241–265. Degen, W., 2002. Cyclides. In: Farin, G., Hoschek, J., Kim, M.-S. (Eds.), Handbook of Computer Aided Geometric Design. Elsevier Science, pp. 575–602. Farouki, R.T., Pottmann, H., 1996. Polynomial and rational Pythagorean-hodograph curves reconciled. In: Mullineux, G. (Ed.), The Mathematics of Surfaces VI. Oxford Univ. Press, pp. 355–378. Jüttler, B., Sampoli, M.L., 2000. Hermite interpolation by piecewise polynomial surfaces with rational offsets. Computer Aided Geometric Design 17, 361–385. Kazakeviˇci¯ut˙e, M., 2005a. Blending of natural quadrics with rational canal surfaces. PhD Thesis, Vilnius University. Kazakeviˇci¯ut˙e, M., 2005b. Classification of pairs of natural quadrics from the point of view of Laguerre geometry. Lithuanian Mathematical Journal 45, 63–84. Krasauskas, R., 2006. Bézier patches on almost toric surfaces. In: Elkadi, M., Mourrain, B., Piene, R. (Eds.), Algebraic Geometry and Geometric Modeling. In: Mathematics and Visualization Series. Springer, pp. 135–150. Krasauskas, R., 2007. Minimal rational parametrizations of canal surfaces. Computing 79, 281–290.
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Krasauskas, R., Kazakeviˇci¯ut˙e, M., 2005. Universal rational parametrizations and spline curves on toric surfaces. In: Computational Methods for Algebraic Spline Surfaces, ESF Exploratory Workshop. Springer, pp. 213–231. Krasauskas, R., Zub˙e, S., 2007. Canal surfaces defined by quadratic families of spheres. In: Proceedings of the COMPASS II Workshop, pp. 138– 150. Peternell, M., 1997. Rational parametrizations for envelopes of quadric families. Ph.D. Thesis, Institute of Geometry, University of Technology, Vienna. Peternell, M., Pottmann, H., 1997. Computing rational parametrizations of canal surfaces. Journal of Symbolic Computation 23, 255–266. Pottmann, H., 1993. Rational curves and surfaces with rational offsets. Computer Aided Geometric Design 12, 175–192. Pottmann, H., Peternell, M., 1998. Application of Laguerre geometry in CAGD. Computer Aided Geometric Design 15, 165–186. Pratt, M.J., 1990. Cyclides in computer aided geometric design. Computer Aided Geometric Design 7, 221–242. Rossignac, J.R., 1987. Constraints in constructive solid geometry. In: Proceedings of the 1986 Workshop on Interactive 3D Graphics. Chapel Hill, North Carolina, 1987, pp. 93–110. Shene, C.-K., 1998. Blending two cones with Dupin cyclides. Computer Aided Geometric Design 15, 643–673.
Branching blend of natural quadrics based on surfaces with rational offsets Rimvydas Krasauskas Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24, LT-03225 Vilnius, Lithuania Received 5 July 2007; received in revised form 1 November 2007; accepted 4 November 2007 Available online 14 December 2007
Abstract A new branching blend between two natural quadrics (circular cylinders/cones or spheres) in many positions is proposed. The blend is a ring shaped patch of a PN surface (surface with rational offset) parametrized by rational bivariant functions of degree (6, 3). The general theory of PN surfaces is developed using Laguerre geometry and a universal rational parametrization of the Blaschke cylinder. The construction is extended via inversion to a PN branching blend of degree (8, 4) between Dupin cyclide and a natural quadric. © 2007 Elsevier B.V. All rights reserved. Keywords: Rational offset; PN surface; Natural quadrics; Dupin cyclides
1. Introduction
In current CAD systems, curves and surfaces are represented in a standard NURBS form, i.e. they are parametrized by rational B-splines. Unfortunately, offsets of rational surfaces arising in practical applications are in general not rational and need to be approximated. On the other hand traditional 3d modeling primitives like natural quadrics (sphere, circular cylinder/cone) or torus surfaces admit rational offsets. According to Rossignac (1987), about 99% of mechanical parts can be represented by combinations of planes and natural quadrics with the possibility of representing blends between them. Since rolling ball blends have rather intuitive shape, they are the most popular in practice, but in general they are irrational or admit rational parametrizations of high degree (see details in Section 2). On the other hand in many cases the smoothness of the blend is more important than its exact shape. This situation in CAD industry motivates us to search for more general rational surfaces with rational offsets that can serve as blending surfaces between natural quadrics. We choose to work with Pythagorean normal (PN) surfaces which were introduced by Pottmann (1993) and are direct generalizations of well-known planar rational
