The Geometry of Carpentry and Joinery

Transcription

1 The Geometry of Carpentry and Joinery P. MORIN McGill University, Canada J. MORRISON Carleton University, Canada Abstract In this paper we propose to model a simplified wood shop. Following the work of Demaine, Demaine and Kaplan in [4] we limit the cutting tools of our carpenter to a circular saw. We extend that previous work to include a model of basic rules of carpentry and joinery. This model is then applied to the problem of building a polygon P by joining together strips of wood and cutting them with a circular saw. We begin with deciding if a specific blueprint can be constructed and then progress to the design of blueprints for specific polygons. 1 Introduction Demaine, Demaine and Kaplan [4] study the problem of cutting a polygon P from a convex polygon Q that contains P using a circular saw. In their model, a circular saw is represented by a line segment of positive length r, called the radius of the saw. A cut is a line segment s, disjoint from the interior of P such that s Ò Q contains a line segment whose length is at least r. When one or more cuts disconnects Q, the component(s) not containing P are removed to obtain a new polygon Q ¼, on which further cuts can be made. Figure 1 illustrates how all this is analogous to cutting a shape from a piece of plywood with a circular saw by making successive cuts and removing the parts of the plywood that become disconnected from the main form. This model is a reasonable mathematical abstraction of several types of hand-held and tabletop saws whose blade is circular and must be spinning before it enters the material to be cut. We say that a polygon P is cuttable with a circular saw of radius r, if for any convex polygon Q containing P, there exists a finite sequence of cuts c 1 c k resulting in a sequence of polygons Q Q 0 Q 1 Q k P where Q i is obtained This research was funded by Natural Sciences and Engineering Research Council of Canada, NCE-GEOIDE and the Government of Ontario. 1

2 2 Proceedings in Informatics Q P Figure 1: Cutting a form with a circular saw. from Q i 1 via cut c i. More simply, we say that P is cuttable by a circular saw if there exists an r 0 such that P is cuttable with a circular saw of radius r. The main result of Demaine, Demaine and Kaplan is Theorem 1 (Demaine, Demaine and Kaplan) A polygon P is cuttable by a circular saw if and only if P does not have two consecutive reflex vertices on its boundary. In this paper we study what happens to Theorem 1 when the model is extended using some basic knowledge of carpentry. The basics we speak of encompass two fundamental areas, aesthetics and robust design. The aesthetic qualities state that in making something from wood it must be made to look as good as possible. Robust design implies that a design should eliminate as many possible sources of error as is feasible. Aesthetics criteria imply that only quality wood can be used and thus plywood and particle board are out. Thus the large convex sheet Q, mentioned by Demaine, Demaine and Kaplan, must be made by joining smaller pieces. Also, wood pieces joined together whose grains have different orientations become a single piece which cannot be sanded. Thus the wood grain must have a specific orientation and desired polygons cannot be joined together with arbitrary pieces. The robust design criteria forces yet more constraints. To eliminate as many sources of error as possible we should join only fixed width planks. This will remove the possibility of accidentally joining the wrong pieces. Since joining is considered an irreversible process this error should be avoided at all costs. We also claim that large pieces made of fixed width planks look nicer and allow wood grains to be easily oriented.

3 Morin and Morrison: The Geometry of Carpentry and Joinery Figure 2: Building a polygon by cutting and joining strips. Finally we make the last robust requirement: all of the pieces which form an edge of the polygon must be joined together before that edge is cut. The rationale for this is to suppose that the pieces are cut before joining, then any portion of the joining process can cause the pieces not to be flush in the final product. This is in direct contradiction to the definition of robust design and therefore shows the need for the restriction. With these ideas in mind it is clear that we study the case in which P is created by joining together regular strips (rectangles) at their edges and cutting them with a circular saw. In this process, there are two rules that must be obeyed. The first is that once two strips are joined together they cannot be unjoined. The second is that, before cutting an edge e of P, all incident strips on e must be joined together. This model appears to be a reasonable facsimile to the process which creates many tabletops, desktops and wood floors. Any polygon that can be cut with a circular saw can be fabricated by cutting and joining. This is because all of the wood strips can be joined into one large sheet and then cut. However, the converse is not true. There are many polygons that cannot be cut by a circular saw, but which can be built by cutting and joining. An example is given in Figure 2. This is due to the fact that we can cut parts of the polygon individually and then join them together. A blueprint B P Cµ is a polygon P together with a set of vertical line segments C, each of which is contained in P and has both endpoints on the boundary of P. The elements of C are called chords of P. In computational terms, a blueprint is represented as a subdivision of P induced by the chords in C. The chords of C partition the interior of P into faces. A join is the process of removing a chord c from C, thereby merging the two faces incident on c. A cut is the process of cutting an edge e of P from the face f that contains e on its boundary. A cut must satisfy the following two rules.

