Minimum area enclosure and alpha hull of a set of freeform planar closed curves

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1 Minimum area enclosure and alpha hull of a set of freeform planar closed curves A. V. Vishwanath, R. Arun Srivatsan, M. Ramanathan Department of Engineering Design, Indian Institute of Technology Madras, Chennai , India Abstract Of late, researchers appear to be intrigued with the question; Given a set of points, what is the region occupied by them? The answer appears to be neither straight forward nor unique. Convex hull, which gives a convex enclosure of the given set, concave hull, which generates nonconvex polygons and other variants such as α-hull, poly hull, r-shape and s-shape etc. have been proposed. In this paper, we extend the question of finding a minimum area enclosure (MAE) to a set of closed planar freeform curves, not resorting to sampling them. An algorithm to compute MAE has also been presented. The curves are represented as NURBS (non-uniform rational B- splines). We also extend the notion of α-hull of a point set to the set of closed curves and explore the relation between alpha hull (using negative alpha) and the MAE. Keywords: concave hull; alpha hull; enclosing curve; region occupied; convex hull; freeform curves; minimum area; 1 Introduction In the domain of point sets, convex hull [39, 14] has been the traditional answer when one asked to find the region enclosed by the set. Quite a few algorithms exist for computing the convex hull of a point set [39], both in R 2 as well as in R 3. Though convex hull has found numerous applications, ranging from interference checking [32] to shape matching [13] of geometric objects, one of the disadvantages of the convex hull is that, at times it does not best represent the area occupied by the input set. Hence, researchers started asking the question of what is the region occupied by the set of points and one of the earlier answer appears to be alpha hull (α-hull) [17] (whose discrete counterpart is α-shape [18]). α-hull for a point-set is a shape that generalizes the notion of convex hull. The alpha shape uses a real parameter α, variations of which leads to a family of shapes. The output of the alpha shape need not necessarily be convex nor connected. Concave hull, which appears to have been introduced in[26](they call it as non-convex footprints) and developed further in [35, 1], is an enclosure for the given set that represents the area occupied by the points by generating non-convex polygons. Figure 1(a) shows a concave hull for a set of points whereas the convex hull of the same set of points, shown in Figure 1(b) does not closely represent the set of points. A tighter enclosure can be achieved using concave hull (Figure 1(a)) than using convex hull (Figure 1(b)). For concave hull of set of points, a user-controlled parameter, called as tuning parameter is used to smooth the concave hull. Figure 1(c) shows a coarse concave hull whereas the one in Figure 1(d) is a smoother one. Corresponding author. emry01@gmail.com, Tel: , Fax:

2 (a) Concave hull (b) Convex hull (c) Coarse concave hull (d) Smoother concave hull Figure 1: Concave hull vs. convex hull [1] (a) A model of Piston and shaft assembly. (b) Convex hull using [21] (c) MAE Figure 2: A model of piston-shaft arrangement, its convex hull and MAE. As opposed to convex hull which is well-defined [39], concave hull does not appear to have a precise definition [35]. In particular, convex hull is the minimum perimeter as well as minimum area convex enclosure of the set of points. However, for non-convex enclosures, such objectives often conflict each other i.e., minimizing area and perimeter is not possible simultaneously [2] and one has to find a common ground, which leads to non-unique solutions called Poly hulls. Chaudhuri et al. [11] have introduced r-shape and s-shape enclosures for a set of points. Recently, χ-shape has been proposed in [16]. A-shape, another variant of α-shape was introduced in [34]. Other parameter-based hulls such as using exterior angle ω [48], depth-based digging from convex hull [4] have also been proposed recently. Computation of minimum area polygon that passes through all the points in the set has been addressed in [24]. So far, the question of region occupation seems to be more on the set of points and not much appear to have been done for other domains, for example, set of closed curves. An algorithm for computing the convex hull of a set of freeform curves has been provided in [21] which was then extended to freeform surfaces [45]. Recently, algorithm s efficiency in [21] was improved using biarc approximation in [31]. For set of straight line and circular segments, an algorithm for computing convex hull has been presented in [49]. Other prominent problems from the point-set that have now been extended to the domain of geometric modeling and CAD involving curves and surfaces as input are medial axis [42], Voronoi computation [12, 27, 23], Delaunay graph for ellipses [22], visibility graph [41], minimum spanning hypershpere [38], and minimum enclosing ellipse [6]. 2

