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Results on nonsingular, cyclide transition surfaces
COMPUTER AIDED GEOMETRIC DESIGN ELSEVIER
Computer Aided Geometric Design 15 (1998) 127-145
Results on nonsingular, cyclide transition surfaces Seth Allen a,1, Debasish Dutta b,, a Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA b Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, M1 48109, USA
Received June 1996; revised April 1997
Abstract
In this paper, we generate nonsingular, transition surfaces between natural quadrics using possibly singular cyclides. Both Dupin cyclides (ring, spindle, and homed) and parabolic cyclides (singular and nonsingular) are considered. Necessary and sufficient conditions for the existence of cyclide transition surfaces and easily implementable tests of these conditions are given in each quadric/quadric case. The relationship of the existence of cyclide transition surfaces to the common inscribed sphere condition is examined. Also, we show that except in one special case, blends and joins cannot simultaneously exist between two quadrics. © 1998 Elsevier Science B.V.
1. I n t r o d u c t i o n
In this paper, we study the conditions for the existence of nonsingular, cyclide transition surfaces connecting two natural quadrics. These surfaces are nonsingular tubes lying on (possibly singular) cyclides and tangent along circles to the quadrics they connect. The study of nonsingular, cyclide transition surfaces fits naturally into the category that has, in the literature, been called "cyclide blending". In the literature, a cyclide blend referred to any cyclide transition surface, including singular surfaces, between two quadrics. Here, we consider only those cyclide transition surfaces that are nonsingular, since surfaces containing singularities are seldom desired by a designer. Further, we make a distinction between blends and joins, since the word "blend" usually has a very specific meaning to a designer (Allen and Dutta, 1997b). See Fig. 1 for our classification of a transition surface T connecting primary surfaces P and Q. Note, the union of the set of * Corresponding author. E-mail: [email protected]. ] E-mail: [email protected]. 0167-8396/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S01 67-8396(97)0002 1-6
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Transition Surface T connecting P and Q
Nonsingular T
Non-intersecting P and Q
Singular T
Intersecting P and Q
T does not follow intersection •
Tfollows intersection i
i
s
Tlies on one side of P and of Q
/
Tis a join
Tis a blend
Fig. 1. The classification of transition surfaces.
blends and the set of joins make up the set of all nonsingular, cyclide transition surfaces. So, a cyclide blend is one subtype of a nonsingular, cyclide transition surface. In this paper, we give necessary and sufficient conditions for the existence of nonsingular, cyclide transition surfaces in each natural quadric/natural quadric case. The proofs of these results can be found in the technical report (Allen and Dutta, 1997a). Although several papers present constructions for various quadric/quadric cases in specific positions, see for example (Pratt, 1990, 1995; Srinivas and Dutta, 1994), conditions for when these constructions are possible are not explicitly given. Thus far only the cone/cone case (Johnstone and Shene, 1994; Pratt, 1990; Shene, 1992) has been thoroughly examined. Further, in all the previous constructions, no effort is made to ensure the construction of nonsingular transition surfaces. This paper completes the investigation of cyclide blends that we began in (Allen and Dutta, 1997b, 1997c) by putting that work into the larger context of nonsingular, cyclide transition surfaces. By explicitly showing (1) which type, blend or join, of nonsingular, cyclide transition surface is constructed in proofs of the conditions, and (2) that except in the case of intersecting spheres, there may exist either a blend or a join, but never both, the designer is able to determine not only when a nonsingular, cyclide transition surface exists, but also when this surface must be a blend and when it must be a join.
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In this paper, a cone is not the double cone obtained from the algebraic equation. Instead, we consider the single cone whose axis is a ray with endpoint at the vertex. This paper assumes an understanding and familiarity of cyclides in CAGD, as can be found is (Allen and Dutta, 1997a, 1997b, 1997c; Chandru et al., 1989; Pratt, 1990; Srinivas and Dutta, i 994). For much more detailed versions of the results found in this paper, including the necessary background material and proofs, see (Allen and Dutta, 1997a). 1.1. Overview
The definition of cyclide transition surfaces is given in Section 2. Section 3 contains necessary and sufficient conditions for the existence of nonsingular, cyclide transition surfaces in each of the ten quadric/quadric cases. Also in this section, we examine the result that blends and joins cannot simultaneously exist between two quadrics in given positions, except in the case of intersecting spheres. Parabolic cyclide transition surfaces are considered in Section 4. The development of the previous sections is followed here. In Section 5, the common inscribed sphere condition and its relationship to the existence of blends and joins is explored. Easily implementable tests for when nonsingular, cyclide and parabolic cyclide transition surfaces exists are presented in Section 6. When a nonsingular, cyclide transition surface exists, these tests can also be used to determine its type, blend or join. Finally, Section 7 contains our conclusions and an example.
2. Cyclide transition surfaces Definition 2.1 (Cyclide transition surface). A cyclide transition surface T between two quadrics Q1 and Q2 is a tube from a cyclide C bounded by two lines of curvature cl and c2 from the same family, so that C is tangent to Q~ along ci. Then T is one of two portions of C lying between Cl and c2. See Figs. 2 and 3 for examples of cyclide transition surfaces bounded by latitudinal and longitudinal circles respectively. T is said to connect Q I and Q2. In this paper we are only interested in nonsingular transitions surfaces. Since a nonsingular, cyclide transition surface can be either a cyclide blend or a cyclide join, we define these next. For more information on the development of these definitions see (Allen and Dutta, 1997b).
Definition 2.2 (Cyclide blend). Given a nonsingular, cyclide transition surface T between two quadrics Ql and Q2 where T is the portion of a cyclide C lying between the circles c~, T is a cyclide blend when (1) the curve of the intersection L of QI and Q2 is nonempty and closed, (2) each ci is a latitudinal line of curvature, and (3) L must wrap around the axis of each axial natural quadric being blended.
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\ Fig. 2. A cyclide blend between a cylinder and a cone.
Fig. 3. A cyclide join between a cylinder and a cone. See Fig. 2. T is said to blend QI and Q2. Definition 2.3 (Cyclide join). Given a nonsingular, cyclide transition surface T between two quadrics Q1 and Q2 where T is the portion of a cyclide C lying between the circles ci, T is a cyclide join when • each ci is a longitudinal lines of curvature, or • each ci is a latitudinal lines of curvature and either Q j and Q2 do not intersect, or the quadrics' intersection curve does not wrap around the axis of the axial natural quadric being joined. See Fig. 3. T is said to join Q1 and Q2. -
-
In our previous investigations of cyclide blends (Allen and Dutta, 1997b, 1997c), we saw that whenever a cyclide blend exists, it is possible to find a blend that lies on a nonsingular cyclide. That we were able to only consider nonsingular cyclides results from the intersection requirement for a blend. However, we are now unable to exclude singular cyclides from consideration. Consider for example, connecting a cone with a
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far away and very small (point-like) sphere. In this case, every cyclide on which the transition surface lies must be singular near the sphere. So some additional work will be required to ensure that when using a singular cyclide, the portion of the cyclide used for the transition surface is nonsingular.
3. Existence results In this section, we indicate necessary and sufficient conditions for the existence of nonsingular cyclide transition surfaces in every natural quadric/natural quadric case. The proofs of these results, which are published in the technical report (Allen and Dutta, 1997a), all follow the same pattern. Each proof is split into a series of subcases according to the positions of the quadrics with respect to each other. In each subcase, the plane of symmetry P of the potential tangent cyclide is identified, and circles in P are chosen. These circles will be the extreme circles of the tangent cyclide when it exists. Figures indicating these circles are given in many cases. First we consider the plane/plane, plane/sphere, and sphere/sphere cases. As Theorem 3.1 shows, we can always find a blend or a join in these cases.
Q~ - . , Q2
a.
IY
Q1
o. =..,
Fig. 4. This figure illustrates the plane/plane and plane/sphere cases.
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1 ~ ~r~
'--fL a.
b.
