cyclides in pure blending i

COMPUTER AIDED

GEOMETRIC DESIGN

ELSEVIER Computer Aided Geometric Design 14 (1997) 51-75

Cyclides in pure blending I

S e t h A l l e n a,l, D e b a s i s h D u t t a b,*

a Department of Mathematics University of Michigan, Ann Arbor, MI 48109, USA Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, M1 48109, USA

Received August 1995; revised March 1996

Abstract

In this paper, we study blends between natural quadrics using both Dupin ring cyclides and parabolic cyclides. We present a new definition of a pure cyclide blend that we believe will help ensure the blend surface is useful to a designer. This definition will force the construction of nonsingular cyclide blends, deliberately excluding cyclide joins and singular surfaces. Further, we study the implications of the cyclide blend definition on the position of the quadrics in a blend and on the properties of the blending cyclide's extreme circles. This is the first part of a two part paper.

Keywords: Dupin cyclide; Blending surface; Shape design

1. Introduct ion

1.1. Background

Blends are important in design both for functional reasons, i.e., reducing stress and simplifying manufacture, and for cosmetic reasons. Much has been written about blends. See (Vida et al., 1994) for a survey of parametric blending techniques and its excellent list of references on both parametric and algebraic blending.

In this paper we examine cyclide blends between the natural quadrics (the plane, sphere, cylinder, and cone). The natural quadrics are an important subset of the primitives used in most CSG systems. Unless otherwise indicated, a cyclide is a Dupin cyclide. This is a degree four surface with circular lines of curvature. The cyclide has been studied by Dupin (1822), Maxwell (1868), and Cayley (1873). A modern treatment of this surface's

* Corresponding author. E-mail: [email protected]. l E-mail:swallen @ math.lsa.umich.edu.

0167-8396/97/$17.00 Copyright © 1997 Published by Elsevier Science B.V. All rights reserved SSDI 0167-8396(96)00021-0

52 S. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75

geometry is presented in (Chandru et al., 1989). Srinivas and Dutta (1995) present tools to help implement cyclides in a CAD/CAM system. The degenerate forms of the cyclide include the parabolic cyclide and the natural quadrics.

The use of cyclides as blending surfaces for natural quadrics has been examined by Pratt (1990), by Srinivas and Dutta (1994), among others. A general method and specific constructions for several cases of natural quadric blends are given in Pratt (1990, 1995). Additional constructions can be found in Srinivas and Dutta (1994), including the blend of a cylinder and a sphere with a parabolic cyclide. Shene (1992) and Johnstone and Shene (1994) examine the cone/cone case in detail, and offer constructive proofs of if and only if conditions for the existence of such blends. If and only if conditions for the existence of cone/cone blends are stated (and in one direction proved) in the case of nonparallel cone axes in (Pratt, 1990). Pratt also shows how this result could be generalized to cylinder/cone and cylinder/cylinder blends.

Also of interest is the use of cyclide patches as blending surfaces. Martin (1982) and DePont (1984) develop the theory of principal cyclide patches. Another good reference on cyclide patches is (Nutbourne and Martin, 1988).

1.2. Overview

Typically a blending surface between two arbitrary surfaces is defined as any transition surface that is tangent to each surface along a curve. Cyclide transition surfaces often contain singularities, fail to follow the intersection curve of the objects being connected (when the objects intersect), lie both inside and outside the objects, etc. Since a designer usually imagines a blend surface as a rolling ball blend or as putty pressed along the intersection curve of the objects being blended, the definition of a cyclide transition surface as a blend does not capture the designer's intent. In Section 2, we examine the characteristics o f a blend surface and find the conditions that ensure a cyclide transition surface possesses these characteristics. Using these conditions, we propose a new definition of a cyclide blend that helps the designer specify the surface he intends.

We consider blends with parabolic cyclides in Section 3. We carry out some of the development in Section 2 for parabolic cyclides to specify a parabolic cyclide blend definition and to verify that this definition satisfies our requirements.

Section 4 examines the properties of the extreme circles that a cyclide blending surface possesses. The results in this section will be used in the second part of this paper to construct cyclide blends. Further, in Section 5, we develop analogous results for parabolic cyclide blends.

The position of the quadrics being blended is the subject of Section 4. The result that we get here will also be used in the second part of this paper.

Finally, Section 6 contains our conclusions. This is a first of a two part paper. In the second part (Allen and Dutta, 1996), we apply

our definition of a cyclide blend to get if and only if conditions (with proofs) for the existence of cyclide blends between any two natural quadrics (ten cases total). We also show how the common inscribed sphere condition relates to our definition of a blend. Further, the second part contains simple, easily implementable tests for when two natural

S. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75

B

53

Fig. 1. A) The latitudinal lines of curvature (latitudinal circles) on a cyclide. B) The longitudinal lines of curvature.

quadrics can be blended by a cyclide. Again the development is carried out for both ring and parabolic cyclides.

2. The cyclide blend

2.1. Preliminaries

In this paper, a cone is not the double cone one gets from the algebraic equation of a cone, but a single cone. So the axis of a cone is a ray with its endpoint at the vertex of the cone.

The following facts and definitions will be useful in the proofs that follow. For more detailed information, see (Chandru et al., 1989; Pratt, 1990; Srinivas and Dutta, 1995).

• Normal offsets of a cyclide are cyclides. • All lines of curvature on the cyclide are circles. • A cyclide is the envelope of two families of spheres, so a cyclide is a canal surface.

The characteristic circles from each family are its lines of curvature. • A cyclide has two perpendicular planes of symmetry. The centers of the spheres of

the families that envelop the cyclide lie in these planes along either an ellipse or a hyperbola depending on the plane. These planes are referred to as the plane of the hyperbola and the plane of the ellipse.

• The lines of curvature that come from the family of spheres with their centers on the hyperbola are referred to as latitudinal circles. The circles coming from the family with their centers on the ellipse are longitudinal circles. See Fig. 1.

• The intersection of each plane of symmetry with the cyclide contains two special circles, called extreme circles. The extreme circles in the plane of the ellipse are the largest and smallest latitudinal circles. Similarly, the extreme circles in the plane of the hyperbola are the largest and smallest longitudinal circles.

• All the planes of the circles from one family intersect in a line called a line o f

similitude. • Suppose we have two circles in a plane with centers cz and c2 and radii rl and r2.