E-mail address: [email protected]. 0167-8396/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cagd.2007.11.005
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Pythagorean hodograph curves (Farouki and Pottmann, 1996). A surface in R3 with a fixed rational parametrization f (t, u) = (f1 (t, u), f2 (t, u), f3 (t, u)) is called a PN surface, if its unit normal ∂f1 ∂f2 ∂f3 ∂f1 ∂f2 ∂f3 ft × fu , ft = , , , fu = , , , N (t, u) = ft × fu ∂t ∂t ∂t ∂u ∂u ∂u is rational. It is clear that the offset of such PN surface at distance d has a rational parametrization f (t, u) + dN (t, u). The paper is organized as follows: Section 2 surveys previous PN surface blending constructions. Section 3 outlines necessary facts about the Blaschke model of Laguerre geometry. Section 4 describes all steps of the main branching blend construction and derives its properties. The construction is generalized to Dupin cyclides using inversions in Section 5. Conclusions are drawn in Section 6. 2. Survey of PN surface blending constructions The majority of known PN surface blending constructions between natural quadrics are pieces of rational canal surfaces. A canal surface is an envelope of a 1-parameter family of spheres in R3 , defined by a spine curve s(t) = (s1 (t), s2 (t), s3 (t)) and a radius function r(t). For the envelope to be real the condition on derivatives ˙s (t)2 − r˙ 2 (t) 0 is necessary. Let us collect this data in a curve γ (t) = (s(t), r(t)) ∈ R4 and denote the canal surface by C(γ ). Peternell and Pottmann (1997) proved that a canal surface C(γ ) defined by a rational curve γ can be rationally parametrized. If γ has degree k then Cγ can be parametrized with degree at least (3k − 2, 2), under a mild condition ˙s (t)2 − r˙ 2 (t) > 0 (Krasauskas, 2007). This degree bound is the minimal possible in general. The first non-trivial blending application of canal surfaces was proposed by Pratt (1990) and Boehm (1990). They used Dupin cyclides, which are quartic canal surfaces defined by special conics γ ∈ R4 . For example, any two circular cones with a common inscribed sphere can be blended by a part of a Dupin cyclide bounded by two circles (see Fig. 1, left). All possible blend surfaces of this type were studied in details by Shene (1998). Canal surfaces defined by general conics γ ∈ R4 can blend natural quadrics in more general positions (Fig. 1, middle) as was shown in Krasauskas and Zub˙e (2007). Results on rational parametrizations (Peternell and Pottmann, 1997; Peternell, 1997; Krasauskas, 2007) enabled Kazakeviˇci¯ut˙e (2005a) to develop a theory of rational variable radius rolling ball blends between natural quadrics in arbitrary positions. Here we will consider just one important case. Example 1. Let Qa and Qb be two cylinders in R3 defined by equations x12 + x22 = ra2 and x22 + x32 = rb2 , where 0 < ra < rb (Fig. 1, right). The condition that a sphere touches both cylinders Qa and Qb defines a quartic surface V ⊂ R4 . Any curve on V generates a canal surface touching both cylinders, i.e. a rolling ball blend. Unfortunately a fixed radius case corresponds to an irrational curve on V . Nevertheless, a certain rational quartic curve γ ⊂ V can be found (Krasauskas and Kazakeviˇci¯ut˙e, 2005). This construction generates a canal surface Cγ of degree (10, 2) which is minimal possible (Krasauskas, 2007). It is impossible to construct such a blending with a boundary circle on the cylinder Qa , since the corresponding curve on V and the associated canal surface are irrational. We are going to search for a more general PN surface blending construction as one piece of an algebraic surface. Other available methods related to surfaces with rational offsets, like patching with cubic cyclides (Pottmann and Peternell, 1998) or linear normal constructions (Jüttler and Sampoli, 2000), are not acceptable for us because they generate a 1–1 Gaussian map, which is too restrictive for our purposes.