Y.._~/Q2,
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Fig. 5. This figure illustrates the sphere/sphere cases.
l
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T h e o r e m 3.1. Let Ql and Q2 be two quadrics in the set of all planes and spheres. Then there always exists a nonsingular, cyclide transition surface T between Q1 and Q2. Figs. 4 and 5 indicate the extreme circles of tangent cyclides constructed for these cases. Now we look at the plane/axial natural quadric cases. Again a nonsingular, cyclide transition surface between a plane and an axial natural quadric always exists, except in two special cases. T h e o r e m 3.2. Let t 9 be a plane and Q be a cylinder. Then there exists a nonsingular, cycIide transition surface T between P and Q if and only if P and Q intersect in an ellipse. When T exists, it is a blend. See Fig. 6 for an indication of why no nonsingular, cyclide transition surfaces exist in the positions indicated by Theorem 3.2. T h e o r e m 3.3. Let P be a plane and Q be a cone. Then there exists a nonsingular, cyclide transition surface T between P and Q if and only if P and Q do not intersect in one or two rays. Fig. 7 illustrates many of the plane/cone cases. Now we consider the sphere/axial natural quadric cases. A nonsingular, cyclide transition surface always exists in these cases except in one special position in the sphere/cone case.
q
ql
q2
q2
b.
a.
Fig. 6. This figure illustrates why no nonsingular, cyclide transition surface exists between a plane P and a cylinder C when they (a) do not intersect, and (b) intersect in lines only.
P
P --
~
/
\
( a.
' ~ / ~
\
b.
\-JQ
/
/
z
2 C c.
Fig. 7. This figure illustrates the plane/cone cases.
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Theorem 3.4. Let S be a sphere and Q be a cylinder. Then there always exists a nonsingular, cyclide transition surface T between 5" and Q. See Fig. 8 for the positions of the extreme circles of tangent cyclides in the sphere/ cylinder cases. Theorem 3.5. Let S be a sphere and Q be a cone. Then there exists a nonsingular, cyclide transition surface T between S and Q if and only if it is not the case that (1) S lies entirely outside the double cone Q o f which Q is one half and (2) 5' is tangent to Q at a point not on Q. When a sphere satisfies the conditions of Theorem 3.5, every cyclide tangent to both quadrics must contain a singularity lying between the quadrics. Fig. 9 demonstrates why this happens. Fig. 10 shows some of the remaining cases where a nonsingular, cyctide transition surface exists. For the axial natural quadric/axial natural quadric cases, we need the following two definitions. Definition 3.6 (Longitudinal conditions). Suppose we are given a sphere or an axial natural quadric Q and a plane P~. Two circles cj and c2 that lie in P~ satisfy the hmgitudinal conditions when (1) both cl and c2 are tangent to the cross section of Q in P~ in the distinct points ql and q2 respectively, (2) when Q is an axial natural quadric, the line qlq~2 is perpendicular to Q's axis, (3) one of the circles, say c2, does not lie entirely outside the other, and
Fig. 8. This figure illustrates the sphere/cylinder cases.
x
tf
"",
\
.x
, f~ \ C . ~" t
I
~
C,.,\ \\ /I
a.
Fig. 9. This figure illustrates the special sphere/cone case where no nonsingular transition surface exists.
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/"-\'
,•
\\
// ', ,'
~2
'
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°k/6
t" ' " , ,'
Fig. 10. This figure illustrates the sphere/cone cases.
(4) in 19e, c2 lies outside and does not contain Q, and • when Q is a sphere, cl contains Q, or • when Q is an axial natural quadric, ct lies partially inside and partially outside
(7. In Definition 3.7 below, we take a given normal to the plane to be the outward pointing normal, so the inside and the outside of the plane refer to the regions of three space that the plane determines.
Definition 3.7 (Latitudinal conditions).
Suppose we are given a natural quadric Q and a plane 19h. Two circles el and e2 that lie in 19h satisfy the latitudinal conditions when (1) both cl and c2 are tangent to the cross section of Q in Ph in the distinct points ql and q2 respectively, (2) when Q is an axial natural quadric, the line ql q~ is perpendicular to Q ' s axis, (3) neither cl nor c2 lies inside the other, and (4) in 19h, one of the following three conditions between the circles and Q holds: • both cl and c2 lie outside and do not contain Q, • both Cl and e2 lie outside and contain Q (implying Q is a sphere), or • both ct and c2 lie inside or partially inside Q. T h e o r e m 3.8. Let Ql and Q2 be axial natural quadrics. Then there exists a nonsingular, cyclide transition surface T between Qt and Q2 if and only if the axes of both quadrics lie in a plane t 9, and there exists two circles el and ¢2 in 19 satisfying either (1) the longitudinal conditions with both quadrics where cl lies partially inside and partially outside each quadric and c2 lies entirely outside both quadrics, or (2) the latitudinal conditions with both quadrics where el is tangent to Ql at a point inside c2 if and only if cj is tangent to Q2 at a point inside c2.
3.1. A summary of existence results Fig. 11 summarizes the existence results for nonsingular, cyclide transition surfaces. Here 0 means "never", 1 means "always", and e is a small number.
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T.S. [I Plane [Sphere Plane 1 1 Sphere 1 1 Cylinder 1 - e 1 Cone 1-
135
Cylinder Cone 1 -e 1 -e 1
1-e
Fig. 11. Summary of existence results for nonsingular, cyclide transition surfaces.
a.
b°
C.
Fig. 12. (a) Two intersecting spheres (b) Blended (c) Joined.
3.2. On simultaneous blends and joins In this section, we state the result that simultaneous blends and joins between two quadrics can only exist in the sphere/sphere case. Again for a proof, we refer the interested reader to (Allen and Dutta, 1997a). T h e o r e m 3.9. Suppose Q l and 0,2 are two natural quadrics and T is a nonsingular,
cyclide transition surface connecting Q, 1 and Q2. Except in the case that Ql and Q,2 are two intersecting spheres where neither sphere contains the other, we get • i f T is a join, then there exists no blends between Ql and Q2, and • i f T is a blend, then there exists no joins between Q1 and Q,2. If Q,I and Q,2 are two intersecting spheres and neither sphere contains the other, then there exists both blends and joins connecting the spheres. For an example of how two intersecting spheres can be both blended and joined, see Fig. 12. Using the results of the previous section, Theorem 3.9, and the results in (Allen and Dutta, 1997c), it is an easy matter to determine • when a nonsingular, cyclide transition surface exists between two quadrics, and • whether a nonsingular, cyclide transition surface is a blend or a join.
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Fig. 13. A parabolic cone/cylinder blend shown in one of the planes of symmetry.
4. Parabolic cyclide transition surfaces Parabolic cyclides can be used to extend the number of cases in which two natural quadrics can be blended or joined. In this section, we examine the possibility of the existence of nonsingular, parabolic cyclide transition surfaces between two natural quadrics. Note, the definition of a parabolic cyclide transition surface follows naturally from Definition 2.1. See Fig. 13 for a picture of a parabolic cyclide blend. We state without proof the necessary and sufficient conditions for the existence of a nonsingular, parabolic cyclide transition surface in each quadric/quadric case. For more detail, we refer the interested reader to (Allen and Dutta, 1997a).
Theorem 4.1. No nonsingular, parabolic cyclide transition surfaces involving planes are possible. Theorem 4.2. Two spheres Sl and $2 can be connected by a nonsingular, parabolic cyclide transition surface T if and only if neither sphere contains the other When Sl and $2 intersect, T is a blend. Theorem 4.3. Two axial natural quadrics can be connected by a parabolic cyclide if and only if the axes of the quadrics lie in a common plane P, and (1) in P, the quadrics are both tangent to the same line l, (2) in P, both quadrics lie on the same side of l, and (3) neither quadric contains the other. As an immediate consequence of Theorem 4.3, we get the following result.