The centers o f similitude of the circles are the two points lying on the line through c] and c2 that satisfy the following equation:

rl _ dist(p,c])

r2 dist(p, c2)


where dist( , ) is the distance function and p is a point on the line through c I and c2. See (Chandru et al., 1989; Shene, 1992).

• The line of similitude for the latitudinal circles lies in the plane of the ellipse, is perpendicular to the plane of the hyperbola, and contains the outer center of similitude of the extreme circles in the plane of the hyperbola.

• The line of similitude for the longitudinal circles lies in the plane of the hyperbola, is perpendicular to the plane of the ellipse, and contains the inner center of similitude of the extreme circles in the plane of the ellipse.

The main tool we use to classify cyclide blends is the following classical theorem.

Theorem 2.1. When a cyclide and a quadric are tangent along their curve of intersection, this curve is a line of curvature of the cyclide.

Proof. See (Jessop, 1916). t2

We make the following remarks which will be useful for much of what follows. These results can be found in (DoCarmo, 1976; Nutbourne and Martin, 1988).

Remark 2.2. Every curve on a plane (respectively sphere) is a line of curvature of the plane (respectively sphere).

Remark 2.3. The lines of curvature of a cone or cylinder Q are lines and circles. The circular lines of curvature are the circles found by intersecting planes perpendicular to the axis of Q with Q. The linear lines of curvature are found by intersecting Q with planes containing Q's axis.

Theorem 2.4 (Joachimstahl's Theorem). Suppose two nonsingular surfaces intersect in a curve L, and L is a line of curvature of one of the surfaces. Then the angle between the two surfaces is constant if and only if L is a line of curvature of the other surface.

The angle between two surfaces that intersect tangentially is zero or 180 degrees along their curve of intersection. So by Joachimstahl's Theorem and Theorem 2.1, we know that the curve of intersection between a cyclide and a tangent natural quadric must be a line of curvature of both objects. Since the longitudinal and the latitudinal lines of curvature of a cyclide are circles, Theorem 2.1 reduces the problem of blending by cyclides to finding conditions under which a cyclide and a natural quadric intersect tangentially in a circular line of curvature of both surfaces. Given Remarks 2.2 and 2.3, we are well on our way to understanding the geometry of a quadric/quadric blend.

2.2. The blend definition

When a designer specifies a blend between two surfaces, typically he imagines pressing putty along the curve of intersection of the surfaces, so the blend resembles a rolling ball blend. First, notice that this blending surface must be between two intersecting surfaces (each of these surfaces could be a composite of many different surface patches), and that

s. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75 55

it will be nonsingular when the surfaces it is tangent to are nonsingular. Second, this surface must follow the intersection curve of the surfaces being blended. So the blend surface should have roughly the same shape as the curve of intersection. Finally notice that the blend surface lies entirely inside or entirely outside each of the surfaces being blended. Here the blend is achieved by adding material only; not by adding material in one region and removing material in another. So a blend surface has three important characteristics:

(1) the blend surface is nonsingular when the surfaces being blended are nonsingular, (2) the blend surface follows the curve of intersection between the surfaces being

blended, and (3) the blend surface stays on one side of each of the surfaces being blended.

Our goal is to specify a blend definition for cyclides that ensures a cyclide blend has these three properties. Note, such a blend definition forces only pure blends, and excludes from consideration joins. A join is any nonsingular transition surface either between two non-intersecting surfaces, or that lacks properties (2) and/or (3) above. Since the cyclide tube shown in Fig. 7 on p. 60 lies both inside and outside the surfaces being connected and hence does not satisfy property (3) above, it is a join. It is not the authors' intention to say that such a surface is not useful, but only that such a surface is better characterized as a join. I For our classification of transition surfaces T connecting primary surfaces P and Q, see Fig. 2.

We begin by finding a condition that ensures that the cyclide blending surface lies entirely inside or entirely outside each of the surfaces being blended. The following lemma and proposition will help us formulate a blend definition that forces the cyclide to have this property.

We will need the following two remarks in the proof of Lemma 2.7.

Remark 2.5, Suppose a nonsingular cyclide C and an axial natural quadric Q are tangent along a line of curvature L. Let Ne be the outward pointing normal to C and Nq be the outward pointing normal to Q. Then along L, either N~ and Nq point in the same direction or they point in opposite directions. Further by Joachimstahl's Theorem, the sign of the dot product of Nc and Nq cannot change along L, since there are no singular points on C.

Remark 2.6, Suppose C is a nonsingular cyclide. Then given any sphere 6: from the family F of spheres enveloping C whose centers lie on C's ellipse, there is another sphere So from F so that 6:0 does not intersect 6:.

Proof. Let P~ be the plane of the ellipse, so in P~ are two extreme circles cl and c 2 where c2 contains cl. Then let L be a line through the center of Cl that does not intersect S. Pick any sphere So in F that lies entirely on the other side of L from 6: in P~, i.e., pick So tangent to L. []

! Since cyclide joins are important surfaces in geometric modeling, a subsequent paper will address cyclide joins and examine conditions for the existence of joins in each of the natural quadric/natural quadric cases.


Transition Surface T connecting P and Q

Nonsingular T Singular T

Non-intersecting Intersecting P and Q P and Q

Tdoes not follow Tfollows intersection intersection

/

,,/ T lies on both sides / of either P or Q

,J

fJ ,"

Tis a join

T lies on one side of P and of Q

Tis a blend

Fig. 2. The classification of transition surfaces.

L e m m a 2.7. Suppose an axial natural quadric (i.e., a cone or a cylinder) is tangent to a nonsingular cyclide along a curve. Then the cyclide lies either entirely inside or entirely outside the quadric if and only if the cyclide and the quadric intersect along a latitudinal line of curvature of the cyclide.

Proof. First suppose a cyclide C and an axial natural quadric Q are tangent along a longitudinal line of curvature L. Consider C 's extreme circles cl and c2 in the plane of the ellipse Pc where cl is contained inside c2. Suppose L intersects ci in the point qi. Then in Pc (using Corollary 5.5), Q looks like two lines, where one line is tangent to cl at ql and the other line is tangent to c2 at q2. See Fig. 3. The line from Q's intersection with Pc tangent to c2 lies entirely outside C, while the line tangent to cl lies inside C near the point ql. So near L, Q lies both inside and outside C.