4 4 Proceedings in Informatics Rule 1 The edge e must be entirely contained on the boundary of f. Rule 2 At least one endpoint of e must be a non-reflex vertex in f. Rule 1 models the constraint that all strips incident on e must be joined together before cutting e. Rule 2 comes from Theorem 1 and the assumption that our cutting tool is a circular saw. A constructionc v 1 v n m of B is a sequence of joins and cuts in which each edge of P and each chord of C appears exactly once. We say that B is feasible if it has a construction. In this paper we give a linear time algorithm to determine whether a blueprint is feasible. We also study the problem of determining whether a polygon has a feasible blueprint in which all chords have integral x-coordinates. Section 2 describes our algorithm for determining whether a blueprint is feasible. Section 3 presents our results on designing blueprints. Section 4 summarizes and concludes with open problems. 2 Testing Feasibility In this section, we study the problem of determining whether a blueprint B P Cµ is feasible and give an algorithm for finding a construction of B. Because of Rule 2, it is intuitively clear that consecutive reflex vertices will be the primary obstacle in finding a construction of B. Therefore, we call an edge e of P a reflex edge if both endpoints of e are reflex vertices. Our algorithm is divided into two steps which are discussed in the next two subsections. In the first step we attempt to determine, for each reflex edge e, the direction the circular saw will travel when e is cut. In the second step, we find an ordering of the joining and cutting operations that gives us a construction of B. We begin by reducing the problem to only include a modified set of chords C ¼ which include only the Steiner chords parallel to the chords in C. These chords are found by shooting vertical rays up and down from every vertex v in polygon P (see Figure 3). For clarity we say a chord is a marked chord if it is in both C ¼ and C and all other chords in C ¼ are unmarked chords. We end this section with a discussion on how to use the construction of B ¼ P C ¼ µ to create a construction of B. To facilitate this last item we impose that the joining of unmarked chords precede the cutting of the edges associated with them. 2.1 Directing Reflex Edges Let e u vµ be an edge of P. Then, during a constructionc we say that e is cut in direction uv if the joining of the chord incident on u is performed before cutting e. The following lemma states that we can join the chords of at most one endpoint of a reflex edge before cutting that edge.

5 Morin and Morrison: The Geometry of Carpentry and Joinery 5 a) Chords C b) Steiner Chords C ¼ Figure 3: Two different chord sets in a polygon. e 1 u v c w c v w e 2 x Figure 4: An overlap. Lemma 1 There is no construction of B ¼ in which a reflex edge e u vµ is cut in direction uv and in direction uv. Proof. Saying that e is cut in both directions is equivalent to saying that the chords incident on both endpoints of e are joined before e is cut. However, once these two chords are joined e is a reflex edge on the face containing e and, by Rule 2, can not be cut. ¾ An overlap consists of two edges e 1 u vµ and e 2 w xµ and two chords c v and c w such that c v is incident on v and on the interior of e 2 and c w is incident on w and on the interior of e 1 (see Figure 2.1). The following lemma shows that reflex edges which overlap have constraints on the directions in which they can be cut. Lemma 2 Let e 1 u vµ, e 2 w xµ, c v and c w form an overlap. Then, in any construction of B ¼ in which e 1 is cut in direction uv, e 2 must be cut in direction wx. Proof. Suppose that this were not the case, and that there is a constructionc of B ¼ in which e 1 is cut in direction uv and e 2 is cut in direction wx. Then, by Rule 1, c w must be joined before e 1 is cut. By Lemma 2, e 2 must be cut before c w is joined. By Rule 1, c v must be joined before e 2 is cut. Finally, by Lemma 2, e 1 must be cut before c v is joined. Symbolically, e 1 c v e 2 c w e 1

