The Intersection of Two Ringed Surfaces and Some Related Problems

Transcription

1 Graphical Models 63, ) doi: /gmod , available online at on The Intersection of Two Ringed Srfaces and Some Related Problems Hee-Seok Heo and Sng Je Hong Department of Compter Science, POSTECH, Pohang , Korea Joon-Kyng Seong and Myng-Soo Kim 1 School of Compter Science and Engineering, Seol National University, Seol , Korea and Gershon Elber Compter Science Department, Technion, IIT, Haifa 3000, Israel Received Agst 6, 001 We present an efficient and robst algorithm to compte the intersection crve of two ringed srfaces, each being the sweep C generated by a moving circle. Given two ringed srfaces C1 and vc v, we formlate the condition C 1 C v i.e., that the intersection of the two circles C1 and C v is nonempty) as a bivariate eqation λ,v) = 0 of relatively low degree. Except for redndant soltions and degenerate cases, there is a rational map from each soltion of λ,v) = 0 to the intersection point C1 C v. Ths it is trivial to constrct the intersection crve once we have compted the zero-set of λ,v) = 0. We also analyze exceptional cases and consider how to constrct the corresponding intersection crves. A similar approach prodces an efficient algorithm for the intersection of a ringed srface and a rled srface, which can play an important role in accelerating the ray-tracing of ringed srfaces. Srfaces of linear extrsion and srfaces of revoltion redce their respective intersection algorithms to simpler forms than those for ringed srfaces and rled srfaces. In particlar, the bivariate eqation λ,v) = 0 is redced to a decomposable form, f ) = gv)or f) gv) = r), which can be solved more efficiently than the general case. c 001 Elsevier Science USA) 1 Corresponding athor. mskim@cse.sn.ac.kr /01 $35.00 c 001 Elsevier Science USA) All rights reserved. 8

2 INTERSECTION OF TWO RINGED SURFACES 9 1. INTRODUCTION A ringed srface is a sweep srface generated by a circle moving nder translation, rotation, and scaling; ths it can be decomposed into a one-parameter family of circles [10]. Srfaces of revoltion, Dpin cyclides, and canal srfaces all belong to the class of ringed srfaces [15, 19]. Additionally, early investigations of generalized cylinders were focsed on the sweeping of circles along a trajectory space crve. In short, we find the ringed srface in varios branches of three-dimensional shape modeling and processing. Ot of many related reslts, we mention van Wijk s ray-tracing algorithm for canal srfaces [] and Nishita and Johan s scan-line algorithm for rendering ringed srfaces [17]; the examples in these two papers inclde the representation of varios interesting three-dimensional shapes by canal and ringed srfaces. The intersection of two freeform srfaces is another problem of general importance in geometric and solid modeling. For example, Boolean operations on bondary models rely on accrate comptation of the intersection crves between the srfaces of the two original solids. Many algorithms have been sggested for compting the intersection of two srfaces. However, it remains a challenging task to develop an algorithm that can compte the intersection crve of two arbitrary freeform srfaces accrately, robstly, efficiently, and withot ser intervention [9]. Becase of the difficlty of dealing with general freeform srfaces, many previos approaches to the intersection problem have been focsed on srfaces of common special types sch as the classic CSG primitives planes, cylinders, cones, and tori) [11, 16, 1]. In this paper, we consider the intersection of two ringed srfaces. The problem is more difficlt than the case of natral qadrics or the sal CSG primitives; in fact, cylinders, cones, and tori may be considered the simplest types of ringed srface. On the other hand, ringed srfaces are simpler than general freeform srfaces, and so it is reasonable to look for a simpler soltion to the intersection problem in this case. Dpin cyclides have both implicit and parametric representations. Martin et al. [15] compted the intersection of cyclides with planes, qadrics, and other cyclides by sbstitting the parametric eqation of one srface into the implicit eqation of the other. Johnstone [10] presented an algorithm for intersecting a ringed srface with a cyclide, sing circle decomposition and a transformation called the inversion map. Johnstone reformlated the intersection of a ringed srface and a cyclide as the intersection of a rled srface and a one-parameter family of cyclides. The parametric eqation for each line of the rled srface is then sbstitted into the implicit eqation of the corresponding cyclide in the family, prodcing an algebraic eqation in two variables. However, Johnstone leaves open the problem of intersecting two ringed srfaces. In this paper, we attack this problem sing an approach similar to the one taken in or previos work on the intersection of two rled srfaces [8]. Given two ringed srfaces C1 and vc v, each represented as a one-parameter family of circles, we formlate the condition C1 Cv sing a bivariate eqation λ,v) = 0 of relatively low degree. There is a rational map from the zero-set of λ,v) = 0tothe intersection crve C1 ) vc v ), except for some redndant or degenerate cases. We consider each circle to be the intersection of a sphere and a plane i.e., C1 = O 1 P 1 and C v = Ov Pv ). From the two spheres O 1 and Ov,wecanfind a plane Pv 1 that contains the intersection circle C1 v = O 1 Ov. Then the three planes P 1, Pv, and Pv 1 determine a common intersection point x,v). When we sbstitte the point eqation x,v) into the implicit eqation of O1 or Ov, we get an eqation λ,v) = 0 of relatively low