Now suppose a cyclide C and an axial natural quadric Q are tangent along a latitudinal line of curvature L. Let S be the sphere in the family of spheres enveloping Q that contributes L. Let F be the family of spheres enveloping C with their centers on the ellipse. Each sphere in F contributes a longitudinal line of curvature, so each sphere in F is tangent to Q, and hence to S, at a point on L (where L and the longitudinal line of curvature intersect). Let So be any sphere in F. Since S and So are tangent some point p, we have one of the following three conditions:

(1) neither S nor So is contained inside the other, (2) S is contained inside 50, or (3) So is contained inside S.

See Fig. 4.

S. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75 57

C2 ~q2 / / '~i

' x/ 6'-

'\ ci

Fig. 3. This figure is drawn in the plane Pc of the cyclide C's ellipse. The axial natural quadric Q is tangent to C along the longitudinal line of curvature L (shown here as a dotted line).

i , , , ,, , / \ , , , \

/ \ ,/ , ,\ / \ \ \ / \ / '\ / f '\ j - - \ ' ,

I

/ / 1) 2) ~ 3)/

Fig. 4. This figure is drawn in the plane that contains Q's axis and p.

First consider case (1). Suppose neither S nor So lies inside the other. Then the normals to C and Q point in opposite directions at p. Since So lies on one side of the tangent plane at p and Q lies on the other, So lies entirely outside Q. Since in Case (1) the normals to Q and to C point in opposite directions and in Cases (2) and (3) the normals point in the same directions, we see using Remark 2.5 that every sphere in F must lie entirely outside Q. So C lies entirely outside Q.

Now consider Case (2). If S is contained inside So, then L is contained inside So. So every sphere in F intersects So. By Remark 2.6, this cannot happen. So Case (2) does not occur.

Now consider Case (3). So is contained inside S, so the normals to C and Q point in the same direction at p. By Remark 2.5, and the fact that Case (2) does not occur, we get that all spheres from F must lie inside S. So C lies inside Q. []

From the proof of Lemma 2.7, we see that when a horned cyclide is tangent to an axial natural quadric along a longitudinal line of curvature, the cyclide still lies both inside and outside the quadric. Further, since all latitudinal lines of curvature pass through the singular points on the horned cyclide, we get:

Remark 2.8. A horned cyclide can never be a blend surface.

58 s. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75

Situations might occur where subsequent boolean operations remove the singularities from a singular surface that will act as a blend surface in the final part. Remark 2.8 does not hold in these cases.

A spindle cyclide can in some instances act as a blend surface, as long as the cyclide's singular points are not part of the blend. However, as we see in the second part of this paper, it is always possible to find a nonsingular cyclide that can just as easily act as the blend surface. Therefore, we concentrate on nonsingular cyclides. So for the remainder of this paper, cyclide and parabolic cyclide always refer to nonsingular cyclides and nonsingular parabolic cyclides respectively, unless otherwise noted.

Notice that when a sphere is tangent to a cyclide along a curve, it must be a member of one of the two families of spheres defining the cyclide, and so must be contained either wholly inside or wholly outside the cyclide. Since a plane is a special case of a sphere (one with infinite radius), using the above together with Lemma 2.7, we get:

Proposi t ion 2.9. Suppose a cyclide and a natural quadric are tangent along a latitudinal line o f curvature. Then the cyclide lies either entirely inside or entirely outside the

quadric.

Given Proposition 2.9 and our intuition of a blend, we restrict blends to be along latitudinal lines of curvature of the blending cyclide. At first this may seem to place an unnecessary restriction on the blends involving planes and spheres, but this is not so. First note that when a plane and a cyclide are tangent along a curve, that curve must be a latitudinal line of curvature of the cyclide, see Corollary 5.3. Second note that for a cyclide blend to make sense, both lines of curvature of the blend must be from the same family. So any blends involving a plane, cone, or cylinder, must be along latitudinal lines of curvature. This leaves only the case of the sphere/sphere blend where the spirit of an engineering blend is not compromised by blending along longitudinal lines of curvature. But here the restriction is no loss, since a sphere/sphere blend along latitudinal lines of curvature is always possible, as we see in Theorem 3.8 in the second part of this paper (Allen and Dutta, 1996).

Since we want the blend to follow the intersection curve, we impose the following additional restriction on the quadrics being blended. We require the quadrics to intersect in a curve that the blending cyclide can "cover", again thinking of forming the blend surface by pressing putty along the intersection curve of the quadrics. So the two quadrics must intersect in a closed curve, and when either of the quadrics is an axial natural quadric, the curve of intersection must wrap around its axis. This second condition follows from the fact that the tangent intersection of a cyclide and an axial natural quadric is a circular line of curvature of the quadric through which the axis of the quadric passes.

If the intersection of a plane with an axial natural quadric is a closed curve (an ellipse), then the curve wraps around the axis of the quadric. Later we will show why the intersection curve of two axial natural quadrics that can be blended automatically wraps around the axis of each quadric. So only in the case of a sphere and an axial natural quadric do we need to specifically verify that the intersection curve wraps around the axis of the quadric. In Fig. 5A, we see a single ring cyclide with its upper portion removed


/ / 13)/

.......

\, \ \

Fig. 5. A) A cyclide transition surface between a cone and a sphere that does not satisfy condition (3) of Definition 2.10. B) A blend surface between the quadrics that is not possible using cyclides.

smoothly connecting a cone and a sphere along latitudinal lines of curvature, where the intersection curve of the quadrics does not wrap around the axis of the cone. Therefore, the cyclide surface smoothly connecting the quadrics, although perhaps useful, is not a blend surface. Further notice that in this case, every cyclide transition surface between the cone and the sphere resembles the surface shown, in that the cyclide must be tangent to the cone along a circular line of curvature. When a designer envisions a blend surface between a cone and a sphere in this position, he imagines the blend surface on the side of the cone, following the intersection curve of the quadrics. Fig. 5B shows what a designer would typically mean by a blend surface here, although this surface is not possible using cyclides. Fig. 5 demonstrates why an extra condition (beyond specifying that the cyclide is tangent to the quadrics along latitudinal circles) will be necessary to ensure that the cyclide surface follows the intersection curve in the sphere/axial natural quadric blending case.