Theorem 4.4. No nonsingular, parabolic cyclide transition surface between two cylinders is possible. In the constructive parts of the proofs of the previous theorems, it is always possible to find a nonsingular, parabolic cyclide that contains the nonsingular, parabolic cyclide
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_ (
a.
,
\
~
_,
C/
o
137
_,
b.
Fig. 14. This figure, drawn in the plane P, illustrates the sphere/cone parabolic cyclide cases.
P.T.S. Plane Sphere Cylinder
IIPlane I Sphere 0 0
0
0 1/2
Cylinder Cone 0 0 0
Fig. 15. Summary of existence results for nonsingular, parabolic cyclide transition surfaces. transition surface we seek. In the sphere/axial natural quadric case, this is not true. See for example, Fig. 14(a). Note, in the blends case, the requirement that the intersection curve wraps around the axis of the axial natural quadric allows us to always find a nonsingular, parabolic cyclide containing the transition surface we seek. See Fig. 14(b).
Theorem 4.5. Given a sphere S and an axial natural quadric Q, let P be a plane containing the center of S and the axis of Q. Then S and Q can be connected by a parabolic cyclide if and only if (1) in P, S and Q are both tangent to the same line l, and (2) in P, S and Q lie on the same side of l. Fig. 15 summarizes the existence results for nonsingular, parabolic cyclide transition surfaces. Here 0 means "never", 1 means "always", and e is a small number. Next we consider the possibility of simultaneous parabolic blends and joins. Since a nonsingular, parabolic cyclide transition surface is a blend if and only if the quadrics being connected satisfy certain intersection requirements, we get the following theorem.
Theorem 4.6. Suppose Q1 and Q2 are two natural quadric and T is a nonsingular, parabolic cyclide transition surface connecting Ql and Q2. Then • if T is a join, then there exists no blends between Q t and Q2, and • if T is a blend, then there exists no joins between Q l and Q2.
5. Inscribed sphere condition In this section, we consider what the inscribed sphere condition means for blending and joining with cyclides. Previous work (Johnstone and Shene, 1994; Pratt, 1990; Shene,
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1992) shows the result that there exists a cyclide or parabolic cyclide transition surface (possibly singular) between two cones with nonparallel axes if and only if the double cones contain a common inscribed sphere. Theorem 5.1 shows when the common inscribed sphere condition implies the existence of a nonsingular, cyclide transition surface. One way to prove this theorem uses the offset property of cyclides closely following the proof presented by Pratt (1990) attributed to M. Sabin. However, some additional work is required to ensure that the transition surface we construct in Theorem 5.1 is nonsingular. T h e o r e m 5.1. Suppose Q1 and Q2 are axial natural quadrics that contain a common inscribed sphere S. Then there exists a nonsingular, cyclide transition surface T between Ql and Q2 if and only if Ql and Q2 are not tangent along a common line. Now we show that a cyclide blend or join of two axial natural quadrics with intersecting and nonparallel axes implies a common inscribed sphere. An outline of the proof of this theorem can be found in (Allen and Dutta, 1997a). T h e o r e m 5.2. Suppose Ql and Q2 are axial natural quadrics that are connected by a nonsingular, cyclide transition surface T. If Ql and Q2 satisfy the following two conditions: • the axes intersect, and • neither quadric contains the other, then they have a common inscribed sphere. Notice that when two axial natural quadrics do not have intersecting axes, or when the axes are parallel and one of the quadrics contains the other, the quadrics cannot have a common inscribed sphere. However, in these cases it may still be possible for the quadrics to be joined by a nonsingular, cyclide transition surface. See Fig. 16. Finally, we state the relationship of the common inscribed sphere condition to the existence of nonsingular, parabolic cyclide transition surfaces.
a.
Fig. 16. This figure is drawn in the plane of the axes of two axial natural quadrics QI and Q2 that have no common inscribed sphere. (a) Shows quadrics with nonintersecting axes. (b) Shows quadrics with intersecting, parallel axes where QI contains Q2.
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Theorem 5.3. Suppose that there exists a nonsingular, parabolic cyclide transition suro .face T between two axial natural quadrics Ql and Q2. Further suppose Q1 and Q2 intersect. Then Ql and Q2 have a common inscribed sphere (and T is a blend). Theorem 5.4. If two axial natural quadrics have a common inscribed sphere and are tangent along a line that is a common line o f curvature o f both quadrics, then there exists a nonsingular, parabolic cyclide transition surface T connecting the quadrics. Further, T is a blend.
6. Tests for existence of cyclide transition surfaces In this section we present tests that determine when a nonsingular, cyclide transition surface between two natural quadrics is possible. These results follow from the work presented in the previous sections. Also, we include refinements that allow the designer that determine when the cyclide transition surface will be a blend. These tests are important in the context of the implementation of cyclide blending and joining in geometric modeling systems.
Plane/Plane: There always exists a nonsingular, cyclide transition surface T between two planes. T is never a blend. There never exists a nonsingular, parabolic cyclide transition surface between two planes. Plane/Sphere: There always exists a nonsingular, cyclide transition surface T between a sphere S and a plane P. Suppose S has center c and radius r. Then T is a blend if and only if the distance from c to P is less than r. There never exists a nonsingular, parabolic cyclide transition surface between a sphere and a plane. Plane/Cylinder: Let N be a normal to the plane Then there exists a nonsingular, the cylinder if and only if N • V There never exists a nonsingular, and a cylinder.
and L be the axis of the cylinder with direction V. cyclide transition T surface between the plane and ~ 0. T is a blend. parabolic cyclide transition surface between a plane
Plane/Cone: Let c~ be the half angle of the cone Q, and/3 be the angle that the axis of Q makes with the normal to the plane P. Then there exists a nonsingular, cyclide transition surface T between Q and P if and only if when Q's vertex lies on P, c~
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and a cone.
Sphere/Sphere: Let SI and $2 be two spheres, where r~ is the radius and ci is the center of S~. There always exists a nonsingular, cyclide transition surface T between $1 and $2. T can be a blend when dist(cl, c2) < rl + r2 (when the spheres intersect). Further suppose rl <~ re. Then there exists a nonsingular, parabolic cyclide transition surface T between SI and $2 if and only if dist(cl, c2) + rl > r2. (T exists when neither sphere contains the other.) T is a blend if and only if dist(cl, c2) < rl + r2.
Sphere/Cylinder: Suppose a sphere S has radius R~ and a cylinder Q has radius Rq. Further suppose that the distance from the center of S to the axis of Q is d. There always exists a nonsingular, cyclide transition surface T between S and Q. T is a blend if and only if Rs - Rq ~ d. There exists a nonsingular, parabolic cyclide transition surface T between S and Q if and only if IRs - Rql = d. T is a blend if and only if R~ - Rq = d.
Sphere/Cone: Given a sphere S with center c and radius r, and a cone Q. Let P be the plane containing c and Q's axis. Let ~) be the double cone of which Q is one half. So in P, Q looks like two rays 11 and 12, and ~) looks like two lines/] and/~. Let N be the unit vector in the direction of Q's axis. Then, there exists a nonsingular, cyclide transition surface T between S and Q, except when all of the following three conditions are true using either/] or/2 in place of line L: (1) The distance from c to L is r. (So S is tangent to Q at a point p. Let V0 be the vector from p to c and let V1 be the vector from Q's vertex to p.) (2) V0 • N > 0. (So S and Q lie on the same side of L.) (3) V1 • N < 0. (So S is tangent to ~) at the point p which is not on Q.) T is a blend if and only if the distance from from c to Ii and the distance from c to 12 are both less than or equal to r. There exists a nonsingular, parabolic cyclide transition surface T between S and Q if and only if the following two conditions are true using either/] or/2 in place of the line L: (1) The distance from c to L is r. (So S is tangent to Q at a point p. Let V0 be the vector from p to c.) (2) V0 • N > 0. (So S and Q lie on the same side of L.) T is a blend if and only if the distance from c to either Ii or 12 equals r and the distance from c to the other ray is less than r.