3 30 HEO ET AL. degree. Ths the intersection problem is reformlated as the problem of finding the zero-set of λ,v) = 0. An efficient and robst algorithm for tracing all branches of the zero-set can be fond in a recent srvey paper of Patrikalakis and Maekawa [18] and the references cited in the paper.) The problem-redction scheme employed in this paper is not limited to the intersection of two ringed srfaces. A similar techniqe has been sed in the intersection of two rled srfaces [8]. In this paper, we also present an algorithm for the intersection of a ringed srface and a rled srface. More generally, Kim and Elber [13] reviewed the paradigm that consists of converting varios interesting geometric problems to a search for the zero-set of constraint eqations in the parameter space of inpt crves and srfaces. Some applications inclde the comptation of sweeps and Minkowski sms of freeform crves and srfaces. Nonrational bisectors for freeform crves and srfaces have also been compted sing the same method of problem redction [4, 5]. Elber and Kim [6] present an efficient algorithm that solves mltivariate rational eqations, the bivariate case of which was sed in the implementation of algorithms presented in this paper. Among all these applications, the intersection of two ringed srfaces is the most interesting, becase at first it looks to be a nonlinear problem, whereas other problems have rather obvios linear strctres in their constraint eqations. The constrction of three planes in a general position is the key idea for redcing this intersection problem to one of finding a zero-set in the parameter space of the two original ringed srfaces. Compting the self-intersection of a freeform srface is sally more difficlt than the case of intersecting two different srfaces. Two identical srfaces are tangential everywhere; ths, a sbdivision-based approach wold not work for this case. On the other hand, or algorithm can deal with the self-intersection of a ringed srface qite easily, with a slight modification to the original algorithm. Let C and v C v be two different parameterizations of the same ringed srface. Clearly, v = 0 is a trivial soltion of λ,v) = 0 since C and C v represent the same circle when = v. This paper shows that the zero-set of λ,v) = λ,v)/ v) 4 = 0 corresponds to the self-intersection of a ringed srface. Maekawa et al. [14] sggested a method of designing a pipe srface a special case of a ringed srface) that has no local or global self-intersection. Thogh the algorithm does not constrct the intersection crve explicitly, the intersection test between two end circles of the same radis) also ses the principle of three planes, where the third plane P1 v is now the bisector plane between the centers of the two circles. Srfaces of linear extrsion and srfaces of revoltion are among the most freqently sed srfaces in designing simple three-dimensional shapes. They belong to the class of rled and ringed srfaces. Using the reslt of this paper and also that of Heo et al. [8] for intersecting two rled srfaces, we can deal with the cases of intersecting srfaces of linear extrsion and srfaces of revoltion. In addition to that, there is a more efficient way of compting the intersection of these srfaces since they provide some special strctres that we can tilize for speeding p the comptation of the zero-set λ,v) = 0. For example, in the case of intersecting two srfaces of linear extrsion, the bivariate eqation can be decomposed in the form of λ,v) = f ) gv) = 0. Moreover, in the case of intersecting two srfaces of revoltion, the zero-set of λ,v) = 0 can be redced to f) gv) = r), where f) and gv) are space crves and r) is a scalar fnction. In the case of intersecting a srface of revoltion and a srface of linear extrsion, the space crve f) redces to a line l), which can frther simplify the zero-set comptation.