Fig. 1. Blending constructions with canal surfaces.
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3. Blaschke model of Laguerre geometry The main idea of Laguerre geometry is to consider oriented spheres in R3 as points in R4 (see Pottmann and Peternell (1998) for details). A sphere c with a center (x1 , x2 , x3 ) and radius x4 is mapped to a point ζ (c) = (x1 , x2 , x3 , x4 ) ∈ R4 (the orientation is encoded by a sign of x4 ). We use also the projective extension RP 4 of R4 with additional coordinate x0 . Each oriented plane e: e0 + e1 x1 + e2 x2 + e3 x3 = 0, e12 + e22 + e32 = 1, in R3 is mapped to the hyperplane ζ ∗ (e) in (RP 4 )∗ , defined as e0 + e1 x1 + e2 x2 + e3 x3 + x4 = 0, i.e. with homogeneous coordinates (e0 , . . . , e3 , 1). A Laguerre transformation of R4 is an affine transformation f (x) = λA(x) + b, where λ = 0 and A is a linear transformation that preserves the Minkowski metric v, wM = v1 w1 + v2 w2 + v3 w3 − v4 w4 . From this projective point of view Laguerre transformations are exactly those projective transformations of RP 4 that preserve the absolute quadric Ω: x0 = 0, x12 + x22 + x32 − x42 . A line in R4 is called isotropic line if v, v = 0 for its directional vector v. All the spheres tangent to a surface X in R3 define an isotropic hypersurface Iso(X) in R4 . It is a union of isotropic lines. Iso(X) is also a union of all offsets of X in the form of hyperplane sections Iso(X) ∩ {x4 = d}. For example, the zero section (d = 0) can be identified with X. It is natural to apply Laguerre transformation L to the surface X in R3 using the zero section L(X) = L(Iso(X)) ∩ {x4 = 0}. In particular, offsetting corresponds to the simple Laguerre transformation—translation in the x4 -axis direction. Pottmann and Peternell (1998) proposed to use the Blaschke model of Laguerre geometry for PN surface modeling. The Blaschke map δ : (RP 4 )∗ → RP 4 maps hyperplanes to points. For an appropriate affine coordinate system δ(ζ ∗ (e)) = (1, e1 , e2 , e3 , e0 ). Therefore the image is lying in a quadric x12 + x22 + x32 = 1 in R4 which is called the Blaschke cylinder B ⊂ R4 . The main observation of Pottmann and Peternell (1998) is the following 1–1 correspondence: {PN surfaces in R3 } ↔ {rational surfaces in the Blaschke cylinder B}. x12
+ x22
+ x32
= x02
RP 3
in can be parametrized by homogeneous coordinates of 2 PS (z0 , z1 ) = |z0 | + |z1 |2 , 2 Re(z0 z¯ 1 ), 2 Im(z0 z¯ 1 ), |z0 |2 − |z1 |2 .