Theorem 2.1, Proposition 2.9, and the previous discussion lead us to the following definition of cyclide blends. We believe this definition captures the essence of what a designer intends for a blend. For completeness, we also define a cyclide join.

Definition 2.10 (Cyclide blend). A nonsingular cyclide blends two intersecting quadrics when

(1) the curve of the intersection of the two quadrics that is being blended is nonempty and closed,

(2) the cyclide is tangent to each quadric along a latitudinal line of curvature, and (3) the intersection curve of the two quadrics must wrap around the axis of each axial

natural quadric being blended. See Fig. 6.

We make the following further distinction between interior, exterior, and mixed blends.

Definition 2.11 (Interior/exterior/mixed cyclide blend). A cyclide blend is an interior (respectively exterior) blend when the cyclide is contained on the inside (respectively outside) of both quadrics it blends. A mixed cyclide blend is one in which the cyclide lies inside one quadric and outside the other.


i

Fig. 6. A cyclide blend between a cylinder and a cone.

/ /

/ / /

/ ,

, / /

Fig. 7. A cyclide join between a cylinder and a cone.

Note, when one of the quadrics being blended is a plane, we will refer to the halfspace in the direction of the plane's normal as the outside of the plane, following the convention of outward pointing normals.

D e f i n i t i o n 2.12 (Cyclide join). A c y c l i d e j o i n s two quadrics when either • the cyclide is tangent to both quadrics along longitudinal lines of curvature, or • the cyclide is tangent to both quadrics along latitudinal lines of curvature and either

- the quadrics do not intersect, or - the quadrics' intersection curve does not wrap around the axis of the axial natural

quadric being joined. See Fig. 7.

3. P a r a b o l i c c y c l i d e b l e n d s

In this section we examine the role of nonsingular parabolic cyclides, which we refer to simply as parabolic cyclides, in the blending of natural quadrics. In particular, this

S. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75

• ; - ~ . -

r . . . . . ~ . .j

A ) ' . . . . . , . . . . . i . ' - ' . ' . ' . I

6 1

, . . . . J , .

B) '.:.:

Fig. 8. The transformation of a ring cyclide into a parabolic cyclide. A) The extreme circles in the plane of the ellipse. B) The extreme circles in the plane of the hyperbola. The shaded region lies inside the cyclide.

section includes a formal definition of a parabolic cyclide blend. Much of what was done in the previous section can be easily extended to parabolic cyclides with little additional work. To see why this is true, we need to understand how a ring cyclide transforms into a parabolic cyclide.

Consider one of the extreme circles in the plane of the hyperbola of a ring cyclide. Let the radius of this circle go to infinity as the circle itself is moved so that its distance from the other circle remains constant. See Fig. 8. The hyperbola will become a parabola as the circle becomes a line. Notice that the radius of the larger extreme circle in the plane of the ellipse must also go to infinity, since this circle must be large enough to enclose both extreme circles in the plane of the hyperbola. The ellipse also becomes a parabola as the large circle becomes a line. So a parabolic cyclide has two lines of curvature that are lines, one in each plane of symmetry. These lines are referred to as extreme lines since they are deformations of extreme circles in a ring cyclide. All the remaining lines of curvature are circles. Also note that the parabolic cyclide is the envelope of two disjoint families of spheres where:

(1) now the centers of the spheres lie on parabolas, and (2) every sphere from one of the families lies entirely outside the parabolic cyclide,

and every sphere from the other family lies inside the cyclide. With these differences between the ring and parabolic cyclides in mind, we will be able to easily outline the development in the earlier sections for parabolic cyclides.

Theorem 2.1 remains true for parabolic cyclides and will again be an important tool in the proofs that follow. The only new complication is that an axial natural quadric can be tangent to a parabolic cyclide along a line. This case is handled in the definition of a parabolic cyclide blend by insisting that the parabolic cyclide is tangent to each object being blended along a closed line of curvature.

Now we introduce our definition of parabolic blending. See Fig. 9 for an example of a parabolic blend between a cone and cylinder. Notice that the axes of the cone and the

62 S. Allen, D. Dutta Computer Aided Geometric Design 14 (1997) 51-75

Fig. 9. A parabolic cone/cylinder blend shown in one of the planes of symmetry.

cylinder intersect and that the cone and the cylinder are tangent along an extreme line of the parabolic cyclide.

Definition 3.1 (Parabolic cyclide blend). A parabolic cyclide blends two intersecting quadrics when

(1) the curve of the intersection of the two quadrics that is being blended is nonempty and closed,

(2) the cyclide is tangent to each quadric along a closed curve, and (3) the intersection curve of the two quadrics must wrap around the axis of each axial

natural quadric being blended. See Fig. 9.

Blending with parabolic cyclides is somewhat simpler than blending with ring cyclides in that we do not need to check that the cyclide and quadric are tangent along latitudinal lines of curvature. When blending with ring cyclides, it was important to insist that the cyclide and the axial natural quadric being blended where tangent along a latitudinal line of curvature, since otherwise the quadric would lie both inside and outside the cyclide. Proposition 3.2 indicates why the family from which the line of curvature comes in a parabolic cyclide blend does not matter. It also shows us that only exterior parabolic cyclide blends are possible.

Proposition 3.2. Suppose a parabolic cyclide and a natural quadric are tangent along a closed line of curvature. Then the parabolic cyclide always lies entirely outside the quadric..

Proof. Call the quadric Q. Corollary 5.7 shows us that Q cannot be a plane. Using Corollary 5.8, we see that when Q is a sphere, the proposition is true. So we may assume that Q is an axial natural quadric.

By Corollary 5.9, we let P be the plane of symmetry of the parabolic cyclide C containing the axis of Q. In P are the extreme line l and the extreme circle c. Q and C are tangent along a circular line of curvature that intersects l and c in the points q0 and ql respectively. See Fig. 10.


Q I

I<S p~ q ()i"

C

r

Fig. 10. An axial natural quadric Q and a parabolic cyclide's extreme circle c and extreme line l shown in the plane of symmetry containing Q's axis. The shaded region lies outside the cyclide.