Cylinder/Cylinder: Let Q1 and Q: be two cylinders with radii r~ such that rl /> Y2. Then there exists a nonsingular, cyclide transition surface T between Q1 and Q2 if and only if the axes of the cylinders are coplanar and
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14t
• when the axes are parallel and d is the distance between the axes, either d + r 2 < rl (so Ql contains Q2) or rl + r2 < d (so that neither cylinder contains the other), or
• when the axes intersect, r~ = r2. T is n e v e r a blend. There n e v e r exists a nonsingular, parabolic cyclide transition surface between two cylinders. Cylinder/Cone: 2
Let Q1 be a cone with half angle ~ and vertex v, and let Q2 be a cylinder with radius r. Let Q I be the double cone of which Q1 is one half. Then there exists a nonsingular, cyclide transition surface T connecting Q1 and Q2 if and only if (1) the quadrics' axes are coplanar and hence lie in a plane P, (2) in /9, neither of the rays of Q l ' s cross section lies on a line of Q2's cross section, (3) either Ql'S and Qz's axes intersect or v lies inside Q2, and (4) Q1 and Q 2 have a common inscribed sphere, so either • Q j ' s axis lies on Q2's axis, or • Q j ' s axis and Q2's axis intersect in exactly one point that is distance d from v, and d sin c~ = r. T is a blend if and only if v lies inside Q2. There exists a nonsingular, parabolic cyclide transition surface T connecting Ql and Q2 if and only if (1) the axes of the quadrics intersect and hence are contained in a plane t9, and (2) in /9, one of the rays of Q l ' s cross section lies on a line from Q2's cross section. T is a blend. Cone/Cone:
Let Qj and Q2 be two cones with half angles cq and c~2 respectively. Let Q~ be the double cone of which Q~ is one half. There exists a nonsingular, cyclide transition surface T connecting Q1 and Qz if and only if (1) the quadrics' axes are coplanar and hence lie in a plane P, (2) in/9, neither of the lines from Q l ' s cross section equals a line from Qz's cross section, (3) one__of the following three conditions is true: • Q l ' s axis equals Q2's axis, • Q ll's axis is parallel but not equal to Q2's axis, and c~1 = c~2, or • Q l ' S axis and Q 2 ' s axis intersect in exactly one point p that is distance d~ from Qi's vertex and dl sin C~l = d2 sin c~2, and
2 A more direct test similar to the test used in the cone/cone case that constructs the extreme circles of a tangent cyclide can be readily derived.
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(4) one of the following two conditions is true: • Q1 and Q2 intersect at their vertices only. So either one cone contains the other or neither cone contains the other. • Q1 and Q2 have distinct vertices. Let 5' be the line through the cones vertices, then either the following three conditions are true: 5' lies entirely outside both cones, - in P, Q1 and Q2 lie on the same side of S, and there exist two circles cl and c2 that satisfy the longitudinal conditions with both cones where c2 lies outside both cones. (See below for how to construct these circles.) or the following two conditions are true: S lies partially inside both cones, and there exist two circles el and c2 that satisfy the latitudinal conditions with both Q1 and Qz where cl is tangent to QI at a point inside c2 if and only if Cl is tangent to Q2 at a point inside c2. (See below for how to construct these circles.) T is a blend if and only if the circles cl and c2 satisfy the latitudinal conditions and in P, Q1 and Q2 intersect in exactly two points. There exists a nonsingular, parabolic cyclide transition surface T connecting Q1 and Q2 if and only if the following four conditions are true: (1) The quadrics' axes are coplanar and hence lie in a plane P. (2) In P, one of the lines from Q l ' s cross section equals a line from Q2's cross section. Call this line L and let N be a vector perpendicular to L. (3) Let di be the direction of Qi's axis. Either Ql'S axis intersects Q2's axis in a point not on L or dl • d2 < 0. (So neither Ql nor Q2 contains the other.) (4) The signs of N . dl and N . d2 are equal. (So Q1 and Q2 lie on the same side of L.) T is a blend if and only if Q l ' s axis intersects Q2's axis. It is not hard to show that, if there exists a cyclide tangent to two axial natural quadrics with its extreme circles in a plane of symmetry in a region determined by the quadrics, then there exists an infinite family of tangent cyclides with extreme circles in the same region. Using this fact, we are able search for the circles in the cone/cone test by simply trying a pair of circles that satisfy the latitudinal or the longitudinal conditions with both quadrics in each region of interest. If a pair does not satisfy the conditions we require in a particular region, then no nonsingular, cyclide transition surface exists in that region. Further, notice that by picking one of the circles, the other circle, if it exists, is completely determined.
7.
Implemented
example
and
summary
This paper was motivated by a need to state explicitly necessary and sufficient conditions for when a nonsingular, cyclide transition surface between two natural quadrics exists. We provide simple, easily implemented tests for these conditions. Except in the
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143
II a)
b)
c)
Fig. 17. Shows two views in the construction of a water faucet using nonsingular, cyclide transition surfaces. (a) Shows the natural quadrics used in the model. (b) Adds joins for the handle and spigot. (c) Adds blends to get the finished faucet.
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cone/cone case, no conditions had been explicitly stated in previous work, although some constructions indicated which tests would need to be carried out before the construction could continue. Here we show that nonsingular, cyclide transition surfaces are only possible when the indicated test conditions are satisfied and which type, blend or join, of nonsingular, cyclide transition surface we get. Further, we show that except in the case of intersecting spheres, two quadrics in given positions cannot be both blended and joined. The work here is carried out for both ring and parabolic cyclide transition surfaces concluding the investigations we began in (Allen and Dutta, 1997b, 1997c). Fig. 17 shows the construction of a water faucet made from nonsingular, cyclide transition surfaces. Each frame contains the same objects shown in two different views. Note, a small portion of the sphere on the spigot is used so that there are no tangent discontinuities where the cyclides meet other objects. As we have shown, except in the case of two axial natural quadrics, we can almost always expect to be able to find a nonsingular, cyclide transition surface between two natural quadrics. This is further evidence of the utility of cyclides in CAD/CAM. Already cyclides have proven useful in a number of engineering applications including cable harness design, motion planning, surface composition, and pipe joining. The results reported in this paper are currently being implemented in the DESIGNBASE modeler of Ricoh Corporation. The methods that the software uses to construct nonsingular, cyclide transition surfaces, when they exist, closely follow the constructive portions of the proofs as found in (Allen and Dutta, 1997a, 1997b, 1997c).
Acknowledgments We gratefully acknowledge the financial support received from AFOSR grant F4962093-1-0419 and AFOSR grant F49620-95-1-0209.
References Allen, S. and Dutta, D. (1997a), Cyclide transition surfaces: blending and joining, Technical report, UM-MEAN-97-05, Department of Mechanical Engineering, University of Michigan. Allen, S. and Dutta, D. (1997b), Cyclides in pure blending I, Computer Aided Geometric Design 14, 51-75. Allen, S. and Dutta, D. (1997c), Cyclides in pure blending II, Computer Aided Geometric Design 14, 77-102. Chandru, V., Dutta, D. and Hoffmann, C. (1989), On the geometry of Dupin cyclides, The Visual Computer 5, 277-290. Johnstone, J.K. and Shene, C.-K. (1994), Dupin cyclides as blending surfaces for cones, in: Fisher, R.B., ed., Mathematics o f Surfaces V, Oxford University Press, 3-29. Pratt, M.J. (1990), Cyclides in computer aided geometric design, Computer Aided Geometric Design 7, 221-242. Pratt, M.J. (1995), Cyclides in computer aided geometric design II, Computer Aided Geometric Design 12, 131-152.
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Shene, C.-K. (1992), Planar intersection and blending of natural quadrics, Ph.D. thesis, Department of Computer Science, Johns Hopkins University. Srinivas, Y.L. and Dutta, D. (1994), Blending and joining using cyclides, ASME Trans. Journal of Mechanical Design 116, 1034-1041.