The unit sphere S:
(1) CP 1 : (2)
The map PS : C2 → R4 is homogeneous in the sense that PS (λz0 , λz1 ) = |λ|2 PS (z0 , z1 ). Bézier curves and patches on S can be lifted to C2 uniquely up to a constant factor. Therefore PS is called a universal rational parametrization of S (see Cox et al. (2003), Krasauskas (2006) for details). From the projective point of view the Blaschke cylinder is a cone over a sphere, i.e. it is a toric threefold. Its universal rational parametrization is slightly more complicated (cf. Cox et al., 2003): (3) PB (d0 , d1 , z0 , z1 ) = d0 , 2d1 Re(z0 z¯ 1 ), 2d1 Im(z0 z¯ 1 ), d1 |z0 |2 − |z1 |2 , d1 |z0 |2 + |z1 |2 . The map PB : R2 × C2 → R5 is homogeneous, PB (ρ|λ|2 d0 , ρd1 , λz0 , λz1 ) = ρ|λ|2 PB (d0 , d1 , z0 , z1 ), and can be useful for studying rational surfaces in the Blaschke cylinder. According to (1), this is equivalent to studying PN surfaces in R3 . 4. Branching blends based on PN surfaces The general scheme of the proposed method consists of three steps: (1) construction of a Gaussian map; (2) definition of a support function; (3) conversion from dual to point representation. Define a branching blend of two circular cylinders/cones as a ring shaped C1 blend between them, such that one of its boundary curve girds Qa and the other lies on the side of Qb . Here we illustrate the new method in the case of branching blend of two cylinders considered in Example 1. Construction 2. Our goal is to generate a branching blend of cylinders Qa and Qb defined in Example 1, which is a PN surface bounded by a circle Ca , x3 = h, on the vertical cylinder Qa and by a rational curve Cb on the upper side of the horizontal cylinder Qb (see Fig. 2, left). Step 1. Normals along Ca and Cb define the following curves on the unit sphere: a circle na on the equator and a circular arc nb on the plane section x1 = 0, see Fig. 2(right). We do not fix endpoints of nb yet: we keep them
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Fig. 2. Boundary curves Ca (in black), Cb (in grey) and their Gaussian images na (in black), nb (in grey) on the unit sphere.
Fig. 3. Blending two cylinders in three Laguerre-different positions.
symmetric with respect to the plane x2 = 0. In order to build a symmetric Gaussian map it remains to find a Bézier representation of the spherical quarter, x1 , x3 0. Consider symmetric quadratic parametrizations of these circles in homogeneous form: na (t) = (1 + t 2 , 2t, 1 − t 2 , 0),
n b (t) = (1 + t 2 , 2t, 0, 1 − t 2 ).
Since the arc nb is traced twice when we go around Cb , we reparametrize the second arc by the formula t → 2kt/(1 + t 2 ),
(4)
i.e. nb (t) = (1 + t 2 )2 n b (2kt/(1 + t 2 )). The endpoints nb (±1) = n b (±k) of nb symmetric with respect to the plane x2 = 0 and depend on the parameter k > 0. Now it is a standard situation where a universal rational parametrization of a sphere can be used as described in Krasauskas (2006) in order to obtain a unique Gaussian map N (t, u) of degree (4, 2). We lift the quadratic and the quartic curves na and nb to a linear and a quadratic curves in the space C2 (i.e. PS (ca ) = na , PS (cb ) = nb ) ca = (1 − it, t − i),
cb = (1 + t 2 , 2kt).
Then the Gaussian map is defined by N (t, u) = PS (Λ(t)(1 − u)ca + ucb ), where Λ(t) = 1 + it is a path between 1 and 1 + i in C (see Krasauskas (2006) for details): N(t, u) = 8 − 12u + 16t 2 − 24ut 2 + 5u2 + 10u2 t 2 + 8t 4 − 12t 4 u + 5t 4 u2 − 16ku2 t 2 + 4k 2 u2 t 2 + 16kut 2 , 4t (1 + t 2 )(2 − u)(ku − 2u + 2), 4(1 − u)(2 − u)(1 − t 4 ), u(4 + 8t 2 − 3u − 6ut 2 + 4t 4 − 3t 4 u + 16kut 2 − 4k 2 ut 2 − 16kt 2 ) . Step 2. Combining Eq. (2), Eq. (3) and taking d1 = 1, the tangent plane formula for a PN surface x = x(t, u) with the Gaussian map N(t, u) has the following form (here we assume x4 = 0) S(t, u)x0 + N1 (t, u)x1 + N2 (t, u)x2 + N3 (t, u)x3 + N0 (t, u)x4 = 0.
(5)
Let us call S(t, u) a support function (it is d0 in Eq. (3)). In fact the usual support function is defined as a distance from the tangent plane to the origin and it is equal to S/N0 . In our case the Gaussian map is fixed, so N0 is not changing.
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Fig. 4. Blending cylinders with different angles between their generators.