4 INTERSECTION OF TWO RINGED SURFACES 31 The rest of this paper is organized as follows. In Section, the intersection of two ringed srfaces is redced to the problem of compting the zero-set of a bivariate fnction λ,v) = 0. Redndant soltions from this zero-set approach are discssed in Section 3, and degenerate cases are analyzed in Section 4. Section 5 introdces an algorithm to compte the self-intersection of a ringed srface. Section 6 presents an algorithm for the intersection of a ringed srface and a rled srface, and Section 7 considers the intersection of two simple sweep srfaces sch as srfaces of revoltion and srfaces of linear extrsion. Finally, Section 8 concldes this paper.. PROBLEM REDUCTION Let S 1, s) and S v, t) be two ringed srfaces, each of which is defined as the nion of a one-parameter family of circles S 1, s) = C 1 s) and S v, t) = v C v t), where C1 s) and Cv t) are circles parameterized by s and t, respectively. Assme that the circle C1 has center p 1) = p 1,x ), p 1,y ), p 1,z )) and radis r 1 ) and, moreover, that C1 is contained in the plane P 1 with normal n 1) = n 1,x ), n 1,y ), n 1,z )). Let O1 denote a sphere with center p 1 ) and radis r 1 ). Then we have C 1 = P 1 O 1. Similarly, the circle C v and sphere Ov have their center p v) = p,x v), p,y v), p,z v)) and radis r v); moreover, the circle C v is contained in the plane Pv with normal n v) = n,x v), n,y v), n,z v)), and so we find that C v = Pv Ov. Ths the intersection S 1 S of or ringed srfaces is shown to be the nion of a twoparameter family of circle circle intersections S 1 S = v C1 Cv = v P 1 O1 ) P v O v ) = v P 1 P v ) O 1 O v ) = v P 1 P v ) C v 1 = v P 1 P v ) P v 1 ) O 1 = v P 1 P v ) Pv 1 O 1 = v P 1 P v ) P v 1 ) Ov = v P 1 P v ) Pv 1 O v, where C1 v is the circle formed by the intersection of the two spheres O 1 and Ov, and Pv 1 is the plane containing the circle C1 v. A simple calclation shows that the plane P1 v contains the following point: p 1 ) + p v) + r 1 ) r v) p 1 ) p v) p v) p 1 )). 1)

5 3 HEO ET AL. Moreover, the difference vector p 1 ) p v) is normal to the plane P1 v. Note that the inner prodct of this normal vector and the position vector of Eq. 1) prodce the scalar qantity p 1 ) p v) + r ) r 1 v). The intersection point ˆx,v), ŷ,v), ẑ,v)) of the three planes P1, Pv the following matrix eqation: n 1 ) ˆx,v) n 1 ), p 1 ) n v) ŷ,v) = n v), p v) p 1 ) p v) ẑ,v) p 1 ) p v) + r v) r 1 ), and Pv 1 satisfies. ) When the three vectors n 1 ), n v), and p 1 ) p v) are linearly independent, the above eqation has a niqe soltion ˆx,v), ŷ,v), ẑ,v)). We now define bivariate fnctions w,v), x,v), y,v), z,v) as follows: n 1 ) w,v) = n v), p 1 ) p v) n 1 ), p 1 ) n 1,y ) n 1,z ) x,v) = n v), p v) n,y v) n,z v), µ,v) p 1,y,v) p 1,z,v) n 1,x ) n 1 ), p 1 ) n 1,z ) y,v) = n,x v) n v), p v) n,z v), p 1,x,v) µ,v) p 1,z,v) n 1,x ) n 1,y ) n 1 ), p 1 ) z,v) = n,x v) n,y v) n v), p v), p 1,x,v) p 1,y,v) µ,v) where and µ,v) = p 1) p v) + r v) r 1 ) p 1,x,v) = p 1,x ) p,x v), p 1,y,v) = p 1,y ) p,y v), p 1,z,v) = p 1,z ) p,z v). By Cramer s rle, the intersection point ˆx,v), ŷ,v), ẑ,v)) can then be compted from ˆx,v) = x,v) w,v), y,v) ŷ,v) = w,v), ẑ,v) = z,v) w,v).

