γ 1 γ 3 γ γ 3 γ γ 1 R (a) an unbounded Yin set (b) a bounded Yin set Fig..1: Jordan curve representation of a connected Yin set M R. A shaded region represents M and the dashed curves its boundary M that consists of a number of Jordan curves. The representation only entails choosing orientations of these Jordan curves so that in both (a) and (b) we have M = int(γ 1 ) int(γ ) int(γ 3 ). In particular, M in subplot (a) is not a manifold because γ and γ 3 have an improper intersection. Note that an isolated intersection point of two curves is improper if at its local neighborhood one curve lies entirely at only one side of the other; otherwise it is a proper intersection [41, Def. 4.1]. By Theorem. and Definition.3, regular open semianalytic/semialgebraic sets form a Boolean algebra since they are the intersection of the universes of two Boolean algebras. Furthermore, Boolean operations on Yin sets preserve the boundedness of a bounded boundary, hence we have Theorem.5. The algebra Y := (Y,,,,, R ) is a Boolean algebra. Recall from Section. that the exterior of P, P := (P ), is the interior of its complement and we define P Q := (P Q). We employ Yin sets instead of bounded regular open semianalytic sets for two reasons. First, Yin sets are slightly more general. Second, by Definition.1, the two distinguished members ˆ and ˆ1 of the Boolean algebra of Yin sets are simply and R, respectively. This is not true if we restrict the regular sets to be bounded. The interior of a Jordan curve γ i, denoted by int(γ i ), is the complement of γ i that always lies at the left of an observer who traverses γ i according to its orientation. It can be shown that a connected Yin set M can always be expressed as M = int(γ 1 ) int(γ ) int(γ l ), (.7) where {γ i : i = 1,..., l} is a set of oriented Jordan curves that are almost pairwise disjoint, i.e. they do not have proper intersections and the number of their improper intersections is finite; see Figure.1 for two examples. Consequently, any Yin set M has the representation M = j i int (γ j,i ), (.8) where j is the index of connected components of M. The significance of (.7) and (.8) is that a Yin set can be tracked by tracking its boundary, which justifies IT. We define the volume of a Yin set S Y as S := dx, (.9) S 8
ipam method. Step (CubiMARS-) is essentially the same as step (ipam-4), except that here new markers have to be located and inserted on the spline instead of a linear edge. Step (CubiMARS-3) precludes ill-conditioning of spline fitting and prevents a number of robustness problems in computational geometry [16]. Similar to (ipam-5), step (CubiMARS-3) enforces a lower bound of distances between adjacent markers. However, in (ipam-5) a short edge is always replaced by its midpoint while in (CubiMARS-3) one of the endpoints is removed, leaving the other one intact. This change stems from properly handling C 1 discontinuities on the interface: if a C 1 discontinuity is known a priori, only C continuity should be enforced at this knot when constructing the spline. As far as spline construction is concerned, the averaging procedure in (ipam-5) may spuriously change the location of a C 1 discontinuity, resulting in an unnecessary loss of accuracy. Hence (CubiMARS-3) is more appropriate than (ipam-5) in representing the interface with splines; see Section 5.1. The last step does not change results of the cubic MARS method; it simply generates a local representation of the solution from the global solution; see Section 4.3 for details of the algorithm. There are two main reasons for having the local solution. First, it makes the cubic MARS method amenable to be coupled to high-order finite-volume methods as material regions inside control volumes must be available in discretizing spatial operators. Second, this design facilitates accurate and efficient treatments of topological changes since such an algorithm should be based on the local material regions instead of the global chain of spline knots. It is also informative to note that, during the process of a topological change, there must exist a crucial time instant where the boundary of the phase is not a manifold; see Figure.1 (a) for an example. This is another reason why we adopted Yin sets to model physically meaningful material regions: a manifold with boundary is not general enough to handle topological changes. In a future paper we will report our algorithms of dealing with topological changes via Yin sets. 4.3. Splinegon clipping. As mentioned in Section, a physically meaningful material region is modeled by a Yin set M. To obtain a local representation of the solution in the last step of Definition 4.3, the intersection of two Yin sets M and C needs to be determined. Without loss of generality, we assume that is positively oriented and that each of M and C has a single Jordan curve as its boundary. Denoting by the oriented boundary of M, we enumerate all possibilities on the relative position of such M = int() and C as follows. (Case-1) =. (Case-) contains at most one point. (Case-3) only contains isolated points and its cardinality is at least two. (Case-4) contains curve segments. Note that the intersection points as above can be either proper or improper; see Figure.1. As a useful result in computational geometry, a point p is in the bounded complement of a Jordan curve if and only if any ray starting from p to infinity has an odd number of proper intersections with the Jordan curve. Out of the above four possible scenarios, (Case-1) is regarded to hold when the numerical value of M C is less than a user-given small positive real number. For (Case-), the single point must be an improper intersection. Consequently, we must have one of the following (i) C M, (ii) M C, (iii) C M =, and (iv) is negatively oriented and is a subset of C. In other words, (iv) means that the boundary of C M consists of both of the two Jordan curves and. Clearly, these scenarios can be easily detected by checking whether a point on belongs to C. 13