6 6 Proceedings in Informatics an impossibility. ¾ Lemma 2 provides a method for assigning directions to the reflex edges of P. We assign directions to edges of P using the following algorithm. 1. If e u vµ is a reflex edge with both endpoints on unmarked chords then, by Lemma 1, there is no construction of B ¼. 2. If e u vµ is a reflex edge having only the endpoint u on a marked chord then, by Lemma 1, every construction of B ¼ cuts e in direction uv. 3. Finally, we iterate the following procedure until no more directions are assigned: For every reflex edge e 2 w xµ. If e 2 overlaps an edge e 1 u vµ which has already been assigned the direction uv and which satisfies the conditions of Lemma 2 then we set the direction of e 2 to be wx. If at any time this procedure attempts to reassign a different direction to an edge whose direction has already been assigned then we can terminate since, by repeated applications of Lemmata 1 and 2, B ¼ is not feasible. 4. Once this procedure is complete, any reflex edge e u vµ that has not yet had a direction assigned to it is assigned the direction uv if u is to the left of v and uv otherwise. If the above algorithm succeeds in assigning directions to all edges of P then we say that the assignment of directions is consistent. The ability to consistently define directions for reflex edges is a necessary condition for B ¼ to be feasible. However, we have not yet proven that it is a sufficient condition because we must also show that there exists a consistent ordering among the joining and cutting operations. 2.2 Ordering Joins and Cuts Define the directed graph G B ¼ µ V Eµ as follows: 1. The vertex set V contains each edge of P and each chord of C ¼. 2. The edge c eµ is present in E if chord c ¾ C ¼ is unmarked and has an endpoint on e ¾ P. 3. The edge c eµ is present in E if chord c ¾ C ¼ is marked and has an endpoint strictly in the interior of e ¾ P. 4. The edge e cµ is present in E if e u vµ, c is marked with an endpoint on u and the direction of e is uv. An example of a blueprint B ¼ for a polygon with one reflex edge along with the corresponding graph G B ¼ µ is shown in Figure 5. The following is the main result of this section.

7 Morin and Morrison: The Geometry of Carpentry and Joinery 7 Figure 5: A blueprint B ¼ and the corresponding graph G B ¼ µ. e 1 u v c w c v w e 2 x Figure 6: A simple cycle in G B ¼ µ corresponds to an overlap that violates the conditions of Lemma 2. Theorem 2 B ¼ is feasible if and only if the reflex edges of P can be assigned consistent directions and G B ¼ µ has no cycle. Proof. We have already shown that if we can not assign consistent directions to the reflex edges of P then B ¼ is not feasible. Therefore, suppose we can assign consistent directions to the edges of P but that G B ¼ µ has a cycle. First note that, since P is a polygon and C ¼ forms a trapezoidation of P, any cycle of length greater than 4 must have repeated vertices. Therefore, G B ¼ µ contains a cycle of length exactly 4. This cycle includes 2 edges e 1 and e 2 of P and two chords c v and c w in C ¼. The edges e 1 and e 2 and chords c v and c w form an overlap, and this overlap contradicts the assumptions of Lemma 2 (see Figure 6). We claim that the directions of e 1 and e 2 were not assigned during Step 4 of the algorithm for assigning directions. Clearly, one of them, say e 1, was not assigned during Step 4 otherwise they would both be directed left-to-right and would not violate Lemma 2. But then e 2 would also have had its direction assigned in Step 3. Therefore, in any construction of B, e 1 and e 2 must be assigned those directions. But then, by Lemma 2, there can be no construction of B ¼. It remains to show the converse, i.e., if directions can be consistently assigned to reflex edges of P and G B ¼ µ is acyclic then there exists a construction of B ¼.