Now we show that c must lie outside Q. To see this consider the outward pointing normals to C at q0 and ql. Assume P is the plane of symmetry that comes from the plane of the ellipse. So the interior of C lies in the half plane of P containing c and outside it. So the outward pointing normal to C at q0 points away from c and the outward pointing normal at ql points into c. Now suppose c lies inside Q. Then the outward pointing normals to Q and C at q0 point in the same direction, and the outward pointing normals at ql point in opposite directions. Since C is has no singular points, this contradicts Joachimstahl's Theorem. So c must lie outside Q. When P is the other plane of symmetry, a similar proof shows the same result, except that the direction of the outward pointing normals to C are reversed.

Now let p be any point on l and on Q. Let Sq be the sphere from the family that envelops Q and that is tangent to I at p. Let Sc be the sphere that is tangent to l at p and that is from the family that envelops C with their centers on P. Both Sq and Sc have their centers on P, and hence are tangent at p. Since c lies outside Q and Sc is tangent to c, S~ contains Sq. So Q either lies entirely outside C or entirely inside C depending on whether S~ is from the family of spheres lying inside C or lying outside C. []

4. The extreme circles

In the second part of this paper, we prove if and only if conditions for the existence of cyclide blends in each of the ten natural quadric/natural quadrics cases. The construction in each proof depends on picking circles to serve as the blending cyclide's extreme circles in the plane of the hyperbola. In this section, we examine the necessary conditions on these circles when the cyclide that the circles determine is a blending cyclide. Then we use these conditions to construct cyclides that are tangent to a given quadric.


4.1. The ring cyclide's extreme circles

The following lemma and remark indicate the conditions that must be satisfied by the extreme circles of the blending cyclide. Using Lemma 4.1, we will be able to show that the cyclide determined by the circles we pick is unique and nonsingular. Remark 4.2 illustrates the necessity of two further conditions on the extreme circles.

L e m m a 4.1. Suppose we are given a plane Ph and two circles Cl and c2 lying in Ph. The circles el and c2 determine a nonsingular cyclide with cl and c2 as the cyclide's extreme circles lying in the plane o f the hyperbola i f and only i f

(1) cl and c2 do not intersect, and (2) neither cl nor e2 is contained inside the other.

The cyclide determined is unique.

Proof. Let ri be the radius of ci and assume without loss of generality that r2 ~> rl > 0. Recall, the Dupin cyclide has three parameters a, c, and m that control its shape,

where a 2 > c 2. The parameter b is given by b = v / ~ - c 2. In its standard position, the cyclide is represented algebraically by

(X 2 + y2 + Z 2 q_ b 2 _ m2)2 = 4(ax -- cm) 2 + 452y 2, (1)

or equivalently by

(x 2 + y2 + z 2 __ b 2 _ m2)2 = 4(cx - am) 2 - 452z 2. (2)

Since changing the sign of a, b, c, or m at most reflects the cyclide through the plane x = 0, we take these constants to be positive (de Pont, 1984). Recall that C is a horned cyclide if and only if c >~ m, and C is a spindle cyclide if and only if m ~> a. Fig. 11 shows the geometric relationship between the cyclide parameters and its extreme circles in the plane of the hyperbola for a cyclide given by either Eq. (1) or (2). For more information, see (Pratt, 1990; Chandru et al., 1989).

To see that the cyclide is uniquely determined, we solve for the cyclide parameters. The parameter a is half the distance from the center of cl to the center of c2. Recall that r2 = m + c and rl = m - c. So parameter m = (rl + r2)/2, and parameter c = (~': - " 1 ) / 2 -

D e f i n e the following coordinate system in Ph, Let the x-axis lie along the vector from the center of c2 to the center of cl, so that the center of c2 is at the point ( - a , 0). Now let ql be the point on c2 nearest the center of cl. Let q2 be the point on cl nearest the center of c2, and let q3 be the point on cl furthest from the center of c2. So

ql = ((rn + c) -- a, 0), q2 = (a -- ( m -- c), 0), and q3 = (a + (m - c), 0).

See Fig. 11. First we show that C is well-defined if and only if c2 does not contain el. This is the

case, since C is well-defined when a > c and since c2 does not contain cl, if and only if, q3 lies to the right of ql, which is true, if and only if,

( m + c ) - a < ( m - c ) + a .

S. Allen, D. Dutta /Computer Aided Geometric Design 14 (1997) 51-75

2 , /" /; t

m+c

-a T

O1 4 - m-c

"¢ O q ~ a , q3

65

Fig. 11. This figure shows the two circles cl and c2 in the plane Ph as well as the geometric meaning the cyclide parameters.

Since rl > 0, we get m > c and so C is not a horned cyclide. Finally notice that a > m if and only if c2 and cl do not intersect, if and only if, q2

lies to the right of ql, if and only if,

( . ~ + c ) - a < a - ( . ~ - c ) .

So C is not a spindle cyclide if and only if c2 and cl do not intersect. []

R e m a r k 4.2. Suppose a cyclide C is tangent to a quadric Q along a latitudinal line of curvature. Let cl and c2 be the extreme circles of C in the plane of the cyclide's hyperbola Ph. Then

(1) both Cl and c2 are tangent to the cross section of Q in Ph, and (2) both Cl and c2 either lie entirely inside or entirely outside Q.

Proof. On a cyclide, every latitudinal line of curvature intersects every longitudinal line of curvature. So cl and c2 intersect the latitudinal line of curvature along which C is tangent to Q. At the point of intersection of ci and this line of curvature, c~ must be tangent to Q. This proves condition (1). Given Proposition 2.9, we see that C must be either entirely inside Q or entirely outside Q. Condition (2) follows. []

Given the previous lemma and remark, we are able to determine a set of conditions that must be satisfied by the extreme circles, lying in the plane of the hyperbola, for a blend. These conditions are summarized in the following definition.

Definition 4.3 (Extreme circles conditions). Suppose we are given a quadric Q and a plane Ph. Two circles cl and c2 that lie in the plane Ph satisfy the extreme circles conditions when

(1) cl and c2 do not intersect, (2) neither cl nor c2 is contained inside the other, (3) both Cl and c2 are tangent to the cross section of Q in Ph, and (4) both cl and c2 together lie either entirely inside or entirely outside Q.


/'

G~ ..... C

C

Fig. 12. This figure shows that if two circles are tangent to a third, then a line passing through the points of tangency contains a center of similitude of the two circles.

Now we present three lemmas that will be helpful in establishing that the cyclide determined by circles satisfying the extreme circles condition is tangent to the quadric in question. We will see that satisfying the extreme circles condition is sufficient for blends of planes and spheres, but that one additional condition must be satisfied for blends of axial natural quadrics.