Algorithm 1: Clipping a splinegon with a simple linear polygon Input: An oriented Jordan curve, a simple linear polygon C with positively oriented boundary, P, a set of isolated intersection points characterizing. Precondition: (i), } (ii) the cardinality of P is at least two. { Γi Output:, a set of simple closed splines with positive orientation. ) Postconditions: (i) C int () = i int ( Γi ; (ii) Γ i Γ j implies that they are almost disjoint; 1 break into curve segments separated by isolated intersection points {γ i } the set of curve segments inside C 3 if {γ i } is an empty } set and is positively oriented then 4 return { Γi = {} 5 else if {γ i } is an } empty set and is negatively oriented then 6 return { Γi = 7 end 8 p s i, pe i the starting and ending points of γ i 9 mark all starting points in {p s i } as undiscovered c // index of closed splines in the output 11 while there exists an undiscovered starting point, say p s r do 1 anchor p s r; front p e r; c c + 1 13 initialize Γ c with γ r, the curve segment with front as an endpoint 14 closed false; 15 while closed = false do 16 γ r cuts C into two splinegons, the one at the left of γ r is C r 17 if front=p s j for some undiscovered ps j and γ j C r then 18 p min p s j 19 else start from front to traverse counterclockwise; set p min as the next undiscovered p s i front or the next vertex of C; if they are equally close to front, choose p s i 1 end if p min front then 3 append into Γ c the line segment from front to p min 4 end 5 front p min 6 if front=anchor then 7 closed true 8 else if front= p s j for some undiscovered ps j then 9 append γ j into Γ c and mark p s j as discovered 3 front p e j 31 end 3 end 33 mark anchor as discovered 34 end 14
p e 1 p s 1 p s 1 p e 1 p s p e v 1 v (a) positively oriented with no improper intersections p e p s v 1 v (b) negatively oriented with no improper intersectons p s 1 p e 1 p e 1 = p s p s 1 = p e p e v 1 v (c) positively oriented with an improper intersection p s v 1 v (d) negatively oriented with an improper intersection Fig. 4.: Intersecting a simple closed splinegon with a simple linear polygon. Referring to Algorithm 1, the hollow and filled circles are the starting and ending points of the curve segments, respectively. They are also proper intersections. In contrast, a hollow pentagon represents an overlapping starting point and ending point, and is thus an improper intersection. The solid squares represent vertices of the linear polygon. The arrows indicate the orientation of the simple closed spline. The shaded regions represent the intersection. In the rest of this subsection, we focus on the nontrivial scenarios (Case-3) and (Case-4). As shown in Algorithm 1, C is intended to be a control volume and M the material region. Besides C and M, a finite point set P is also designed as another input parameter to characterize. For (Case-3), P contains the isolated intersection points, proper or improper. For (Case-4), P contains the endpoints of the curve segments in, in addition to the isolated intersection points. Since C is a linear polygon and the Jordan curve is represented by cubic (or lower-order) splines, these isolated intersection points can be easily calculated by explicit algebraic formulas. In particular, the formulas are simple for a rectangle as it consists of four line segments of the form x = x, x 1, y [y, y 1 ] or y = y, y 1, x [x, x 1 ]. Also because of this simplicity, any one-dimensional intersection of a spline to C is easily detected. Thus the design of taking P as an input parameter neither incurs any loss of generality nor avoids any difficulties. The preconditions listed in Algorithm 1 follows directly from the above discussions. At line 1 of Algorithm 1, we divide into a number of spline segments. Then line is implemented by checking whether or not the midpoint of γ i is in the interior of 15
. The lines 3-7 deal with the special case that all γ i s are outside of C, which also implies that all points in P are isolated improper intersections. Due to the return statements at line 4 and line 6, the points p s i, pe i in line 8 do exists. We illustrate the rest of Algorithm 1 by four typical cases in Figure 4.. For subplot (a), suppose p s 1 is picked as the first undiscovered point p s r on line 11 of Algorithm 1. Then p s 1 is always the anchor during the construction of Γ 1. After the execution of lines 1-13, Γ 1 only contains the spline segment from p s 1 to p e 1. The condition front= p s j for some undiscovered ps j at line 17 clearly does not hold and the execution of line yields p min = p s. Since p s front, the linear segment p e 1p s is appended into Γ 1 at line 3. After setting p min = p s as the new front at line 5, the condition at line 8 holds. Then at line 9 we append γ into Γ 1 and mark p s as discovered. At the end of the first inner while loop, p e is set as the new front, with Γ1 containing the spline segment from p s 1 to p e 1, the line segment p e 1p s, and the spline segment from p s to p e. During the second execution of the inner while loop, p min is set to p s 1 at line, and hence the two conditions at lines and 6 hold whereas that at line 8 does not. Now that Γ 1 is closed and all undiscovered starting points have been discovered, the algorithm terminates within a single outer loop and returns { Γ 