8 8 Proceedings in Informatics Suppose therefore that there is a consistent assignment of directions and G B ¼ µ does not contain a cycle. Then we topologically sort G B ¼ µ to obtain a total ordering v 1 v O nµ on V, i.e., the edges of P and the chords in C ¼. We claim that there exists a construction of B in which the cuts and joins occur in the order in which they appear in v 1 v O nµ. We prove this by showing that, by the time an edge e of P appears in the sequence v 1 v O nµ, e it is entirely contained on the boundary of a single face f and is not a reflex edge on f. Therefore, the construction satisfies Rules 1 and 2. Because of the edges added during Step 1 in the construction of G Bµ, all chords incident on e must appear before e in the total order. Therefore, Rule 1 is satisfied. Next we need to show that the edge e is not reflex in f. If e is not a reflex edge in P then e is still not a reflex edge in f. If e is a reflex edge in P then it has been assigned a direction, and during Step 2 in the construction of G B ¼ µ an edge was added that guarantees e appears in the total order before one of the chords, say c v, incident on one of the endpoints, say v, of e. Therefore, on the face f, v is not a reflex vertex and e is not a reflex edge. Thus, Rule 2 is also satisfied. ¾ If P is a polygon with n edges then C ¼ contains O nµ chords and there is an algorithmic version of Theorem 2 that runs in O nµ time, provided that the the blueprint B ¼ is given in a topological data structure (e.g., a doubly-connected edge list [5]). If the input is not given in this form, then such a data structure can be computed in O n log nµ time. Step 1, 2, and 4 of the algorithm for directing reflex edges are easily implemented in O nµ time. Step 3 can be implemented as a limited breadth-first traversal of the graph having reflex edges of P as vertices and an edge between two vertices if the corresponding edges of P are part of an overlap. This graph has size O nµ since each reflex edge overlaps at most 2 other reflex edges and hence this step of the algorithm can be completed in O nµ time. Once the reflex edges of P have been assigned directions, the graph G B ¼ µ can easily be constructed in time linear in the size of G B ¼ µ. Since G B ¼ µ has O nµ vertices and is planar, the construction of G B ¼ µ can be completed in O nµ time. Topologically sorting the vertices of G B ¼ µ again takes O nµ time [3]. 2.3 Summary Notes The previous sections provide an algorithm for testing feasibility of a blueprint B ¼ P C ¼ µ in O nµ time where P is a polygon with n edges and C ¼ contains only Steiner chords. We claim that given a construction for B ¼, finding a construction for B is obvious. Recall that B contains the chords in B ¼ plus some extra non-steiner chords both of whose endpoints are not on vertices of P. These additional chords have no effect on the direction assigned to edges of P during the first step of the algo-

9 Morin and Morrison: The Geometry of Carpentry and Joinery 9 rithm. Furthermore, if we compute the graph G Bµ, then these non-steiner chords become sources, i.e., they have in-degree 0. Thus, a topological sort of G B ¼ µ can be extended to a topological sort of G Bµ by placing the non-steiner chords before any other vertices of G B ¼ µ. Since no path in G Bµ has a non-steiner chord in its interior, the result is guaranteed to be a valid topological sort. In summary, a blueprint B P Cµ where P is a polygon with n edges and C contains m chords can be tested for feasibility in O nµ time. If B is feasible, a construction of B can be output in O nµ time if only steiner chords are considered and in O n mµ time otherwise. 1 3 Designing Blueprints The algorithm in Section 2 provides a means of testing whether a given blueprint is feasible. An obvious question that arises is that of finding a blueprint for a given polygon P. It should be noted at this point that if the constraint of fixed width strips is removed then designing a feasible blueprint becomes easy. By adding all of the steiner chords to the set C and removing Step 2 of the construction of G Bµ then we claim the blueprint becomes feasible. In this case, every reflex edge of P is incident on two chords in C and hence has its direction assigned in Step 4 of the algorithm for assigning directions to reflex edges. Then all reflex edges are directed left-to-right and it is clear that G does not contain any cycles, hence P Cµ is a feasible blueprint. More interesting problems arise when we keep the constraints on the blueprint of P. In order to express the constraint of fixed width strips we assume that the polygon P has been scaled relative to the size of the strips and that all chords have integral spacing. It is important to note that the chords C are represented as a single position and an orientation. Since chords have integral spacing and are all parallel then only a single point and orientation are necessary to represent the chords and avoids redundant, possibly large space use. 3.1 Fixed Width Strips without Rotations If the orientation of P is given (e.g., the grain of the wood must run in a certain direction) and all chords must lie on integer x coordinates then there is an optimal algorithm for designing and testing a blueprint. This algorithm is runs in time O nµ where n is the number of edges in the polygon P. The first problem we encounter is that of computing a blueprint for P given a particular translation of P. To do this, we first partition P into trapezoids by adding all Steiner chords C ¼. This can be done in O nµ time [2] and gives us a 1 A construction of B only describes the order in which chords are joined and edges are cut. It does not actually provide a plan for cutting the edges. It is possible to compute such a plan, in linear time [4].