A support function Sa (t, u) of a sphere touching the vertical cylinder along the circle Ca is derived from the same Eq. (5) by substituting x = (1, 0, 0, h, ra ). The other substitution x = (1, 0, 0, 0, rb ) will define a support function Sb (t) of the horizontal cylinder along the curve Cb . Finally we define a support function for our blending surface as a combination S(t, u) = Sa (t, u) + u2 (Sb (t) − Sa (t, 1)), which preserves C 1 contact with S(t, u) along the boundary u = 0. Initial data depend on three parameters ra , rb and h. Assume that h = ra + rb (see Remark 3). If we choose k = ra /rb then ra = kr and h = (k + 1)r, where we denote r = rb , and the support function S(t, u) depends only on k and r: S(t, u) = r (1 + t 2 )2 u(4 − 3u)/4 + 2k(1 − 2t 2 u + 2t 2 + t 4 )(1 − u) + k 2 t 2 u2 . Step 3. From the plane representation we obtain the point representation by solving the following linear system (where t and u indicate partial differentiation with respect to t and u, respectively) Sx0 + N1 x1 + N2 x2 + N3 x3 = 0, St x0 + (N1 )t x1 + (N2 )t x2 + (N3 )t x3 = 0, Su x0 + (N1 )u x1 + (N2 )u x2 + (N3 )u x3 = 0. If the degree of (S, N1 , N2 , N3 ) is (dt , du ) then the degree of the solution (x0 , x1 , x2 , x3 ) is (3dt − 2, 3du − 2) in general. Since (dt , du ) = (4, 2) we can expect a solution of degree (10, 4). Fortunately, the solution expressed in terms of minors of the corresponding 3 × 4 matrix has the following common factor g = gcd(x0 , x1 , x2 , x3 ) = 2(1 + t 2 )2 (1 − u) + k(u + 6t 2 u + t 4 u − 8t 2 ), that enables us to drop the degree down to (6, 3) by factoring X = x/g: X(t, u) = 2(1 + t 2 )(2 − u) (1 + t 2 )2 (8 − 12u + 5u2 ) + 16kt 2 u(1 − u) + 4k 2 t 2 u2 , 8rkt (2 − u)(3u2 − 6u + 4)(1 + t 2 )2 + 4ku(1 − u)(t 4 − t 4 u + 2t 2 + 1 − u) , 4kr(1 − t 2 ) (1 + t 2 )2 (2 − u)(3u2 − 6u + 4) + 8kt 2 u2 (1 − u) + 4k 2 t 2 u3 , 2r(1 + t 2 )(2 − u) (8 − 12u + 5u2 )(1 + t 2 )2 + 8k(1 − u + 2t 2 − t 4 u + t 4 )(1 − u) − 4k 2 t 2 u2 . Remark 3. Offsetting in a distance d means the transformation of the support function S → S + dN0 . This adds the linear variable d to the first column of the system of equations (6), and therefore, also to the solution x(t, u). The common factor g will be the same and the solution X(t, u, d) will be of degree (6, 3, 1) in variables (t, u, d). It follows that offsetting can be used to extend the construction to other cases with arbitrary parameters ra , rb , h using simple expressions r = (h − ra + rb )/2, k = h/r − 1, d = (h − ra − rb ). Any circular cone/cylinder defines a line with a positive signature in the Minkowski space R4 . Pairs of such lines in R4 are classified from the point of view of Laguerre geometry in Kazakeviˇci¯ut˙e (2005b). Any class can be characterized by three invariants (σ2 , σ3 , α):