6 INTERSECTION OF TWO RINGED SURFACES 33 a) FIG. 1. The intersection of two cylinders: a) two cylinders and b) λ fnction. The bivariate fnctions w,v), x,v), y,v), and z,v) are polynomial if the position crves p 1 ) and p v), the radis fnctions r 1 ) and r v), and the normal vectors n 1 ) and n v) are all polynomial; if the position crves, radis fnctions, and normal vectors are rational, the bivariate fnctions are rational. The condition for the intersection point x,v) w,v), y,v) w,v), z,v) w,v) ) to be located on the sphere O 1 can be formlated as follows: λ,v) = x,v), y,v), z,v)) w,v)p 1 ) w,v)r1 ) = 0. When the vectors n 1 ), n v), and p 1 ) p v) are linearly independent, the eqation λ,v) = 0 is a necessary and sfficient condition for two circles C1 and Cv to have a nonempty intersection point. For each point,v) in the zero-set of λ,v) = 0, the corresponding intersection point can be compted by the rational map x,v) w,v), y,v) w,v), z,v) w,v) ), which is the soltion of Eq. ). Figres 1 and show some examples of intersecting two ringed srfaces. a) b) FIG.. Intersecting two ringed srfaces.

7 34 HEO ET AL. EXAMPLE 1. Let two ringed srfaces S 1, s) = C1 and S v, t) = v C v be defined by the following fnctions: p 1 ) = 0, 4, 0), r 1 ) = 1, n 1 ) = 0, 1, 0), p v) = 4v, 0, 1), r v) = 1, n v) = 1, 0, 0). Note that S 1, s) and S v, t) are two cylinders intersecting orthogonally see Fig. 1a). From these definitions we can formlate the matrix eqation v 4 1 x,v) w,v) y,v) w,v) z,v) w,v) 4 = 4v, 8 8 8v + 8v 1 where the bivariate fnctions w,v), x,v), y,v), and z,v) are defined as follows w,v) = 1, x,v) = 4v, y,v) = 4, z,v) = 8 8 8v + 8v + 1. Finally, the bivariate fnction λ,v) may be compted by means of the eqation: λ,v) = 4v ) v + 8v = 0. ) From the zero-set of λ,v) = 0, we can now constrct the intersection crve x,v) ) see Fig. 1). z,v) w,v) w,v), y,v) w,v), 3. REDUNDANT SOLUTIONS When the three vectors n 1 ), n v), and p 1 ) p v) are linearly dependent, we have w,v) = n 1 ) n v), p 1 ) p v) =0. Ths the condition λ,v) = 0 redces to λ,v) = x,v), y,v), z,v)) w,v)p 1 ) w,v)r1 ) = x,v), y,v), z,v)) = 0, which is eqivalent to x,v) = y,v) = z,v) = 0.

8 INTERSECTION OF TWO RINGED SURFACES 35 As we discss below, when w,v) = 0, there are some redndant cases where λ,v) = 0 holds bt the two circles C1 and Cv have no intersection. Conseqently, when the three vectors n 1 ), n v), and p 1 ) p v) are linearly dependent, λ,v) = 0 is only a necessary, bt not a sfficient, condition for the two circles C1 and Cv to have intersection at a real affine point. The condition w,v) = x,v) = y,v) = z,v) = 0 means that the three planes P1, Pv, and Pv 1 meet either at infinitely many real affine points or at points at infinity; ths the following two cases are the only ones that we need to consider for intersection at real affine points: 1. The two planes P1 and Pv are coplanar.. The three planes P1, Pv, and Pv 1 intersect in a line. The first case can be classified by the following two eqations:,v) = n 1 ) n v) = 0, δ,v) = n 1 ), p 1 ) p v) =0. The circle circle intersection problem is then essentially redced to the planar case. When the two planes P1 and Pv are identical and when, additionally, p 1) = p v), the two circles C1 and Cv intersect if and only if r 1 ) = r v); in this case, the two circles are identical. When the two planes P1 and Pv overlap and p 1) p v), the third plane P1 v is orthogonal to both P1 and Pv. The circle circle intersection points are then located on the intersection line P1 Pv 1 of two planes. This line is parallel to n 1) p 1 ) p v)), and it contains the point m,v) = p 1) + p v) + r 1 ) r v) p 1 ) p v) p v) p 1 )), which is also the foot point of the perpendiclar from p 1 ) to the intersection line. The two intersection points C1 Cv are compted as follows P v m,v) ± r1 ) m,v) p 1) Now, when the three planes P1, Pv are not parallel, we have n 1 ) p 1 ) p v)) n 1 ) p 1 ) p v))., and Pv 1 intersect in a real affine line, bt P 1 and λ,v) = 0, w,v) = 0, and,v) > 0. The common intersection line parameterized by t) can be written L v t) = m,v) + t n 1 ) n v), where m,v) is the foot point of the perpendiclar from p 1 ) to the line. Note that the foot point m,v) satisfies the following eqations: n 1 ), m,v) p 1 ) =0, n v), m,v) p v) =0, n 1 ) n v), m,v) p 1 ) =0.