10 10 Proceedings in Informatics trapezoidal decomposition of P. It only remains to distinguish between marked and unmarked chords. Given the O 1µ space representation of C it is possible to check if a given chord c ¾ C ¼ is marked in O 1µ time. This is done by checking if the distance from the position representing C to the chord c is an integer. Since there are O nµ chords in C ¼ then it takes O nµ time to check all chords once we assign the position to C. To find a feasible blueprint for P, we proceed as follows. If P has no reflex edges, then P is cuttable by a circular saw, so any blueprint is valid. If P has at least one reflex edge e then by Rule 2 in any feasible blueprint e must have at least one endpoint on a chord. Due to the regular spacing of chords we can select either endpoint of the edge e to lie on a chord and the remainder of the chords are specified. Thus there are only two blueprints that need to be tested. Each blueprint can be created and tested for feasibility in O nµ time, yielding an overall running time of O nµ. 3.2 Fixed Width Strips with Rotations If we allow the polygon P to be rotated, then the situation becomes significantly more complicated. With rotations a feasible position and orientation for C must be found. The complication lies in the number of possibly feasible orientations. In the case without rotations we are able to limit the number of feasible solutions to a constant by using Rule 2. As before each possible solution is tested in O nµ time and therefore the total time of the algorithm is O snµ where s is the number of solutions tested. Thus the goal of this algorithm is to minimize the number of possible solutions which we test. As before, the rotation case will also use Rule 2 to limit the number of possibly feasible solutions. If P has no reflex edges then any blueprint will do. Also, if P has only one reflex edge e then any blueprint which has at least one chord through an endpoint of e will also be feasible. From here on we will only consider the case where P contains at least two reflex edges e 1 and e 2. By Rule 2 both e 1 u vµ and e 2 w xµ must have at least one endpoint with a marked steiner chord. There are four combinations where this is true: uw ux vw vx. It remains to be determined how many different blueprints there are if u and w have an endpoint with a marked steiner chord. Let d u wµ be the distance between the vertices u and w. Then there can always be a single line passing through them. Since chords are parallel and equally spaced, then for d u wµ 1 there can be two different lines passing through u and v with up to d u wµ 1 lines between them. Thus there are d u wµ 1 possible orientations of C. When we apply this to all combinations we are left with Equation 1 which determines the number of possible orientations given the two reflex edges e 1 and e 2.

11 Morin and Morrison: The Geometry of Carpentry and Joinery 11 s e 1 e 2 µ d u wµ 1 d u xµ 1 d v wµ 1 d v xµ 1 (1) Now we can perform our algorithm. First, we calculate s e 1 e 2 µ for all pairs of reflex vertices e 1 and e 2 and determine which of these pairs has the minimum value. This will take O r 2 µ where r is the number of reflex edges. We trivially bound this by O n 2 µ. Then, for the minimal pair of reflex edges, µ 1 and µ 2, we compute each of the s s µ 1 µ 2 µ orientations and test for feasibility. This results in a time complexity of O snµ as testing each orientation requires only O nµ time. Thus the algorithm runs in a total time of O r 2 snµ time. If we note that s O dµ where d is the diameter of P and bound r n then the algorithm with run in O n 2 dnµ time. While this algorithm is sensitive to the number of reflex edges we are not aware of any lower bound on this problem. Thus we cannot comment on the overall efficiency of the algorithm. However we do feel that it may be possible to do better if more geometric observations are used rather than only using the combinatoric restrictions of Rule An Output Sensitive Algorithm for Fixed Width Strips with Rotations With a bit more work, we can find a blueprint B P Cµ such that the number of cut and join operations is within a constant factor of the minimum number of cut and join operations required in any blueprint for P. Let m denote the number of (steiner and non-steiner) chords in the feasible blueprint for P with minimum number of chords. We describe an algorithm with running time O n n mµµ that computes a blueprint with O n mµ chords. In cases where m is small compared to the diameter of P, this can result in a significant savings both in terms of algorithmic running time and real construction time. The previous algorithm enumerates a set of potential orientations of P and tests each orientation in turn. To achieve a running time dependent on m, we perform the same algorithm but order the orientations according to the number of chords induced by each orientation. More specifically, we begin by testing orientations that induce fewer chords and proceed to test orientations that induce more chords. The difficult part of all this is ordering the orientations without computing all of them explicitly beforehand. We begin by noting that, given the orientation θ of P, the number of chords incident on an edge e i of length l i and oriented in direction α i is l i cos θ α i µ 1. Therefore, once the orientation θ of P is fixed, it is possible to determine the number of chords in a blueprint for P to within n. Unfortunately, trigonometric functions are computationally difficult to cope with. However, the function axs e i θµ l i 1 2 θ α i µ π