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• σ2 = (+, +), (+, 0), (+, −) is a signature of a 2-plane generated by directional vectors of the lines; • σ3 = (+, +, +), (+, +, 0), (+, +, −) is a signature of the 3-dimensional affine span of both lines in R4 ; • α is a Euclidean (resp. hyperbolic) angle between lines in case σ2 = (+, +) (resp. σ2 = (+, −)). A condition σ2 = (+, +) in the theorem below defines almost half of general positions. Indeed, σ2 = (+, −) defines the other half, and σ2 = (+, 0) is a lower dimensional subset of all possible configurations. In particular, for any pair of two cylinders σ2 = (+, +). Theorem 4. A PN surface construction of a branching blend of two circular cones/cylinders of degree (6, 3) is possible in all positions with σ2 = (+, +). These parametrization degrees are stable under Laguerre transformations and are the minimal possible ones with this property. Proof. A key point of Construction 2 is degree reduction in Step 3 that depends on the Gaussian map, i.e. Step 1. Our plan is to generalize this particular blend to all other positions. At first we check that Construction 2 can be adapted to branching blends for all cases of two cylinders. It is enough to modify the Gaussian map. Let us consider just two examples (others can be defined as their compositions): (1) If the cylinder Qa is translated in distance l in direction of the x2 -axis (see Fig. 3) then we change the quadratic transformation Eq. (4) by the following one ra − rb + h + l ra − rb + h − l (k1 + k2 )t k1 − k2 , k1 = , k2 = , + 2 ra − rb − h + l ra − rb − h − l 1 + t2 combined with the linear projective transformation t + w − 1 + wt 2−l , w= . t → 1 + w − t + wt 2+l (2) In case when the cylinder Qb is rotated about x2 -axis by the angle α (see Fig. 4) we denote e = tan(α/2), then rotate n b (t) to n b (t) = (1 + e2 )(1 + t 2 ), 2(1 + e2 )t, −2e(1 − t 2 ), (1 − e2 )(1 − t 2 ) , t →
and then define nb (t) by the following parameter transformation in n b (t) k − e2 Kt , K = . 1 + t2 1 − ke2 Finally Λ(t) is also modified to Λ(t) = (1 − e) + (1 + e)ti. t →
A case of two cones with σ2 = (+, +) can be reduced to the previous case of cylinders using Laguerre transformation, that is a composition of suitable Lorenz rotations (see details in Kazakeviˇci¯ut˙e, 2005b). For example, a Lorenz rotation with a fixed (x2 , x3 )-plane has the following simple form H (x0 , . . . , x4 ) = (1 − v 2 )x0 , (1 + v 2 )x1 + 2vx4 , x2 , x3 , 2vx1 + (1 + v 2 )x4 , where v = tanh(α/2), α is a hyperbolic angle (see Fig. 5). It remains to prove that degree (dt , du ) = (6, 3) is the minimal one. Stable degree du = 2 means that X is a canal surface, and Example 1 shows that this is impossible in general case. From topology of the curve Cb position on the side of Qb follows that dt 4. Unit normals to the cylinder along Cb define a ruled surface R in R4 with two directrices in general position of degree 2 (the axis of the cylinder traced twice) and 4 (a conic at infinity traced twice). Therefore deg R = 6 and Laguerre stable degree dt = 6 as it is a general hyperplanar section. 2 Let X be the branching blend surface from Construction 2. Then a parametrization X(t, u, d) (see Remark 3) is in fact rational parametrization of the associated isotropic hypersurface Iso(X) in R4 . We decompose the 3d variety Iso(X) as a union of 1-parameter family of surfaces in two ways: Rt = X(t, u, d) for fixed t and Ru = X(t, u, d) for fixed u. They are ruled surfaces of degrees deg Rt = 3 and deg Ru = 6, since by Theorem 4 their general hyperplane sections are of the same degrees.
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Fig. 5. A result of hyperbolic rotation.
Fig. 6. Four tangent spheres of the symmetric branching blend.