9 36 HEO ET AL. Since the three vectors n 1 ), n v), and n 1 ) n v) are linearly independent, the foot point m,v) has a rational representation in and v. The two circles C1 and Cv intersect in two real affine points if and only if r 1 ) m,v) p 1) = r v) m,v) p v). And the two intersection points C1 Cv are then compted as m,v) ± r1 ) m,v) p n 1) 1 ) n v) n 1 ) n v). 4. DEGENERATE CASES In general, the soltion-set of λ,v) = 0 corresponds to a planar algebraic crve in the v-plane. However, in some degenerate cases, the soltion-set may contain the whole v-plane i.e., λ,v) 0). When each circle C1 of the first ringed srface S 1 intersects all the circles C v of the second ringed srface S,wehaveλ,v) 0. For example, the tors may be represented as a nion of cross-sectional circles; at the same time, it can also be represented as a nion of profile circles [11]. There are two more ways of representing the tors as a nion of circles, i.e., as a nion of Yvone-Villarcea circles.) When we intersect the two representations of a single tors with each other, each cross-sectional circle on one tors intersects all profile circles on the other, and vice versa. Conseqently, the reslting λ-fnction vanishes identically: λ,v) 0. Similarly, there are for different ways of decomposing a Dpin cyclide into a nion of circles: we can take either of the original srfaces as the intersection of the two ringed srfaces. Consider a ringed srface with a singlar point where all the circles intersect. A spindle tors, for example, has two sch singlar points.) When we place two ringed srfaces of this type so that their respective singlar points share the same position, each circle from one ringed srface will intersect all the circles on the other ringed srface, and vice versa. Conseqently, λ,v) 0. Two ringed srfaces may intersect at some points in addition to the singlar point. The zero-set of λ,v) 0 is of no se for the constrction of these intersection points, other than the singlar point. When two circles C1 and Cv intersect at a point other than the singlar point, the three planes P1, Pv, and Pv 1 intersect in a straight line throgh the intersection point and the singlar point. Ths we can apply the techniqes introdced in the previos section to sch degenerate cases. There are some other degenerate cases where the three normal vectors n 1 ), n v), and p 1 ) p v) are linearly dependent for all,v): that is, cases where λ,v) 0 and w,v) 0, and conseqently x,v) y,v) z,v) 0. As we have discssed in the previos section, there are essentially two cases to consider: 1. The two planes P1 and Pv overlap, for all and v.. The three planes P1, Pv, and Pv 1 intersect in a line, for all and v. It is easy to detect the first case, where the two ringed srfaces degenerate to planar regions contained in the same plane. The second case is more interesting. An example incldes the case of intersecting two tori with the same axis of rotation and the same major circle, bt with different minor radii.