3 In this paper, we explore the computation of minimal area enclosure (MAE) of a set of curves. Moreover, wealsoextendthenotion ofα-hull, usuallydefinedforaset ofpoints, toasetofcurvesand compare it with MAE. This paper is the manuscript version of the abstract, accepted for presentation track in SIAM GD/SPM11 [47] and no manuscript has been published so far based on the abstract. MAE has many potential applications. A major challenge in applications such as computational fluid dynamics (CFD) is to create a mesh of the CAD model for analysis. The CAD model may consists of overlapping edges/faces, non-manifold surfaces etc., making the meshing a very long and difficult task. Hence, surface wrappers are being used for the purpose of creating a usable CFD model from the CAD model [10]. For example, for an external flow analysis in CFD, only the outer surfaces, that is, a single continuous surface that wraps the complete model is required and no internal geometry and components are needed. However, the outer surface should represent the outer geometry accurately, including sharp corners. Meshing can then be performed on the wrapper, reducing the overall time for analysis considerably. The idea of MAE can be considered as one such wrapper, albeit in two dimensions. Other abstractions such as midsurface [44] or medial axis transform [8, 43], aid in defeaturing [33], a process where not-so necessary features are removed from a model. However, these abstractions have traditionally proved difficult to automate. Figure 2 shows an example model and the utility of its MAE. A set of disjoint curves is a useful representation in computing Voronoi diagram, α-hull, convex hull and in applications such as profile cutting, pocket machining. The construction of MAE for a set of disjoint curves is based on the minimal spanning tree, which has perhaps been introduced here for a set of curves for the first time. It does given an indication about the connectivity of the curves in general (or how it is arranged in the plane). Using MAE as a starting connectivity information, an algorithm to find the curves that lie in the convex hull has been developed (where again, the only known work for arbitrary curves seems to be [21]). Assuming MAE as a rubber band with zero area and then expand them (inverse of compressing the band) along with capturing the change in the connectivity while expansion, the algorithm gives the curves that lie on the convex hull of the set [46]. In the case of contour (or profile) cutting of materials such as foam glass, all the profiles have to be machined and hence the order of the profiles can be used to reduce the total length of tool travel. MAE, depicted as MST for the disjoint curves can be used to determine the tool path and travel across profiles. In the case of pocket machining, modification of the contours are done using the connection points for obtaining simplified tool path. The connection points are essentially minimum distance points and then MST is employed (please see Section 2.3.1, p22 in [28]). In general, as MST of points has several applications, MST of curves will also have, which have to be explored further. MAEs are also more suited for fencing applications. In case of fencing of houses with structures such as trees, the bounding area will be greatly reduced when it is applied around the MAE. Other areas of application for MAE include geographic information processing, image processing, pattern recognition, and feature detection [3]. Applications are also discussed in bit more detail in Section 7.1. In this paper, the input curves are parametric freeform curves represented using NURBS [40], and are closed C 1 -continuous (while implementation of MAE of intersecting curves, this condition is relaxed, please see Section 7), non self-intersecting. We assume that as we travel along a curve in the increasing direction of parameterization, its interior lies to the left. All the curves are assumed to be simply-connected (i.e. each one does not have holes in it) and no portions of any two curves overlap each other. Moreover, the input curves are used as such, i.e. without sampling them. One approach to find the MAE of freeform curves is to generate set of points on the curves and then use an algorithm such as in [35]. However, this approach may result in a very coarsely approximated MAE (depending on the sampling on each curve) of the input curves which might 3

4 impede the accuracy of the results when used in applications. This has been shown for algorithms such as Delaunay graph [23] and medial axis [42]. It is also further discussed in Section 7.1. Following are the contributions of the paper: For set of curves that intersect, it is shown that the MAE can be obtained similar to a Boolean union operation. For set of curves that do not intersect, antipodal points are identified and a minimum spanning tree approach has been proposed for obtaining MAE. Alpha hull, using a negative alpha, has been shown to be related to the external Voronoi diagram for the set of curves. For α =, it is shown that the boundary of the alpha hull is a subset of MAE when curves do not intersect each other. It is also shown that for α = for curves that intersect each other, the boundary of the alpha hull is a superset of MAE. Remainder of this paper is structured as follows: Section 2 presents some basic definitions employed in this paper. Section 3 describes the approach for determining the MAE when only two curves are present. Section 4 extends the approach presented in Section 3 for a given set of curves. Section 5 discusses the alpha hull of points as well as curves. Relation between alpha and and MAE for a set of curves is presented in Section 6. Results and discussion are presented in Section 7. Section 8 concludes the paper. C N p k D B E q Nk F A Figure 3: Concave and convex points in a freeform curve. 2 Definitions Definition 1 The osculating circle [15] of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p. The curvature of the curve at p is then defined to be the curvature of that circle, whose center is S. Curvature vector is then defined as S p. Definition 2 An inflection point is a point on a curve at which the curvature vector changes direction. Definition 3 A point in a freeform curve is convex, if the outward normal at that point and the curvature vector are in opposite directions. Otherwise, the point is called concave. 4