Corollary 5. The blending surface generated by Construction 2 is the envelope of a rational family of circular cones. Proof. For every t a surface Rt is a ruled cubic. Hence Rt necessarily contains a linear directrix Lt that defines a circular cone (or cylinder), which is tangent to X. It is enough to calculate Lt for every Laguerre class separately and then make use of the Laguerre invariance of the construction. 2 5. Shape of the blending surface The branching blend of the cylinders Qa and Qb generated by Construction 2 has two planes of symmetry x1 = 0 and x2 = 0. Consider the quarter of the blend parametrized by X(t, u) with a parameter domain 0 t, u 1. Straightforward calculations show that vertical boundary curves X(0, u) and X(1, u) are circles. Hence the blending surface is tangent to four spheres along circles, and these spheres are touching the both cylinders as it is illustrated in Fig. 6. Moreover, these properties are preserved under Laguerre transformations. The surface X(t, u) behaves very much like a blend generated by a rolling ball that traces a circle Ca on the cylinder Qa . Such a rolling ball blend is irrational (see Example 1). Therefore, Construction 2 and its Laguerre transformations can be treated as a “rational PN approximations” of the corresponding rolling ball blends. The symmetric Construction 2 is unique for given data ra , rb and h in the sense that the first two steps (Gaussian map and dual representation) are unique of degree (4, 2). Nevertheless one can introduce additional degree of freedom by destroying a symmetry with respect to a plane x2 = 0. Let a modified Gaussian map be defined as N (t, u) = PS (Λ (t)(1 − u)ca (t) + ucb (t)), where ca (t) = m(1 − t)ca (−1) + (1 + t)ca (1) /2, Λ (t) = m−1 (1 − t)Λ(−1) + (1 + t)Λ(1) /2. This is essentially a projective reparametrization of the circle Ca . The rest of the construction and the resulting degree (6, 3) will be the same but the symmetry with respect to x2 = 0 will not be preserved when m = 1. This additional parameter m can be useful in asymmetric situations. For example, by changing m one can improve shape of a blend in Fig. 3(right) as it is shown in Fig. 7.
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Fig. 7. Shape dependence on parameter values m = 1, 1.5, 2.
6. Generalization to Dupin cyclides The usual inversion in R3 can be naturally extended to the Minkowski space R4 by the formula: Invp (x) = p + (x − p)/x − p, x − pM ,
(6)
where p ∈ R4 is the center of the inversion. Lemma 6. Any inversion Invp , p ∈ R4 , transforms PN surfaces to PN surfaces. Proof. By translations in any R3 direction and offsetting the point p can be moved to the origin, and the problem is reduced to the classical inversion Inv(x) = x/x2 in R3 . Let f = f (t, u) be a rational parametrization of a PN surface. Hence f 2 is rational. Then partial derivatives of F = Inv(f ) = f/f 2 can be described in the form Ft = Prf (ft )/f 2 ,
Fu = Prf (fu )/f 2 ,
where Pra (x) = x − 2(a, x)/a2 )a is a reflection in R3 . It is clear that Pra preserves cross products and distances, therefore Ft × Fu = ft × fu /f 4 is rational.
2
In order to generalize Construction 2 we need two properties of Dupin cyclides (see e.g. Degen, 2002): • a Dupin cyclide is uniquely defined by its three inscribed/circumscribed spheres; • an inversion of a circular cylinder/cone is a Dupin cyclide. Construction 7. Let Q be a cylinder/cone with a fixed circle C on it and let D be a Dupin cyclide. Then one can take any sphere S inscribed in D and find the other Dupin cyclide D , which contains S (as inscribed sphere) and is tangent to Q along the circle C. If S corresponds to a point s ∈ R4 then we apply an inversion Invs to D and D and get two cylinders/cones Invs (D) and Invs (D ) which can be blended using a suitable branching blend X according to Theorem 4 (if the required conditions are satisfied). Then we apply the inversion Invs again and get a certain blending Invs (X) between Q and D. Construction 7 is illustrated in Fig. 8, where S is a point, i.e. a sphere with zero radius. In Fig. 9 an inversion with a center in a sphere with the non-zero radius (on the left) is shown, where a blend of two cylinders of the same radii are transformed to a torus/cylinder blend (on the right). Theorem 8. Construction 7 produces a blending PN surface Y = Invp X of degree (8, 4) between a cylinder/cone Q with a fixed circle on it and a Dupin cyclide D. The parametrization degrees are stable under Laguerre transformations. Proof. In general case when the inversion center p is not incident with a curve ρ there is a simple formula for degree calculation deg Invp (ρ) = 2 deg ρ − #(ρ ∩ Ω),
(7)
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Fig. 8. Blending construction between a cylinder and a Dupin cyclide.