1 } as the output. The example illustrated in subplot (b) of Figure 4. is complementary to that in subplot (a). Suppose p s is picked as the first undiscovered point p s r on line 11 of Algorithm 1. Then p s is the anchor. At the end of the first and second execution of the inner while loop both front is set to v 1 and v, respectively. After another execution of the inner loop, p s is discovered and Γ 1 is closed. Thus Γ 1 is constructed after three consecutive executions of the inner while loop. However, p s 1 remains as an undiscovered starting point, and the outer while loop is invoked again to construct Γ. After another three analogous execution of the inner while loop, the algorithm terminates and the set { Γ 1, Γ } is returned. The subplots (c) and (d) in Figure 4. illustrates lines 16-18 in Algorithm 1 for handling improper intersections. In the case of subplot (c), cut C with γ 1 and we have two splinegons, the one on the left of γ 1 is C r = C 1, the splinegon with its edges as the spline segment γ 1 and the directed linear segments p s v 1, v 1 v, v p s 1. Since γ C 1, the condition at line 17 holds and p min is set to p s. The condition at line 8 holds while none of the conditions at lines and 6 holds. Hence at the end of the first inner while loop, p e is set as the new front, with Γ 1 containing the spline segment from p s 1 to p e 1 and the spline segment from p s to p e. In contrast, in subplot (d), the splinegon at the left of γ is C r = C, the splinegon with its edges as the spline segment γ and the directed linear segments p s 1v 1, v 1 v, v p s. Since γ 1 C, the condition at line 17 does not hold and p min is set to v 1. Then the condition at line holds while none of the conditions at lines 6 and 8 holds. Hence at the end of the first inner while loop, v 1 is set as the new front, with Γ 1 containing the spline segment from p s 1 to p e 1 and the linear segment p s 1v 1. Theorem 4.4. Consider a simple linear polygon C with positively oriented boundary and an oriented Jordan curve. If consists of two or more points, Algorithm 1 generates a set of positively oriented and pairwise almost disjoint Jordan } curves { Γi such that ) C int () = i int ( Γi. (4.1) 16
Proof. By Theorem.5, Y := C int () must be a Yin set because both C and int() are Yin sets. By (.7) and (.8), the boundary of each connected component of Y is uniquely represented by Jordan curves that are almost pairwise disjoint. By enumerating all possibilities on the relative position of and, one can show that the conditions and #{ } imply that C int () are the regularized union of a finite set of pairwise disjoint Yin sets, each of which is homeomorphic to the open ball. In other words, the form in (4.1) is a simplification of (.8), due to the fact that each of C and int() has their boundary as a single Jordan curve. The boundary of Y must be a subset of. In addition, the isolated points given in P divide the input Jordan curve into a chain of directed curve segments; this also holds for. Each Jordan curve Γ i in the output is a concatenation of some of these directed curve segments. In addition, the segments of that are outside C cannot be part of Y whereas those inside C must be. Similarly, for segments of, those outside int () cannot be part of Y whereas those inside int () must be. Because both and are simple curves, there are at most four incident segments at each of their intersection points, two from and the other two from. Hence the key of Algorithm 1 is to connect the directed segments of inside int () and those of inside C to form oriented Jordan curves in the output. The correctness of this core aspect of Algorithm 1 can be shown by an induction on the number of segments appended into Γ c. As the induction basis, the first segment γ r clearly exists and is inside C. In the inductive step, the choice of the next segment to be appended consists of several cases as follows. (NS-1) front is not an intersection point, hence it must be a vertex of C. This case is determined by line : the next segment must be a line segment on. (NS-) front= p e r is a proper intersection. Hence the other incident segment in cannot belong to the boundary of C int (); this justifies lines 19-1 and lines -4 of Algorithm 1. (NS-3) p e r is an improper intersection. Then by definition the other incident segment in might also belong to the boundary of C int (). However, if this is the case, both incident segments must be in the same splinegon C r as defined at line 16, because the inner while loop in lines 15-3 is for assembling a single Jordan loop. This justifies lines 16-18 and lines 8-31 of Algorithm 1. To sum up, the inner while loop that corresponding to the inductive step maintains an invariant that each segment appended to Γ i is a subset of Y. The induction terminates when anchor is discovered again, which will eventually happen because the propagation of the front is among a closed sequence of a finite number of points on. The postcondition (ii) of Algorithm 1 holds because of the following. Intersection points of the output Jordan curves form a subset of those of the input Jordan curves. A proper intersection of the input Jordan curves is never an intersection of the output Jordan curves because two of the four incident edges at this intersection, one from and one from, are not even in the output Jordan curves any more. By the discussions in (NS-3), an improper intersection of the input Jordan curves must either be an improper intersection of the output Jordan curves or a non-intersection. Finally, (4.1) holds because each Jordan curve Γ i has positive orientation and their boundary are pairwise almost disjoint. 17