10 INTERSECTION OF TWO RINGED SURFACES 37 The detection of degenerate cases involves checking whether certain bivariate fnctions vanish identically: F,v) 0. Assme that F,v) is a polynomial fnction of degree m, n) and that a niform sampling is made on the,v)-parameter domain: i,v j ), for i = 0,...,m and j = 0,...,n. When we have F i,v j ) = 0 at these m + 1)n + 1) parameter vales, we can garantee that F,v) 0. Elber and Kim [3] sed similar tests for the shape recognition of freeform crves and srfaces of special types sch as srfaces of revoltion, srfaces of linear extrsion, rled srfaces, and developable srfaces. 5. SELF-INTERSECTION OF A RINGED SURFACE Assme that a ringed srface S, t) = C t)isdefined by the center p) and radis r) of the circle C and by the normal n) of the plane P that contains the circle C. Then S, t) has a self-intersection if and only if C C v, for v. Assming frther that the two circles C and C v intersect at a real affine point ˆx,v), ŷ,v), ẑ,v)), we obtain the matrix eqation n) ˆx,v) nv) ŷ,v) = p) pv) ẑ,v) n), p) nv), pv) p) pv) + r v) r ) which is eqivalent to n) ˆx,v) n), p) n) nv) ŷ,v) = n), p) nv), pv). p) pv) ẑ,v) p) pv) + r v) r ) When the fnctions p), n), and r) are polynomial or rational, each of the terms n) nv), p) pv), n), p) nv), pv), and p) pv) + r v) r ) has a factor v), and we can dedce that v) n) n) nv) v p) pv) v ˆx,v) ŷ,v) = v) ẑ,v) n), p), n),p) nv),pv) v p) pv) + r v) r ) v) From the above eqation, it is clear that the terms w,v), x,v), y,v), and z,v)have a factor of v). Ths the following bivariate fnction has a factor of v) 4 : λ,v) = x,v), y,v), z,v)) w,v)p) w,v)r ). Now let s define a simplified fnction λ,v) = λ,v) v) 4..

11 38 HEO ET AL. a) FIG. 3. A self-intersecting ringed srface: a) a ringed srface and b) λ,v). The zero-set of λ,v) = 0 is the same as that of λ,v) = 0 except the soltions on the diagonal line = v. Moreover, since λ,v) is symmetric with respect to and v, the zero-set of λ,v) = 0 is also symmetric with respect to the diagonal line = v. Ths we need to compte the zero-set of λ,v) = 0 only when >v. Let a ringed srface S, t) = C t) bedefined by the following fnc- EXAMPLE 3. tions p) = r) = 3 i=0 ) 3 i 1 ) 3 i p i, i 1 ) + 1 4, n) = 1 6 p ), where p 0 = 1, 0, 0), p 1 =,, 1 4 ), p =,, 1 4 ), and p 3 = 1, 0, 1 ) are the control points of the cbic Bézier crve p). Figre 3 shows a self-intersecting ringed srface and a λ-fnction of the ringed srface. Note that the λ-fnction is symmetric with respect to and v, and there is a connected component of the zero-set in the region >v. 6. THE INTERSECTION OF A RINGED SURFACE AND A RULED SURFACE Let S 1, s) be a ringed srface defined by S 1, s) = C s), where the circle C has center p) = p x ), p y ), p z )) and radis r) and, moreover, that C is contained in the plane P with normal n) = n x ), n y ), n z )). Let O

12 INTERSECTION OF TWO RINGED SURFACES 39 denote a sphere with center p) and radis r). Then we have C = P O. Now, let S v, t) be a rled srface defined by S v, t) = qv) + t dv), where qv) is a generating crve and dv) 0 is a direction vector of the rled srface. When the two srfaces S 1, s) and S v, t) intersect, the rling line qv) + tdv), parameterized by t, intersects the plane P ; n), qv) + t dv) p) =0, which prodces the parameter vale of t at the intersection point: t,v) = The intersection point itself is given as n), p) qv). n), dv) n), dv) qv) + n), p) qv) dv). n), dv) When this point is located on the sphere O,wehave λ,v) = n), dv) qv) n), qv) dv) + n), p) dv) n), dv) p) n), dv) r ) = 0. There are some redndant soltions of λ,v) = 0 which do not correspond to real affine intersection points. For example, when the line qv) + t dv) is totally contained in the plane P bt does not intersect the circle C, there is no real affine intersection point even thogh the condition λ,v) = 0 is satisfied. Cylinders and cones have dal representation as rled srfaces as well as ringed srfaces. When we intersect the two different representations of the same srface, a degenerate case occrs, where λ,v) 0. There are also some other degenerate cases, a complete classification of which will be discssed in Seong et al. [0]. An efficient algorithm for intersecting a ringed srface and a rled srface can be sed for the ray-tracing of ringed srfaces []. Figre 4 shows an example of intersecting a Utah teapot approximated by for ringed srfaces) and a rled srface representing a set of reflected rays). Rendering applications involve many tangential intersections, as a reslt of rays passing throgh the silhoette of ringed srfaces. It is a challenge to develop an efficient algorithm that can deal with a large nmber of tangential intersections; we will investigate this direction of research in Seong et al. [0]. 3)