5 C 1 (t) C 2 (r) Figure 4: Outer curve is the MAE when one curve is completely inside the other. Point p in Figure 3 is convex, since N (outward normal) and k (curvature) are opposite to each other whereas point q is concave. Definition 4 A convex (concave) portion of a curve is a contiguous portion of the curve where all points are convex (concave). When traveled in the increasing direction of parameterization, portion DC in Figure 3 is a convex portion whereas F to E is concave. The portions can be identified by first computing points of inflection on the curve. A and B are examples of inflection points. MAE, similar to convex hull [39], indicates that the boundary of a MAE will be simply-connected i.e. will not have holes in it. Typically, MAE implies the boundary along with the underlying area, thoughwhilecomputingwedocomputetheboundaryonly(inawaysimilartomostofthealgorithms for convex hull). 3 MAE of two curves MAE of a single closed curve is the curve itself. When two curves are involved, the following cases have to be considered to determine the MAE: one curve lying completely inside the other; one curve lying completely outside the other; when they intersect. 3.1 One curve is completely inside the other For this case, it is obvious that the curve which is completely outside is the MAE between the two curves (say C 1 (t) and C 2 (r)) (Figure 4). To determine whether one curve is completely inside the other, there are several approaches. We adopted the following approach: first it is required to identify if the curves intersect or not. A quick check of this condition can be done using the convex hull of the control polygons of the two curves. One can use the left predicate (Chapter 1, [39]) to dothis operation. If they donot intersect, thecurves for suredonot intersect each other. However, if they intersect, the curves are then processed for intersection, which typically amounts to polynomial root finding of the two curves (in this paper, a constraint solver has been used [20] for root-finding). Once it is determined that there is no intersection, we take a point P 1 on the curve C 1 (t) and using the winding number [36], determine whether the point lies completely inside the curve C 2 (r). If so, 5

6 (a) A pair of curves outside each other (b)convexhullofthe curves (c) Pushing the convex Hull of the curves (d) MAE Corresponding Figure 5: Rubber band analogy to get MAE for two curves that do not intersect. the curve C 1 (t) lies inside the curve C 2 (r), and the MAE is now C 2 (r). If not, we take a point (say P 2 ) on C 2 (r) and do a similar check. It is to be noted that a point is outside the curve only when the winding number is 0. When the outer curve is convex, the MAE then becomes convex hull. 3.2 One curve is completely outside the other Let us consider two curves that do not intersect and are outside each other (Figure 5(a)). We also assume that the curves contain no straight line segments. Imagine a rubber band that is tightly enclosing them, such as the convex hull between the two (Figure 5(b). As one keeps pushing the convex hull and still enclosing the curves (Figure 5(c)), it will reach a point where the area of the enclosure between the curves tends to become null. Typically, in a rubber-band analogy, from the convex hull to polygonization (such as minimumarea polygon passing through all points), it is implicitly assumed that the rubber-band does not break (the analogy for α-hull, which breaks, comes from ice-cream scooping). In the case of disjoint curves, the limiting condition on the rubber band without breaking is fusing of the rubber band. This fusion will happen not only along the curves but also between them. The fusion between the curves can be represented as a straight line. The initial curve and the zero area line between them will form the minimum area enclosure (Figure 5(d)). It is to be noted that one can find different lines that can amount to zero area and hence infinitely many solutions are possible (it is similar to the fact that minimum area polygon for a set of points is not unique and NP-complete [24]). A minimum distance line has been chosen, as this measure can then be used to mimic MST behaviour arising out of such lines. Initially we find the points on either curve that form the minimum distance between the two curves. The minimal distance occurs when the points on the respective curves are antipodal to each other (Lemma 1). Lemma 1 The distance between two closed C 1 non intersecting curves is minimum only when the normals of the corresponding points are opposite to each other (i.e., antipodal.) Proof Let us assume that we have found points P 1 and P 2 on the two curves that are minimal distant but not antipodal. Construct a circle with these two points as the diameter. If this circle intersects any of the curves at any point other than the minimal distant point, say P, then it implies that distance P 1 P or P 2 P is less than P 1 P 2 as diameter is the largest distance between two points on a circle. This further implies that P 1 P 2 cannot be the minimum distance points at the first place. Hence this directly means that the circle with P 1 and P 2 as diameter cannot intersect the curves at any point other than P 1 and P 2, and hence is tangential at these two points. The fact that 6