Fig. 9. An inversion generates blending between a cylinder and a torus.
where #(ρ ∩ Ω) is a number of intersection points with the absolute quadric Ω. Cubic isoparametric curves ρt on X are hyperplanar sections of ruled surfaces Rt that intersect Ω along conics. Similarly sextic isoparametric curves ρu on X are hyperplanar sections of ruled surfaces Ru that intersect Ω by quartic curves. Hence, from Eq. (7) follows deg Invp (ρt ) = 2 · 3 − 2 = 4 and deg Invp (ρu ) = 2 · 6 − 4 = 8. 2 7. Conclusions We proposed a new PN surface construction for branching blend of two circular cones and cylinders in quite general positions relatively to each other. The construction is Laguerre invariant (in particular offset-stable) and has the minimal possible parametrization degree (6, 3). The construction can be generalized to the blending of a circular cone/cylinder and a Dupin cyclide of degree (8, 4). These results show that the dual approach to PN surfaces is a natural one which can generate blendings of reasonable degree. Future research will be devoted to Bézier representation, shape analysis and singularity structure of the introduced branching blend. Also it will be interesting to extend the proposed construction to other cylinders/cones positions that are not presented in Theorem 4. References Boehm, W., 1990. On cyclides in geometric modeling. Computer Aided Geometric Design 7, 243–255. Cox, D., Krasauskas, R., Musta¸taˇ , M., 2003. Universal rational parametrizations and toric varieties. Topics in Algebraic Geometry and Geometric Modeling, Contemporary Mathematics, vol. 334, pp. 241–265. Degen, W., 2002. Cyclides. In: Farin, G., Hoschek, J., Kim, M.-S. (Eds.), Handbook of Computer Aided Geometric Design. Elsevier Science, pp. 575–602. Farouki, R.T., Pottmann, H., 1996. Polynomial and rational Pythagorean-hodograph curves reconciled. In: Mullineux, G. (Ed.), The Mathematics of Surfaces VI. Oxford Univ. Press, pp. 355–378. Jüttler, B., Sampoli, M.L., 2000. Hermite interpolation by piecewise polynomial surfaces with rational offsets. Computer Aided Geometric Design 17, 361–385. Kazakeviˇci¯ut˙e, M., 2005a. Blending of natural quadrics with rational canal surfaces. PhD Thesis, Vilnius University. Kazakeviˇci¯ut˙e, M., 2005b. Classification of pairs of natural quadrics from the point of view of Laguerre geometry. Lithuanian Mathematical Journal 45, 63–84. Krasauskas, R., 2006. Bézier patches on almost toric surfaces. In: Elkadi, M., Mourrain, B., Piene, R. (Eds.), Algebraic Geometry and Geometric Modeling. In: Mathematics and Visualization Series. Springer, pp. 135–150. Krasauskas, R., 2007. Minimal rational parametrizations of canal surfaces. Computing 79, 281–290.
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Krasauskas, R., Kazakeviˇci¯ut˙e, M., 2005. Universal rational parametrizations and spline curves on toric surfaces. In: Computational Methods for Algebraic Spline Surfaces, ESF Exploratory Workshop. Springer, pp. 213–231. Krasauskas, R., Zub˙e, S., 2007. Canal surfaces defined by quadratic families of spheres. In: Proceedings of the COMPASS II Workshop, pp. 138– 150. Peternell, M., 1997. Rational parametrizations for envelopes of quadric families. Ph.D. Thesis, Institute of Geometry, University of Technology, Vienna. Peternell, M., Pottmann, H., 1997. Computing rational parametrizations of canal surfaces. Journal of Symbolic Computation 23, 255–266. Pottmann, H., 1993. Rational curves and surfaces with rational offsets. Computer Aided Geometric Design 12, 175–192. Pottmann, H., Peternell, M., 1998. Application of Laguerre geometry in CAGD. Computer Aided Geometric Design 15, 165–186. Pratt, M.J., 1990. Cyclides in computer aided geometric design. Computer Aided Geometric Design 7, 221–242. Rossignac, J.R., 1987. Constraints in constructive solid geometry. In: Proceedings of the 1986 Workshop on Interactive 3D Graphics. Chapel Hill, North Carolina, 1987, pp. 93–110. Shene, C.-K., 1998. Blending two cones with Dupin cyclides. Computer Aided Geometric Design 15, 643–673.