Computer Aided Geometric Design 15 (1998) 127-145
Results on nonsingular, cyclide transition surfaces Seth Allen a,1, Debasish Dutta b,, a Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA b Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, M1 48109, USA
Received June 1996; revised April 1997
Abstract
In this paper, we generate nonsingular, transition surfaces between natural quadrics using possibly singular cyclides. Both Dupin cyclides (ring, spindle, and homed) and parabolic cyclides (singular and nonsingular) are considered. Necessary and sufficient conditions for the existence of cyclide transition surfaces and easily implementable tests of these conditions are given in each quadric/quadric case. The relationship of the existence of cyclide transition surfaces to the common inscribed sphere condition is examined. Also, we show that except in one special case, blends and joins cannot simultaneously exist between two quadrics. © 1998 Elsevier Science B.V.
1. I n t r o d u c t i o n
In this paper, we study the conditions for the existence of nonsingular, cyclide transition surfaces connecting two natural quadrics. These surfaces are nonsingular tubes lying on (possibly singular) cyclides and tangent along circles to the quadrics they connect. The study of nonsingular, cyclide transition surfaces fits naturally into the category that has, in the literature, been called "cyclide blending". In the literature, a cyclide blend referred to any cyclide transition surface, including singular surfaces, between two quadrics. Here, we consider only those cyclide transition surfaces that are nonsingular, since surfaces containing singularities are seldom desired by a designer. Further, we make a distinction between blends and joins, since the word "blend" usually has a very specific meaning to a designer (Allen and Dutta, 1997b). See Fig. 1 for our classification of a transition surface T connecting primary surfaces P and Q. Note, the union of the set of * Corresponding author. E-mail: [email protected]. ] E-mail: [email protected]. 0167-8396/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S01 67-8396(97)0002 1-6
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Transition Surface T connecting P and Q
Nonsingular T
Non-intersecting P and Q
Singular T
Intersecting P and Q
T does not follow intersection •
Tfollows intersection i
i
s
Tlies on one side of P and of Q
/
Tis a join
Tis a blend
Fig. 1. The classification of transition surfaces.
blends and the set of joins make up the set of all nonsingular, cyclide transition surfaces. So, a cyclide blend is one subtype of a nonsingular, cyclide transition surface. In this paper, we give necessary and sufficient conditions for the existence of nonsingular, cyclide transition surfaces in each natural quadric/natural quadric case. The proofs of these results can be found in the technical report (Allen and Dutta, 1997a). Although several papers present constructions for various quadric/quadric cases in specific positions, see for example (Pratt, 1990, 1995; Srinivas and Dutta, 1994), conditions for when these constructions are possible are not explicitly given. Thus far only the cone/cone case (Johnstone and Shene, 1994; Pratt, 1990; Shene, 1992) has been thoroughly examined. Further, in all the previous constructions, no effort is made to ensure the construction of nonsingular transition surfaces. This paper completes the investigation of cyclide blends that we began in (Allen and Dutta, 1997b, 1997c) by putting that work into the larger context of nonsingular, cyclide transition surfaces. By explicitly showing (1) which type, blend or join, of nonsingular, cyclide transition surface is constructed in proofs of the conditions, and (2) that except in the case of intersecting spheres, there may exist either a blend or a join, but never both, the designer is able to determine not only when a nonsingular, cyclide transition surface exists, but also when this surface must be a blend and when it must be a join.
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In this paper, a cone is not the double cone obtained from the algebraic equation. Instead, we consider the single cone whose axis is a ray with endpoint at the vertex. This paper assumes an understanding and familiarity of cyclides in CAGD, as can be found is (Allen and Dutta, 1997a, 1997b, 1997c; Chandru et al., 1989; Pratt, 1990; Srinivas and Dutta, i 994). For much more detailed versions of the results found in this paper, including the necessary background material and proofs, see (Allen and Dutta, 1997a). 1.1. Overview
The definition of cyclide transition surfaces is given in Section 2. Section 3 contains necessary and sufficient conditions for the existence of nonsingular, cyclide transition surfaces in each of the ten quadric/quadric cases. Also in this section, we examine the result that blends and joins cannot simultaneously exist between two quadrics in given positions, except in the case of intersecting spheres. Parabolic cyclide transition surfaces are considered in Section 4. The development of the previous sections is followed here. In Section 5, the common inscribed sphere condition and its relationship to the existence of blends and joins is explored. Easily implementable tests for when nonsingular, cyclide and parabolic cyclide transition surfaces exists are presented in Section 6. When a nonsingular, cyclide transition surface exists, these tests can also be used to determine its type, blend or join. Finally, Section 7 contains our conclusions and an example.
2. Cyclide transition surfaces Definition 2.1 (Cyclide transition surface). A cyclide transition surface T between two quadrics Q1 and Q2 is a tube from a cyclide C bounded by two lines of curvature cl and c2 from the same family, so that C is tangent to Q~ along ci. Then T is one of two portions of C lying between Cl and c2. See Figs. 2 and 3 for examples of cyclide transition surfaces bounded by latitudinal and longitudinal circles respectively. T is said to connect Q I and Q2. In this paper we are only interested in nonsingular transitions surfaces. Since a nonsingular, cyclide transition surface can be either a cyclide blend or a cyclide join, we define these next. For more information on the development of these definitions see (Allen and Dutta, 1997b).
Definition 2.2 (Cyclide blend). Given a nonsingular, cyclide transition surface T between two quadrics Ql and Q2 where T is the portion of a cyclide C lying between the circles c~, T is a cyclide blend when (1) the curve of the intersection L of QI and Q2 is nonempty and closed, (2) each ci is a latitudinal line of curvature, and (3) L must wrap around the axis of each axial natural quadric being blended.
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\ Fig. 2. A cyclide blend between a cylinder and a cone.
Fig. 3. A cyclide join between a cylinder and a cone. See Fig. 2. T is said to blend QI and Q2. Definition 2.3 (Cyclide join). Given a nonsingular, cyclide transition surface T between two quadrics Q1 and Q2 where T is the portion of a cyclide C lying between the circles ci, T is a cyclide join when • each ci is a longitudinal lines of curvature, or • each ci is a latitudinal lines of curvature and either Q j and Q2 do not intersect, or the quadrics' intersection curve does not wrap around the axis of the axial natural quadric being joined. See Fig. 3. T is said to join Q1 and Q2. -
-
In our previous investigations of cyclide blends (Allen and Dutta, 1997b, 1997c), we saw that whenever a cyclide blend exists, it is possible to find a blend that lies on a nonsingular cyclide. That we were able to only consider nonsingular cyclides results from the intersection requirement for a blend. However, we are now unable to exclude singular cyclides from consideration. Consider for example, connecting a cone with a
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far away and very small (point-like) sphere. In this case, every cyclide on which the transition surface lies must be singular near the sphere. So some additional work will be required to ensure that when using a singular cyclide, the portion of the cyclide used for the transition surface is nonsingular.
3. Existence results In this section, we indicate necessary and sufficient conditions for the existence of nonsingular cyclide transition surfaces in every natural quadric/natural quadric case. The proofs of these results, which are published in the technical report (Allen and Dutta, 1997a), all follow the same pattern. Each proof is split into a series of subcases according to the positions of the quadrics with respect to each other. In each subcase, the plane of symmetry P of the potential tangent cyclide is identified, and circles in P are chosen. These circles will be the extreme circles of the tangent cyclide when it exists. Figures indicating these circles are given in many cases. First we consider the plane/plane, plane/sphere, and sphere/sphere cases. As Theorem 3.1 shows, we can always find a blend or a join in these cases.
Q~ - . , Q2
a.
IY
Q1
o. =..,
Fig. 4. This figure illustrates the plane/plane and plane/sphere cases.
j
e
..,
C2\
(
I
~\
--.
1 ~ ~r~
'--fL a.
b.
Y.._~/Q2,
c.