Lemma 4.4. Suppose we are given two p lanes /9 and/gh such that /9 is perpendicular to/gh. Further suppose that in Ph are two circles cl and c2 satisfying the extreme circles conditions with P. Then the cyclide determined by cl and c2 is tangent to/9.

Proof. Let C be the cyclide determined by the circles. Suppose ci is tangent to P at the point qi. Then since the line qlq2 is tangent to both circles, it contains a center of similitude s of cl and c2 that lies outside the circles (not between). The line L through s perpendicular to Ph is the line of similitude of C in the plane of the ellipse. Since Ph is perpendicular to P, P contains L. So P intersects C in a line of curvature of C. P also intersects C in a line of curvature o f / 9 by Remark 2.2. So by Joachimstahl's Theorem, the angle between C and/9 is constant.

Since ci is an extreme circle of C, all normals to C along ci point out from c~'s center. Since /9 and /gh are perpendicular, the normal to /9 at the point qi points toward c~'s center. So /9 and C are tangent at the points ql and q2. Since the angle between C and /9 is constant, C and/9 must be tangent. []

We make use of the following fact proven by Shene (1992).

Remark 4.5. Given two distinct fixed circles C1 and C2 and a third circle C tangent

to C1 and C2 at points P1 and P2 respectively, the line/:'1 P2 passes through one of the centers of similitude S of C1 and C2. See Fig. 12.

Lemma 4.6. Suppose we are given a sphere S and a plane Ph such that Ph contains the center o f S. Further suppose that in /9h are two circles cl and c2 satisfying the extreme circles conditions with S. Then the cyclide determined by cl and c2 is tangent to S.

s. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75 67

Proof. Let C' be the cyclide determined by the circles. Suppose c~ is tangent to S at the •

point qi. Then using Remark 4.5, we see that the hne qlq2 contains a center of similitude s of Cl and c2 that lies outside the circles. Again the line L through s perpendicular to Pn is the line of similitude of (7 in the plane of the ellipse. Consider the plane P that

is perpendicular to Ph containing L and qlq2 • P intersects S in a circle whose diameter is given by the line segment qlq2, since Ph contains the center of S and is perpendicular to P. P intersects (7 in a line of curvature of (7, which is the same circle• By Remark 2.2, it is also a line of curvature of S. So C and S intersect in a shared line of curvature. Then by Joachimstahl's Theorem, the angle between (7 and S is constant.

All normals to (7 along c~ point out from ci's center. Since Ph contains the center of S, the normal to S at the point qi points toward ci's center. So S and (7 are tangent at the points ql and q2. Since the angle between (7 and S is constant, C and S must be tangent. []

Now consider a cyclide C that is tangent to an axial natural quadric Q along a latitudinal line of curvature. Let Cl and c2 be the extreme circles of C in the plane of the cyclide's hyperbola Ph, and suppose that ci intersects Q in the point qi. As we will see in Corollary 5.5, Ph contains the axis of Q. So since the circular lines of curvature of Q lie

¢ : - - . . +

in planes perpendicular to the axis of Q, we see that the line qlq2 must be perpendicular to the axis of Q. This leads us to the following strengthening of the extreme circles conditions for axial natural quadrics.

Definition 4.7 (Axial extreme circles conditions). Suppose we are given an axial natural quadric Q and a plane Ph containing the axis of Q. Two circles cl and c2 that lie in Ph satisfy the axial extreme circles conditions when

(1) cl and c2 satisfy the extreme circles conditions with Q, and + - - - - . + .

(2) if ci intersects Q in the point qi, then the line qlq2 is perpendicular to the axis of Q.

L e m m a 4.8. Suppose we are given an axial natural quadric Q and a plane Ph such that Ph contains the axis of Q. Further suppose that in Ph are two circles Cl and c2 satisfying the axial extreme circles conditions with Q. Then the cyclide determined by cl and c2 is tangent to Q.

Proof. Let C be the cyclide determined by the circles. Suppose ci is tangent to Q at

the point qi. Let P the be plane containing the line qlq2 that is perpendicular to the axis of Q, so P intersects Q in a line of curvature of Q and contains the points ql and q2. Consider Q as a canal circle. Then let S be the sphere from the family of spheres defining Q whose characteristic circle lies in the plane P. Note that since the circles cl and c2 satisfy the extreme circles condition with S, C is tangent to S along the circular line of curvature that lies in the plane P (Lemma 4.6). Since Q and S are tangent along this same curve, Q and C are also tangent• This completes the proof• []

Note, that by picking both the extreme circles to lie outside (respectively inside) the quadric, we ensure the cyclide will lie entirely outside (respectively inside) that quadric.

68 S. Allen, D. Dutta/ Computer Aided Geometric Design 14 (1997) 51-75

This allows the designer to easily pick the type of blend required, i.e., interior, exterior, or mixed.

4.2. The parabolic cyclide's extreme line and circle

Now we consider the necessary conditions on a parabolic cyclide's extreme line and extreme circle when the parabolic cyclide that the circle and the line determine is a blending surface. Then using these conditions, we show how to construct parabolic cyclides that are tangent to a given quadric.

Lemma 4.9 is used to pick a nonsingular parabolic cyclide by specifying a coplanar line and circle. The proof of this lemma is very similar to the proof of Lemma 4.1 and so is omitted. The main difference between these proofs is that a parabolic cyclide has only two parameters to solve for instead of three in the ring cyclide case. So by picking an extreme circle, an extreme line, and an interior region, a unique parabolic cyclide is determined.

L e m m a 4.9. Suppose we are given a plane P and a line l and a circle c lying in P, where 1 and c do not intersect. Let R be either the region in the halfplane determined by I containing but outside c or the region that is the union of the interior o f c and the halfplane determined by l that does not contain c. Then l, c, and R determine a nonsingular parabolic cyclide with l and c as the cyclide's extreme line and extreme circle respectively lying in the plane of symmetry 19, where R indicates the interior of the cyclide.

In the theorems that follow, we will specify a circle and a line, but not a region when applying Lemma 4.9. By using this lemma without a specified region, we get one of two parabolic cyclides where the specified circle and line become the cyclide's extreme circle and extreme line. Since both parabolic cyclides can be used in the blend, this will cause no trouble.