13 40 HEO ET AL. FIG. 4. The intersection of a Utah teapot and a rled srface of reflected rays. 7. THE INTERSECTION OF TWO SIMPLE SWEEP SURFACES Srfaces of linear extrsion and srfaces of revoltion belong to the class of rled and ringed srfaces. There is a more efficient way of compting the intersection of these simple sweep srfaces since they provide some special strctres that we can tilize for speeding p the comptation of the zero-set λ,v) = 0. This section smmarizes the analysis reported in Kim [1]: 1. In the case of intersecting two srfaces of linear extrsion, the bivariate eqation can be decomposed in the form of λ,v) = f ) gv) = 0.. In the case of intersecting two srfaces of revoltion, the zero-set of λ,v) = 0 can be redced to f) gv) = r), where f) and gv) are space crves and r)is a scalar fnction. 3. In the case of intersecting a srface of revoltion and a srface of linear extrsion, the space crve f) redces to a line l), which can frther simplify the zero-set comptation. Using the reslt of Heo et al. [8], we may reformlate the intersection of two rled srfaces as a zero-set finding problem for a bivariate eqation λ,v) = d 1 ) d v), p 1 ) p v) =0, 4) where p 1 ) and p v) are generating crves and d 1 ) and d v) are direction vectors of the two rled srfaces. Srfaces of linear extrsion have fixed direction vectors d 1 and d ; ths the bivariate eqation can be represented in a decomposable form where λ,v) = f ) gv) = 0, 5) f ) = d 1 d, p 1 ) and gv) = d 1 d, p v). 6) Figre 5 shows an example of intersecting two srfaces of linear extrsion. Now we consider the intersection of two srfaces of revoltion, which is a special case of intersecting two ringed srfaces. Let S 1, s) = C 1 s) and S v, t) = v C v t)betwo srfaces of revoltion; then we may assme that the circle C 1 has center p 1) = p 1 + n 1

14 INTERSECTION OF TWO RINGED SURFACES 41 FIG. 5. The intersection of two srfaces of linear extrsion. and is contained in the plane P1 with a fixed nit normal n 1. Similarly, the circle C v has center p v) = p + vn and is contained in the plane P v with a fixed nit normal n. Eqation ) is then redced to the matrix eqation where n 1 ˆx,v) n 1, p 1 + n ŷ,v) = n, p +v, 7) p 1 p + n 1 vn ẑ,v) µ,v) µ,v) = p 1 + n 1, p 1 + p v n, p v + r v) r 1 ). By mltiplying the first row by and the second row by v and sbtracting them from the third row of the above matrix eqation, we get the following eqation: n 1 ˆx,v) n ŷ,v) = p 1 p ẑ,v) n 1, p 1 + n, p +v p 1 p r1 ) + r v) + v Let a constant w be defined as n 1 w = n p 1 p = n 1 n, p 1 p 0.. 8) By Cramer s rle, the coordinate fnction ˆx,v) is then compted as n 1, p 1 + n 1,y n 1,z ˆx,v) = w 1 n, p +v n,y n,z, 9) p 1 p r1 ) + r v) + v p 1,y p,y p 1,z p,z

15 4 HEO ET AL. FIG. 6. The intersection of two srfaces of revoltion. where n i = n i,x, n i,y, n i,z ) and p i = p i,x, p i,y, p i,z ), for i = 1,. Note that ˆx,v) can be represented in the form f ) gv). Similarly, ŷ,v) and ẑ,v) can also be represented in this form. The condition for the intersection point ˆx,v), ŷ,v), ẑ,v)) to be located on the sphere O1 can be formlated as follows λ,v) = x,v), y,v), z,v)) p 1 n 1 r 1 ) = 0. 10) Ths the zero-set of λ,v) = 0 can be redced to f) gv) = r), where f) and gv) are space crves and r) is a scalar fnction. Figre 6 shows an example of intersecting two srfaces of revoltion. In the case of intersecting a srface of revoltion and a srface of linear extrsion, the space crve f) redces to a line l). This can be shown by applying the reslt of Section 6 to or case. We may assme that n) n, p) = p + n, and dv) d, for some fixed direction vectors n and d. Then Eq. 3) of Section 6 is redced to λ,v) = n, d qv) n, qv) d + n, p + n d n, d p + n) n, d r ) = 0, where the term n, p + n d n, d p + n) represents a line l). 8. CONCLUSIONS We have presented an efficient and robst algorithm for the intersection of two ringed srfaces and an algorithm for the intersection of a ringed srface and a rled srface. The intersection problem has been reformlated as the search for the zero-set of a bivariate fnction. The overall comptation procedre is based on the B-spline sbdivision techniqe and is nmerically stable [4]. An important advantage of this approach is that the self-intersection of a ringed srface can be similarly reformlated as the zero-set of a bivariate fnction. Thogh this paper has not presented the details, the same approach can be applied to the self-intersection of a rled srface [7]. In the special cases of intersecting srfaces of linear extrsion and srfaces of revoltion, the bivariate eqations are simplified to certain decomposable forms, f ) gv) = 0 or