"k.~32--"
Fig. 5. This figure illustrates the sphere/sphere cases.
l
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T h e o r e m 3.1. Let Ql and Q2 be two quadrics in the set of all planes and spheres. Then there always exists a nonsingular, cyclide transition surface T between Q1 and Q2. Figs. 4 and 5 indicate the extreme circles of tangent cyclides constructed for these cases. Now we look at the plane/axial natural quadric cases. Again a nonsingular, cyclide transition surface between a plane and an axial natural quadric always exists, except in two special cases. T h e o r e m 3.2. Let t 9 be a plane and Q be a cylinder. Then there exists a nonsingular, cycIide transition surface T between P and Q if and only if P and Q intersect in an ellipse. When T exists, it is a blend. See Fig. 6 for an indication of why no nonsingular, cyclide transition surfaces exist in the positions indicated by Theorem 3.2. T h e o r e m 3.3. Let P be a plane and Q be a cone. Then there exists a nonsingular, cyclide transition surface T between P and Q if and only if P and Q do not intersect in one or two rays. Fig. 7 illustrates many of the plane/cone cases. Now we consider the sphere/axial natural quadric cases. A nonsingular, cyclide transition surface always exists in these cases except in one special position in the sphere/cone case.
q
ql
q2
q2
b.
a.
Fig. 6. This figure illustrates why no nonsingular, cyclide transition surface exists between a plane P and a cylinder C when they (a) do not intersect, and (b) intersect in lines only.
P
P --
~
/
\
( a.
' ~ / ~
\
b.
\-JQ
/
/
z
2 C c.
Fig. 7. This figure illustrates the plane/cone cases.
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133
Theorem 3.4. Let S be a sphere and Q be a cylinder. Then there always exists a nonsingular, cyclide transition surface T between 5" and Q. See Fig. 8 for the positions of the extreme circles of tangent cyclides in the sphere/ cylinder cases. Theorem 3.5. Let S be a sphere and Q be a cone. Then there exists a nonsingular, cyclide transition surface T between S and Q if and only if it is not the case that (1) S lies entirely outside the double cone Q o f which Q is one half and (2) 5' is tangent to Q at a point not on Q. When a sphere satisfies the conditions of Theorem 3.5, every cyclide tangent to both quadrics must contain a singularity lying between the quadrics. Fig. 9 demonstrates why this happens. Fig. 10 shows some of the remaining cases where a nonsingular, cyctide transition surface exists. For the axial natural quadric/axial natural quadric cases, we need the following two definitions. Definition 3.6 (Longitudinal conditions). Suppose we are given a sphere or an axial natural quadric Q and a plane P~. Two circles cj and c2 that lie in P~ satisfy the hmgitudinal conditions when (1) both cl and c2 are tangent to the cross section of Q in P~ in the distinct points ql and q2 respectively, (2) when Q is an axial natural quadric, the line qlq~2 is perpendicular to Q's axis, (3) one of the circles, say c2, does not lie entirely outside the other, and
Fig. 8. This figure illustrates the sphere/cylinder cases.
x
tf
"",
\
.x
, f~ \ C . ~" t
I
~
C,.,\ \\ /I
a.
Fig. 9. This figure illustrates the special sphere/cone case where no nonsingular transition surface exists.
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/"-\'
,•
\\
// ', ,'
~2
'
'", ,' //
°k/6
t" ' " , ,'
Fig. 10. This figure illustrates the sphere/cone cases.
(4) in 19e, c2 lies outside and does not contain Q, and • when Q is a sphere, cl contains Q, or • when Q is an axial natural quadric, ct lies partially inside and partially outside
(7. In Definition 3.7 below, we take a given normal to the plane to be the outward pointing normal, so the inside and the outside of the plane refer to the regions of three space that the plane determines.
Definition 3.7 (Latitudinal conditions).
Suppose we are given a natural quadric Q and a plane 19h. Two circles el and e2 that lie in 19h satisfy the latitudinal conditions when (1) both cl and c2 are tangent to the cross section of Q in Ph in the distinct points ql and q2 respectively, (2) when Q is an axial natural quadric, the line ql q~ is perpendicular to Q ' s axis, (3) neither cl nor c2 lies inside the other, and (4) in 19h, one of the following three conditions between the circles and Q holds: • both cl and c2 lie outside and do not contain Q, • both Cl and e2 lie outside and contain Q (implying Q is a sphere), or • both ct and c2 lie inside or partially inside Q. T h e o r e m 3.8. Let Ql and Q2 be axial natural quadrics. Then there exists a nonsingular, cyclide transition surface T between Qt and Q2 if and only if the axes of both quadrics lie in a plane t 9, and there exists two circles el and ¢2 in 19 satisfying either (1) the longitudinal conditions with both quadrics where cl lies partially inside and partially outside each quadric and c2 lies entirely outside both quadrics, or (2) the latitudinal conditions with both quadrics where el is tangent to Ql at a point inside c2 if and only if cj is tangent to Q2 at a point inside c2.
3.1. A summary of existence results Fig. 11 summarizes the existence results for nonsingular, cyclide transition surfaces. Here 0 means "never", 1 means "always", and e is a small number.
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T.S. [I Plane [Sphere Plane 1 1 Sphere 1 1 Cylinder 1 - e 1 Cone 1-
135
Cylinder Cone 1 -e 1 -e 1
1-e
Fig. 11. Summary of existence results for nonsingular, cyclide transition surfaces.
a.
b°
C.
Fig. 12. (a) Two intersecting spheres (b) Blended (c) Joined.
3.2. On simultaneous blends and joins In this section, we state the result that simultaneous blends and joins between two quadrics can only exist in the sphere/sphere case. Again for a proof, we refer the interested reader to (Allen and Dutta, 1997a). T h e o r e m 3.9. Suppose Q l and 0,2 are two natural quadrics and T is a nonsingular,
cyclide transition surface connecting Q, 1 and Q2. Except in the case that Ql and Q,2 are two intersecting spheres where neither sphere contains the other, we get • i f T is a join, then there exists no blends between Ql and Q2, and • i f T is a blend, then there exists no joins between Q1 and Q,2. If Q,I and Q,2 are two intersecting spheres and neither sphere contains the other, then there exists both blends and joins connecting the spheres. For an example of how two intersecting spheres can be both blended and joined, see Fig. 12. Using the results of the previous section, Theorem 3.9, and the results in (Allen and Dutta, 1997c), it is an easy matter to determine • when a nonsingular, cyclide transition surface exists between two quadrics, and • whether a nonsingular, cyclide transition surface is a blend or a join.
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Fig. 13. A parabolic cone/cylinder blend shown in one of the planes of symmetry.
4. Parabolic cyclide transition surfaces Parabolic cyclides can be used to extend the number of cases in which two natural quadrics can be blended or joined. In this section, we examine the possibility of the existence of nonsingular, parabolic cyclide transition surfaces between two natural quadrics. Note, the definition of a parabolic cyclide transition surface follows naturally from Definition 2.1. See Fig. 13 for a picture of a parabolic cyclide blend. We state without proof the necessary and sufficient conditions for the existence of a nonsingular, parabolic cyclide transition surface in each quadric/quadric case. For more detail, we refer the interested reader to (Allen and Dutta, 1997a).
Theorem 4.1. No nonsingular, parabolic cyclide transition surfaces involving planes are possible. Theorem 4.2. Two spheres Sl and $2 can be connected by a nonsingular, parabolic cyclide transition surface T if and only if neither sphere contains the other When Sl and $2 intersect, T is a blend. Theorem 4.3. Two axial natural quadrics can be connected by a parabolic cyclide if and only if the axes of the quadrics lie in a common plane P, and (1) in P, the quadrics are both tangent to the same line l, (2) in P, both quadrics lie on the same side of l, and (3) neither quadric contains the other. As an immediate consequence of Theorem 4.3, we get the following result.