Following the development of the ring cyclide case, we now give a set of conditions that must be satisfied by the extreme circle and the extreme line, lying in a plane of the symmetry, for a parabolic blend. These conditions are summarized in the following definition.

Definition 4.10 (Extreme circle-line conditions). Suppose we are given a natural quadric Q and a plane P. A circle c and a line l that lie in the plane P satisfy the extreme circle- line conditions when

(1) c and l do not intersect, (2) both c and l are tangent to the cross section of Q in P , and (3) c lies outside and does not contain Q.

We need the following generalization of Remark 4.5 about the center of similitude for a line and a circle. Note, the exterior center of similitude of a line and a circle that do not intersect is the point on the circle furthest from the line. To see this, consider how the center of similitude of two circles must move as the radius of one of the circles increases.

s. Allen, D, Dutta / Computer Aided Geometric Design 14 (1997) 51-75 69

/ S

C

i .

"i

Fig. 13. A line l and a circle c that satisfy the extreme circle-line conditions with a sphere S. A center of similitude s of l and ¢ is also shown. This figure is drawn in the plane P which contains the center of S.

R e m a r k 4.11. Given a line l and a circle c, suppose a third circle C is tangent to 1 and

c at points p and q respectively. Then the line pq passes through one of the centers of similitude of 1 and c.

The following two lemmas are needed in the constructions found in the second part of this paper.

L e m m a 4.12. Suppose we are given a sphere S and a plane P such that 19 contains the center o f S. Further suppose that in 19 are a line 1 and a circle c satisfying the extreme circle-line conditions with S. Then either parabolic cyclide determined by l and c is

tangent to S.

Proof. Using Lemma 4.9, let C be the one of the parabolic cyclides determined by l and c. Suppose 1 is tangent to S at the point p and c is tangent to S at the point q.

Then using Remark 4.11 and the fact c lies outside S, we see that the line pq contains a center of similitude s of l and c that lies furthest from I. See Fig. 13. The line L through s perpendicular to 19 is a line of similitude of C'. Consider the plane P0 that is

perpendicular to P containing L and +p-~. P0 intersects S in a circle whose diameter is given by the line segment ~-~, since 19 contains the center of S and is perpendicular to 19o. 190 intersects C in a line of curvature of C, which is the same circle. By Remark 2.2, it is also a line of curvature of S. So C and S intersect in a shared line of curvature. Then by Joachimstahl's Theorem, the angle between C and S is constant.

All normals to C' along e point out from c's center. Since P contains the center of S, the normal to S at the point q points toward c's center. So S and C are tangent at the point q. Since the angle between C and S is constant, C and S must be tangent. []

L e m m a 4.13. Suppose we are given an axial natural quadric Q and a plane P such that 19 contains the axis o f Q. Further suppose that in P are a line 1 and a circle c satisfying the extreme circle-line conditions with Q. Then both o f the parabolic cyclides determined by 1 and c are tangent to Q.


. ' . ' •

• • •

, • • .

Fig. 14. A line l and a circle c that satisfy the extreme circle-line conditions with a cone Q. This figure is drawn in the plane P which contains the axis of Q.

Note, Q looks like either two lines or two rays in the plane P. We refer to these lines or rays as the skeleton of Q in P. Since the line 1 is tangent to Q in P, l contains one of the lines or rays of Q.

Proof. Let C be one of the parabolic cyclides determined by l and c. Suppose Q is tangent to c at the point p. Let S be the sphere whose center in on P and that is tangent to c at p and is tangent to I. See Fig. 14. Since l and the skeleton of Q overlap in P, S is tangent to Q at two points in P. Since the axis of Q lies in P, S must be a sphere from the envelope defining Q. By Lemma 4.12 and Joachimstahl's Theorem, S and C are tangent along a line of curvature L. Since S and Q are also tangent along L, we get that Q and C are tangent. []

5. The position of the quadrics

5.1. Tangent ring cyclides

The following corollaries of Theorem 2.1 tell us the positions of a quadric with respect to a blending cyclide. These four corollaries will be the main tools used to prove the results in the second part of this paper.

Corollary 5.1. Suppose a cyclide C and a cylinder Q are tangent along a curve L. Then L is one of C's extreme circles and Q's axis is perpendicular to the plane of symmetry P containing L.

Proof. By Theorem 2.1, L must be a circular line of curvature of both C and Q. By Remark 2.3, we see that the plane P containing L is perpendicular to Q's axis. Since all the normals to Q along L lie in P, using Maxwell's definition of the cyclide, we see that one of C's anticonics lies in P. Note, when C is a torus, all longitudinal lines of curvature are extreme circles, since they all lie in planes of symmetry. This proves the corollary. []

S. Allen, D. Durra / Computer Aided Geometric Design 14 (1997) 51-75 71

A.

K ~

B.

IW

Fig. 15. The four possible positions for a cyclide and a tangent cylinder. The cylinder is indicated by two parallel lines.

Using the corollary we see that when a cyclide and a cylinder are tangent along a curve they must be in one of the four positions shown in Fig. 15.

Note that our definition of a cyclide blend leads us to exclude from further consideration the positions indicated in Figs. 15A and 15C. In other words, when a cyclide blends a cylinder, the axis of the cylinder must be perpendicular to the plane of symmetry of the cyclide containing the ellipse, and the circle of intersection must be the one of the extreme circles lying in this plane. Since the center of an extreme circle lying in the plane of the ellipse is a vertex of the hyperbola, the axis of the cylinder is also contained in the plane of the hyperbola and intersects the hyperbola tangentially at one of the hyperbola's vertices.

Definition 5.2 (Bounding planes). Consider the family of spheres that define a cyclide with centers on the cyclide's hyperbola. As these spheres move away from the plane of the ellipse, their radii increase. In the limit, the spheres become one of two planes. These planes are called the bounding planes of the cyclide.

Corollary 5.3. Suppose a cyclide C and a plane P are tangent along a curve. Then P must be one o f C's bounding planes, and so must contain the line o f similitude L that lies in the plane o f the ellipse 19e.

Proof. Again by Theorem 2.1, we get that the intersection of P and C is a circular line of curvature. Since P contributes one of C's characteristic circles and is a special case of a sphere, P must be member of the family of spheres on the hyperbola that defines C'. Note, when C is a torus, Pe and L intersect at infinity. The corollary follows. []

72 S. Allen. D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75

Fig. 16. This figure is drawn in the plane Ph of C's hyperbola.