16 INTERSECTION OF TWO RINGED SURFACES 43 f) gv) =r), which are considerably easier to compte than the general bivariate eqations. We believe that this approach has mch potential in accelerating the collision detection among simple sweep srfaces, more details of which will be investigated in ftre work. ACKNOWLEDGMENTS The athors thank the anonymos referees for their comments which were very sefl in improving the presentation of this paper. All the algorithms and figres presented in this paper were implemented and generated sing the IRIT solid modeling system [1], developed at the Technion, Israel. This research was spported in part by the Korean Ministry of Information and Commnication nder the University Basic Research Program of the year 1999, in part by the Korean Ministry of Science and Technology nder the National Research Lab Project, and also in part by the Korean Ministry of Edcation nder the Brain Korea 1 Project. REFERENCES 1. G. Elber, IRIT 7.0 User s Manal, The Technion IIT, Haifa, Israel, Available at G. Elber, J. J. Choi, and M.-S. Kim, Rled tracing, Visal Compt. 13, 1997, G. Elber and M.-S. Kim, Geometric shape recognition of freeform crves and srfaces, Graphical Models Image Process. 59, 1997, G. Elber and M.-S. Kim, Bisector crves of planar rational crves, Compt. Aided Design 30, 1998, G. Elber and M.-S. Kim, A comptational model for nonrational bisector srfaces: crve-srface and srfacesrface bisectors, in Proc. of Geometric Modeling and Processing 000, Hong Kong, April 10 1, 000, pp G. Elber and M.-S. Kim, Geometric constraint solver sing mltivariate rational spline fnctions, in Proc. of the Sixth ACM Symposim on Solid Modeling and Applications 000, Ann Arbor, Michigan, Jne 6 8, 001, pp H.-S. Heo, The Intersection of Rled and Ringed Srfaces, Ph.D. Thesis, Dept. of Compter Science, POSTECH, Febrary H.-S. Heo, M.-S. Kim, and G. Elber, The intersection of two rled srfaces, Compt. Aided Design 31, 1999, J. Hoschek and D. Lasser, Fndamentals of Compter Aided Geometric Design, A. K. Peters, Wellesley, MA, J. Johnstone, A new intersection algorithm for cyclides and swept srfaces sing circle decomposition, Compt. Aided Geom. Design 10, 1993, K.-J. Kim, M.-S. Kim, and K. Oh, Tors/sphere intersection based on a configration space approach, Graphical Models Image Process. 60, 1998, M.-S. Kim, The intersection of simple sweep srfaces, in Proc. of the Riken Symposim on Geometric Processing for Innovative Applications, Tokyo, Jly 1, 000, pp M.-S. Kim and G. Elber, Problem redction to parameter space, in The Mathematics of Srfaces R. Cipolla and R. Martin, Eds.), Vol. 9, pp. 8 98, Springer-Verlag, London, T. Maekawa, N. M. Patrikalakis, T. Sakkalis, and G. Y, Analysis and applications of pipe srfaces, Compt. Aided Design 15, 1998, R. Martin, J. de Pont, and T. Sharrock, Cyclide srfaces in CAD, in The Mathematics of Srfaces I J. Gregory, Ed.), pp , Oxford University Press, Oxford, J. Miller and R. Goldman, Geometric algorithms for detecting and calclating all conic sections in the intersection of any two natral qadric srfaces, Graphical Models Image Process. 57, 1995, T. Nishita and H. Johan, A scan line algorithm for rendering crved tblar objects, Proc. Pacific Graphics 99, Seol, Korea, October 5-7, 1999, pp

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