12 12 Proceedings in Informatics approximates l i cos θ α i µ to within a factor of π 2 in the range 0 π. Therefore, the function acc P θµ n i 1 axs e i θµ should be a good approximation of the number of chords in P. Indeed, if cc P θµ is the number of chords in some blueprint for P that has P oriented in direction θ then it straightforward to verify that acc P θµ n cc P θµ π acc P θµ (2) 2 With P fixed, the function acc P θµ is a piecewise linear function of θ, and its pieces have endpoints at θ ¾ α i : 1 i n. An explicit representation of acc P θµ can be computed fairly easily in O nlognµ time by sorting and then scanning the values of α i. The result is a θ-monotone polygonal chain A whose intersection with any vertical line through some value θ gives the value of acc P θµ. We will use the chain A to order the possible orientations of P. The tool we use for this is an upwards vertical plane sweep [1]. We create a priority queue Q that will contain triplets of the form θ y sµ, where θ is a value in the range 0 πµ, y is the value of acc P θµ and s is the segment of A that spans θ. The elements of Q are ordered by increasing value of y. In the following, we assume that P has at least two reflex edges e 1 and e 2. The cases where P has 0 or 1 reflex edges are easily handled. As before, we make use of the fact that for a blueprint to be feasible, each of the two reflex edges e 1 and e 2 must have a chord passing through one of their endpoints. In particular, an orientation θ is a candidate if, after rotation by θ, the difference between the x-coordinates of endpoints, one from e 1 and one from e 2 is integral. We refer to this as the integral constraint. To begin, we add to Q one triplet θ y sµ for each segment s of A. The point θ yµ is the point on s with smallest y value such that θ satisfies the integral constraint. The plane sweep consists of repeatedly removing the first elements θ y sµ from Q and testing if there exists a feasible blueprint for P in which P is oriented in direction θ. If so, we are done and we report the blueprint. If not, we add to Q the record θ ¼ y ¼ sµ, where θ ¼ y ¼ µ is the point on s with smallest y ¼ y such that θ ¼ satisfies the integral constraint. If P has a feasible blueprint with m chords then Equation 2, combined with the fact that the above algorithm considers values of θ by increasing value of acc P θµ guarantees that the above algorithm finds a blueprint containing O n mµ chords. Furthermore, it is not difficult to verify that the number of values of θ satisfying the integral constraint and having the property that acc P θµ ¾ O mµ is O n mµ. Therefore, the above algorithm terminates after O n mµ tests, each of which takes O nµ time, resulting in a total running time of O n n mµµ. The number of elements in Q never exceeds n and therefore the space complexity is O nµ.

13 Morin and Morrison: The Geometry of Carpentry and Joinery 13 4 Conclusions We have studied the problems related to cutting strips of wood and joining them together to form a polygon P with n vertices. We show that given a blueprint for a polygon with n vertices using m strips of wood, we can test if the blueprint is feasible and, if so, give a construction in optimal O nµ time. We have also studied the problem of constructing a feasible blueprint for a given polygon P using unit-width strips of wood. For the case where the orientation of P is given (e.g., the grain of the wood must run in a certain direction) we give an optimal O nµ time algorithm. In the significantly more complicated case where P can be rotated, we have given an algorithm that runs in O n 2 dnµ time and output a blueprint of size O nµ if it is feasible. Here d is the diameter of P with the assumption that the strips of wood are of unit width. With slightly more work, the algorithm can be made to output a blueprint with O n mµ chords in O n n mµµ time, where m is the minimum number of chords in any feasible blueprint for P. Unfortunately, in the case where P has no feasible blueprint, the running times of both these algorithms are not bounded by any function of the input or output size. This remains an open problem. References [1] BENTLEY, J. L., AND OTTMAN, T. A. Algorithms for reporting and counting geometric intersections. IEEE Transactions on Computing C-28 (1979), [2] CHAZELLE, B. Triangulating a Simple Polygon in Linear Time. Discrete and Computational Geometry 6 (1991), [3] CORMEN, T. H., LEISERSON, C. E., AND RIVEST, R. L. Introduction to Algorithms. MIT Press, Cambridge, [4] DEMAINE, E. D., DEMAINE, M. L., AND KAPLAN, C. S. Polygons cuttable by a circular saw. Computational Geometry: Theory & Applications Special Issue of Selected Papers from the 12 th Annual CCCG (to appear). [5] PREPARATA, F. P., AND SHAMOS, M. I. Computational Geometry. Springer-Verlag, New York, P. Morin is with the School of Computer Science, McGill University, 3480 University Street, Montreal, PQ, Canada H3A 2A7. morin@cgm.cs.mcgill.ca J. Morrison is with the School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada K1S 5B6. morrison@scs.carleton.ca