Theorem 4.4. No nonsingular, parabolic cyclide transition surface between two cylinders is possible. In the constructive parts of the proofs of the previous theorems, it is always possible to find a nonsingular, parabolic cyclide that contains the nonsingular, parabolic cyclide
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_ (
a.
,
\
~
_,
C/
o
137
_,
b.
Fig. 14. This figure, drawn in the plane P, illustrates the sphere/cone parabolic cyclide cases.
P.T.S. Plane Sphere Cylinder
IIPlane I Sphere 0 0
0
0 1/2
Cylinder Cone 0 0 0
Fig. 15. Summary of existence results for nonsingular, parabolic cyclide transition surfaces. transition surface we seek. In the sphere/axial natural quadric case, this is not true. See for example, Fig. 14(a). Note, in the blends case, the requirement that the intersection curve wraps around the axis of the axial natural quadric allows us to always find a nonsingular, parabolic cyclide containing the transition surface we seek. See Fig. 14(b).
Theorem 4.5. Given a sphere S and an axial natural quadric Q, let P be a plane containing the center of S and the axis of Q. Then S and Q can be connected by a parabolic cyclide if and only if (1) in P, S and Q are both tangent to the same line l, and (2) in P, S and Q lie on the same side of l. Fig. 15 summarizes the existence results for nonsingular, parabolic cyclide transition surfaces. Here 0 means "never", 1 means "always", and e is a small number. Next we consider the possibility of simultaneous parabolic blends and joins. Since a nonsingular, parabolic cyclide transition surface is a blend if and only if the quadrics being connected satisfy certain intersection requirements, we get the following theorem.
Theorem 4.6. Suppose Q1 and Q2 are two natural quadric and T is a nonsingular, parabolic cyclide transition surface connecting Ql and Q2. Then • if T is a join, then there exists no blends between Q t and Q2, and • if T is a blend, then there exists no joins between Q l and Q2.
5. Inscribed sphere condition In this section, we consider what the inscribed sphere condition means for blending and joining with cyclides. Previous work (Johnstone and Shene, 1994; Pratt, 1990; Shene,
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1992) shows the result that there exists a cyclide or parabolic cyclide transition surface (possibly singular) between two cones with nonparallel axes if and only if the double cones contain a common inscribed sphere. Theorem 5.1 shows when the common inscribed sphere condition implies the existence of a nonsingular, cyclide transition surface. One way to prove this theorem uses the offset property of cyclides closely following the proof presented by Pratt (1990) attributed to M. Sabin. However, some additional work is required to ensure that the transition surface we construct in Theorem 5.1 is nonsingular. T h e o r e m 5.1. Suppose Q1 and Q2 are axial natural quadrics that contain a common inscribed sphere S. Then there exists a nonsingular, cyclide transition surface T between Ql and Q2 if and only if Ql and Q2 are not tangent along a common line. Now we show that a cyclide blend or join of two axial natural quadrics with intersecting and nonparallel axes implies a common inscribed sphere. An outline of the proof of this theorem can be found in (Allen and Dutta, 1997a). T h e o r e m 5.2. Suppose Ql and Q2 are axial natural quadrics that are connected by a nonsingular, cyclide transition surface T. If Ql and Q2 satisfy the following two conditions: • the axes intersect, and • neither quadric contains the other, then they have a common inscribed sphere. Notice that when two axial natural quadrics do not have intersecting axes, or when the axes are parallel and one of the quadrics contains the other, the quadrics cannot have a common inscribed sphere. However, in these cases it may still be possible for the quadrics to be joined by a nonsingular, cyclide transition surface. See Fig. 16. Finally, we state the relationship of the common inscribed sphere condition to the existence of nonsingular, parabolic cyclide transition surfaces.
a.
Fig. 16. This figure is drawn in the plane of the axes of two axial natural quadrics QI and Q2 that have no common inscribed sphere. (a) Shows quadrics with nonintersecting axes. (b) Shows quadrics with intersecting, parallel axes where QI contains Q2.
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139
Theorem 5.3. Suppose that there exists a nonsingular, parabolic cyclide transition suro .face T between two axial natural quadrics Ql and Q2. Further suppose Q1 and Q2 intersect. Then Ql and Q2 have a common inscribed sphere (and T is a blend). Theorem 5.4. If two axial natural quadrics have a common inscribed sphere and are tangent along a line that is a common line o f curvature o f both quadrics, then there exists a nonsingular, parabolic cyclide transition surface T connecting the quadrics. Further, T is a blend.
6. Tests for existence of cyclide transition surfaces In this section we present tests that determine when a nonsingular, cyclide transition surface between two natural quadrics is possible. These results follow from the work presented in the previous sections. Also, we include refinements that allow the designer that determine when the cyclide transition surface will be a blend. These tests are important in the context of the implementation of cyclide blending and joining in geometric modeling systems.
Plane/Plane: There always exists a nonsingular, cyclide transition surface T between two planes. T is never a blend. There never exists a nonsingular, parabolic cyclide transition surface between two planes. Plane/Sphere: There always exists a nonsingular, cyclide transition surface T between a sphere S and a plane P. Suppose S has center c and radius r. Then T is a blend if and only if the distance from c to P is less than r. There never exists a nonsingular, parabolic cyclide transition surface between a sphere and a plane. Plane/Cylinder: Let N be a normal to the plane Then there exists a nonsingular, the cylinder if and only if N • V There never exists a nonsingular, and a cylinder.
and L be the axis of the cylinder with direction V. cyclide transition T surface between the plane and ~ 0. T is a blend. parabolic cyclide transition surface between a plane
Plane/Cone: Let c~ be the half angle of the cone Q, and/3 be the angle that the axis of Q makes with the normal to the plane P. Then there exists a nonsingular, cyclide transition surface T between Q and P if and only if when Q's vertex lies on P, c~
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and a cone.
Sphere/Sphere: Let SI and $2 be two spheres, where r~ is the radius and ci is the center of S~. There always exists a nonsingular, cyclide transition surface T between $1 and $2. T can be a blend when dist(cl, c2) < rl + r2 (when the spheres intersect). Further suppose rl <~ re. Then there exists a nonsingular, parabolic cyclide transition surface T between SI and $2 if and only if dist(cl, c2) + rl > r2. (T exists when neither sphere contains the other.) T is a blend if and only if dist(cl, c2) < rl + r2.
Sphere/Cylinder: Suppose a sphere S has radius R~ and a cylinder Q has radius Rq. Further suppose that the distance from the center of S to the axis of Q is d. There always exists a nonsingular, cyclide transition surface T between S and Q. T is a blend if and only if Rs - Rq ~ d. There exists a nonsingular, parabolic cyclide transition surface T between S and Q if and only if IRs - Rql = d. T is a blend if and only if R~ - Rq = d.
Sphere/Cone: Given a sphere S with center c and radius r, and a cone Q. Let P be the plane containing c and Q's axis. Let ~) be the double cone of which Q is one half. So in P, Q looks like two rays 11 and 12, and ~) looks like two lines/] and/~. Let N be the unit vector in the direction of Q's axis. Then, there exists a nonsingular, cyclide transition surface T between S and Q, except when all of the following three conditions are true using either/] or/2 in place of line L: (1) The distance from c to L is r. (So S is tangent to Q at a point p. Let V0 be the vector from p to c and let V1 be the vector from Q's vertex to p.) (2) V0 • N > 0. (So S and Q lie on the same side of L.) (3) V1 • N < 0. (So S is tangent to ~) at the point p which is not on Q.) T is a blend if and only if the distance from from c to Ii and the distance from c to 12 are both less than or equal to r. There exists a nonsingular, parabolic cyclide transition surface T between S and Q if and only if the following two conditions are true using either/] or/2 in place of the line L: (1) The distance from c to L is r. (So S is tangent to Q at a point p. Let V0 be the vector from p to c.) (2) V0 • N > 0. (So S and Q lie on the same side of L.) T is a blend if and only if the distance from c to either Ii or 12 equals r and the distance from c to the other ray is less than r.