Corollary 5.4. Suppose a cyclide and a sphere are tangent along a curve. Then the sphere must be a member of the one of the two families of spheres that envelops the cyclide.

Proof. By Theorem 2.1, the sphere and the cyclide intersect along a line of curvature of the cyclide. Therefore the sphere contributes a characteristic circle to the cyclide and hence must be a member of the one of the families of spheres that defines the cyclide. []

By using Corollary 5.4 and our definition of cyclide blending, we get that when a cyclide is used to blend a sphere, the sphere must be a member of the family of spheres with their centers on the hyperbola.

Corollary 5.5. Suppose a cyclide C and an axial natural quadric Q are tangent along a curve. Then the intersection is a circular line of curvature of C, and Q's axis lies in one of the C's planes of symmetry.

Proof. By Theorem 2.1, Q and C intersect in a line of curvature of both surfaces, which is a circle c. Recalling Remark 2.3, we see c is the intersection of Q and some plane P perpendicular to the Q's axis. Suppose without loss of generality that the intersection of Q and C is one of C 's latitudinal circles, so that c's center lies in the plane of symmetry of the hyperbola Ph. See Fig. 16. We will show that Q's axis also lies in Ph. The axis is perpendicular to P through c's center. Further, P contains the line of similitude L that lies in the plane of symmetry of the ellipse Pe. Since L is normal to Ph and since Q's axis is perpendicular to L (it is perpendicular to a plane containing L), the axis lies in a plane parallel to Ph. But the axis contains a point on Ph, so the axis lies in Ph. If c had been a longitudinal circle, then a similar proof shows that the axis lies in Pe. []

So when a cyclide blends a cone, the axis of the cone must lie in the plane of symmetry of the cyclide containing the hyperbola. Using our definition of blending, we come to a specialization of a fact previously proven in (Shene, 1992):

Theorem 5.6. I f a cyclide blends two axial natural quadrics, then the axes of the quadrics are coplanar In fact, both axes must lie in the plane of symmetry of the cyclide containing the hyperbola.

Proof. Follows from Corollary 5.5. D


5.2. Tangent parabolic cyclides

Modified versions of the corollaries presented above remain true for parabolic cyclides. These are presented next.

First we show that when a parabolic cyclide and a plane are tangent along a curve, that curve must be a line.

Corollary 5.7. A parabolic cyclide and a plane can never be tangent along a closed curve.

Proof. Suppose a parabolic cyclide and a plane are tangent along a curve. Since the parabolic cyclide is a degree 3 surface, by Bezout's Theorem, the intersection of the plane and the parabolic cyclide is degree 3. The component of the this intersection along which the cyclide and the plane are tangent must be a curve with multiplicity of at least 2. Therefore the plane and the cyclide are tangent along a line. (A geometric proof follows: Consider the bounding planes of a ring cyclide and the circular lines of curvature in these planes. As the ring cyclide transforms into a parabolic cyclide, the bounding planes move into correspondence and the circles in these planes stretch into the same line.) []

The relationship between parabolic cyclides and tangent spheres is shown next.

Corollary 5.8. Suppose a parabolic cyclide and a sphere are tangent along a curve. Then the sphere must be a member of one of the two families of spheres that envelops the parabolic cyclide.

Proof. See the proof of Corollary 5.4. []

Now we look at how an axial natural quadric and a parabolic cyclide can be tangent along a closed curve.

Corollary 5.9. Suppose a parabolic cyclide C and an axial natural quadric Q are tangent along a closed curve. Then C and Q intersect in a circle c and an extreme line I. Q and C are tangent along both l and c, and l and c are lines of curvature of both surfaces. Further the axis of Q lies in the plane of symmetry t ~ of C containing 1.

Proof. By Theorem 2.1, the intersection of Q and C must contain a line of curvature of both Q and C. Since they are tangent along a closed curve, this line of curvature must be a circle c. The center of c lies in a plane of symmetry P of C, where P is perpendicular to the plane containing e.

First we show that the axis of Q lies in P. Q's axis is perpendicular to the plane containing c and this plane contains the line of similitude of C that is perpendicular to P. So Q's axis is perpendicular to C's plane of symmetry that is not P. This shows that Q's axis is parallel to P. But since the axis also contains a point on P, namely e's center, the axis lies in P.

Now we show that Q and C are tangent along the extreme line l in P. Since P contains the axis of Q, Q looks like either two rays (half-lines) or lines in P, depending

74 s. Allen, D. Dutta / Computer Aided Geometric Design 14 (1997) 51-75

on whether Q is a cone or a cylinder. One of these lines or rays, say lo, intersects c where l does. Since Q and 6' are tangent at this point, l and lo must be the same line (or overlap when lo is a ray). Since l is a line of curvature of both Q and C', and since Q and C' are tangent at the point where 1 and c intersect, by Joachimstahl's Theorem, Q and C' are tangent along l. []

6. Summary

Previous papers on cyclide blends known to the authors use a very loose definition of blends. This definition allows more freedom in when a cyclide blend exists, but often the blend obtained is not what the designer expects. Worse yet, it is difficult for the designer to determine beforehand whether or not the blend will be useful. In this paper we propose a new definition of a cyclide blend that ensures that the blend resembles a variable radius rolling ball blend in that it contains no singular points, it lies either entirely inside or entirely outside the quadrics being blended, and it follows the curve of intersection between the objects being blended. We believe these are the characteristics that a surface must possess to qualify as a blend.

Next, using our definition of a blend, we examine conditions on the extreme circles (or extreme circle and extreme line in the case of a parabolic cyclide) of a cyclide blending surfaces. Using these conditions, we show how a cyclide tangent to a natural quadric can be constructed. We also investigate the position of natural quadrics relative to a blending cyclide.

In the second part of this paper (Allen and Dutta, 1996), we use the definitions of the ring and the parabolic cyclide blends and the results developed here on the extreme circles and on the position of quadrics in blends to examine each of the ten natural quadric/natural quadric blend cases in detail.

Acknowledgments

We gratefully acknowledge the financial support received from AFOSR grant F49620- 93-1-0419 and AFOSR grant F49620-95-1-0209.

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