Cylinder/Cylinder: Let Q1 and Q: be two cylinders with radii r~ such that rl /> Y2. Then there exists a nonsingular, cyclide transition surface T between Q1 and Q2 if and only if the axes of the cylinders are coplanar and
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• when the axes are parallel and d is the distance between the axes, either d + r 2 < rl (so Ql contains Q2) or rl + r2 < d (so that neither cylinder contains the other), or
• when the axes intersect, r~ = r2. T is n e v e r a blend. There n e v e r exists a nonsingular, parabolic cyclide transition surface between two cylinders. Cylinder/Cone: 2
Let Q1 be a cone with half angle ~ and vertex v, and let Q2 be a cylinder with radius r. Let Q I be the double cone of which Q1 is one half. Then there exists a nonsingular, cyclide transition surface T connecting Q1 and Q2 if and only if (1) the quadrics' axes are coplanar and hence lie in a plane P, (2) in /9, neither of the rays of Q l ' s cross section lies on a line of Q2's cross section, (3) either Ql'S and Qz's axes intersect or v lies inside Q2, and (4) Q1 and Q 2 have a common inscribed sphere, so either • Q j ' s axis lies on Q2's axis, or • Q j ' s axis and Q2's axis intersect in exactly one point that is distance d from v, and d sin c~ = r. T is a blend if and only if v lies inside Q2. There exists a nonsingular, parabolic cyclide transition surface T connecting Ql and Q2 if and only if (1) the axes of the quadrics intersect and hence are contained in a plane t9, and (2) in /9, one of the rays of Q l ' s cross section lies on a line from Q2's cross section. T is a blend. Cone/Cone:
Let Qj and Q2 be two cones with half angles cq and c~2 respectively. Let Q~ be the double cone of which Q~ is one half. There exists a nonsingular, cyclide transition surface T connecting Q1 and Qz if and only if (1) the quadrics' axes are coplanar and hence lie in a plane P, (2) in/9, neither of the lines from Q l ' s cross section equals a line from Qz's cross section, (3) one__of the following three conditions is true: • Q l ' s axis equals Q2's axis, • Q ll's axis is parallel but not equal to Q2's axis, and c~1 = c~2, or • Q l ' S axis and Q 2 ' s axis intersect in exactly one point p that is distance d~ from Qi's vertex and dl sin C~l = d2 sin c~2, and
2 A more direct test similar to the test used in the cone/cone case that constructs the extreme circles of a tangent cyclide can be readily derived.
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(4) one of the following two conditions is true: • Q1 and Q2 intersect at their vertices only. So either one cone contains the other or neither cone contains the other. • Q1 and Q2 have distinct vertices. Let 5' be the line through the cones vertices, then either the following three conditions are true: 5' lies entirely outside both cones, - in P, Q1 and Q2 lie on the same side of S, and there exist two circles cl and c2 that satisfy the longitudinal conditions with both cones where c2 lies outside both cones. (See below for how to construct these circles.) or the following two conditions are true: S lies partially inside both cones, and there exist two circles el and c2 that satisfy the latitudinal conditions with both Q1 and Qz where cl is tangent to QI at a point inside c2 if and only if Cl is tangent to Q2 at a point inside c2. (See below for how to construct these circles.) T is a blend if and only if the circles cl and c2 satisfy the latitudinal conditions and in P, Q1 and Q2 intersect in exactly two points. There exists a nonsingular, parabolic cyclide transition surface T connecting Q1 and Q2 if and only if the following four conditions are true: (1) The quadrics' axes are coplanar and hence lie in a plane P. (2) In P, one of the lines from Q l ' s cross section equals a line from Q2's cross section. Call this line L and let N be a vector perpendicular to L. (3) Let di be the direction of Qi's axis. Either Ql'S axis intersects Q2's axis in a point not on L or dl • d2 < 0. (So neither Ql nor Q2 contains the other.) (4) The signs of N . dl and N . d2 are equal. (So Q1 and Q2 lie on the same side of L.) T is a blend if and only if Q l ' s axis intersects Q2's axis. It is not hard to show that, if there exists a cyclide tangent to two axial natural quadrics with its extreme circles in a plane of symmetry in a region determined by the quadrics, then there exists an infinite family of tangent cyclides with extreme circles in the same region. Using this fact, we are able search for the circles in the cone/cone test by simply trying a pair of circles that satisfy the latitudinal or the longitudinal conditions with both quadrics in each region of interest. If a pair does not satisfy the conditions we require in a particular region, then no nonsingular, cyclide transition surface exists in that region. Further, notice that by picking one of the circles, the other circle, if it exists, is completely determined.
7.
Implemented
example
and
summary
This paper was motivated by a need to state explicitly necessary and sufficient conditions for when a nonsingular, cyclide transition surface between two natural quadrics exists. We provide simple, easily implemented tests for these conditions. Except in the
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II a)
b)
c)
Fig. 17. Shows two views in the construction of a water faucet using nonsingular, cyclide transition surfaces. (a) Shows the natural quadrics used in the model. (b) Adds joins for the handle and spigot. (c) Adds blends to get the finished faucet.
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cone/cone case, no conditions had been explicitly stated in previous work, although some constructions indicated which tests would need to be carried out before the construction could continue. Here we show that nonsingular, cyclide transition surfaces are only possible when the indicated test conditions are satisfied and which type, blend or join, of nonsingular, cyclide transition surface we get. Further, we show that except in the case of intersecting spheres, two quadrics in given positions cannot be both blended and joined. The work here is carried out for both ring and parabolic cyclide transition surfaces concluding the investigations we began in (Allen and Dutta, 1997b, 1997c). Fig. 17 shows the construction of a water faucet made from nonsingular, cyclide transition surfaces. Each frame contains the same objects shown in two different views. Note, a small portion of the sphere on the spigot is used so that there are no tangent discontinuities where the cyclides meet other objects. As we have shown, except in the case of two axial natural quadrics, we can almost always expect to be able to find a nonsingular, cyclide transition surface between two natural quadrics. This is further evidence of the utility of cyclides in CAD/CAM. Already cyclides have proven useful in a number of engineering applications including cable harness design, motion planning, surface composition, and pipe joining. The results reported in this paper are currently being implemented in the DESIGNBASE modeler of Ricoh Corporation. The methods that the software uses to construct nonsingular, cyclide transition surfaces, when they exist, closely follow the constructive portions of the proofs as found in (Allen and Dutta, 1997a, 1997b, 1997c).
Acknowledgments We gratefully acknowledge the financial support received from AFOSR grant F4962093-1-0419 and AFOSR grant F49620-95-1-0209.
References Allen, S. and Dutta, D. (1997a), Cyclide transition surfaces: blending and joining, Technical report, UM-MEAN-97-05, Department of Mechanical Engineering, University of Michigan. Allen, S. and Dutta, D. (1997b), Cyclides in pure blending I, Computer Aided Geometric Design 14, 51-75. Allen, S. and Dutta, D. (1997c), Cyclides in pure blending II, Computer Aided Geometric Design 14, 77-102. Chandru, V., Dutta, D. and Hoffmann, C. (1989), On the geometry of Dupin cyclides, The Visual Computer 5, 277-290. Johnstone, J.K. and Shene, C.-K. (1994), Dupin cyclides as blending surfaces for cones, in: Fisher, R.B., ed., Mathematics o f Surfaces V, Oxford University Press, 3-29. Pratt, M.J. (1990), Cyclides in computer aided geometric design, Computer Aided Geometric Design 7, 221-242. Pratt, M.J. (1995), Cyclides in computer aided geometric design II, Computer Aided Geometric Design 12, 131-152.
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Shene, C.-K. (1992), Planar intersection and blending of natural quadrics, Ph.D. thesis, Department of Computer Science, Johns Hopkins University. Srinivas, Y.L. and Dutta, D. (1994), Blending and joining using cyclides, ASME Trans. Journal of Mechanical Design 116, 1034-1041.