7 the circle is tangential to both the curves and the points are on diametric ends results in a condition where the normals are in opposite directions. Hence the Lemma Antipodal constraint of two curves in R 2 Consider two planar closed C 1 -continuous curves C 1 (t) and C 2 (r). The antipodal constraint [38] is: C 1(t),C 1 (t) C 1(t)+C 2 (r) 2 C 2(r),C 2 (r) C 1(t)+C 2 (r) 2 = 0, = 0. (1) where C 1 (t) and C 2 (r) denote the tangent of the curves at the respective parameters t and r. Clearly, the two equations have two unknowns (t and r). Figure 6(a) illustrates the antipodal constraint for two planar curves. In practice, the solution of the constraint equations (Equations in (1)) results in a finite set of candidates(an infinite set of candidates is possible in certain degeneracies, such as when curves have straight line portions. They are not considered in this paper). From this set (which gives aset of tandcorrespondingr values), evaluate thepoints onthecurvesandthenfind the distance (Figure 6(a) shows a set of points on respective curves satisfying antipodal conditions, but not minimum in distance). Points on the respective curve that correspond to the minimum distance are then chosen. Assuming that there is only one set of points contributing to minimum distance between the two curves, MAE is then the two curves and the line between the minimum distance points. Figure 6(b) shows the MAE for the two curves. Definition 5 The line connecting the two antipodal points that is minimum in distance is called minimum antipodal line (MAL) and the points as minimum antipodal points (MAP). C 2 (r) C 2 (r) C 1 (t) C 1 (t) (a) (b) Figure 6: Antipodal constraint for two freeform planar curves in R Two intersecting curves Consider two curves that intersect each other (Figure 7(a)). Imagine a rubber band that is tightly enclosing them, such as the convex hull between the two (Figure 7(b). As one keeps pushing the convex hull and still enclosing the curves (Figure 7(c)), it will reach a point where the enclosure cannot be moved beyond the points where the intersection of curves happen, which will then yield minimum area enclosure (Figure 5(d)). When two curves are intersecting, the minimum distance between them is zero. Do note that the antipodal condition need not be satisfied at the points of intersection. As the curves are intersecting, 7

8 (a) A pair of curves that intersect each other (b) Convex Hull of the curves (c) Pushing the convex hull of the curves (d) Corresponding MAE Figure 7: Rubber band analogy to get MAE for two curves that intersect. C 1 (t) C 1 (t) C 1 (t) C 1 (t) C 2 (r) C 2 (r) C 2 (r) C 2 (r) (a) Two curves intersecting with each other (b) Remaining portion ofthecurvec 1(t) (c) Remaining portion of the curve C 2(r) (d) Remaining portions of the curves C 1(t) and C 2(r), forming a MAE. Figure 8: MAE for two intersecting free-form planar curves in R 2. points of intersections are then determined. The curves are then delineated between the points of intersection. Portions of one curve that are lying completely inside the other are then removed. This again amounts to checking if a curve is inside another. The resultant will be a set of curves forming the MAE. Figure 8(a) shows two curves that are intersecting with each other. Figure 8(b) shows the remaining portions of C 1 (t) after trimming the portion of C 1 (t) lying inside C 2 (r). Figure 8(c) shows the remaining portion of C 2 (r). MAE for the intersecting curves is shown in Figure 8(d). Please note that the process of computing MAE of two intersecting curves emulates Boolean union between the curves. As Boolean union yields a unique result, so will be the MAE, for intersecting curves. It is to benoted that the intersection between two curves can lead to the formation of inner loops, which are then removed to maintain simply-connectedness. Figure 9(a) shows two curves, where the elimination of portions of curves inside each other resulted in the formation of an inner loop as shown in Figure 9(b). The loops are eliminated using the fact that, for the curves parameterized in the anticlockwise direction (interior of the curve is to the left), the loops will form a clockwise direction when traveled in the increasing direction. This can be determined as follows: Let ab be a line completely contained in the loop. Using the tangent vectors Ta and Tb (Figure 9(b)), we can 8

9 C 2 (r) C 1 (t) Figure 9: Two intersecting curves C 1 (t) and C 2 (r) in (a) may produce inner loop(s) as in (b). find if they form a clockwise orientation. All the inner loops are then removed. 4 MAE for a set of curves Section 3 described the approach to compute MAE, given two curves. In this section, we extend the approach to obtain the MAE for a set of curves. We also assume that there is only one MAL between two curves. We first check for the curves that intersect and eliminate all the curves that lie totally inside other curves, as they do not contribute to MAE. After this process, there will be curves that either intersect or lie completely outside. For curves that intersect with each other, we first find the points of intersections between each pairofcurves. Weapplytheprinciplestated earliertofindthemae forapairofcurves(section 3.3). This will result in MAE for a pair of curves. We keep repeating this step till all the intersecting curves are processed. This leaves us with clusters of curves (some obtained using Boolean). For example, for the set of curves in Figure 10(a), the processing of intersecting curves will results in Figure 10(b). After processing Boolean union, this may result in curves that lie totally outside one another. For a curve (say C(t)), lying completely outside, compute the MAP between C(t) and all other curves (using Equation (1)). We repeat this process for all the curves that are outside (Figure 10(c) shows the MAL s for the set of curves). To compute MAE, the minimum spanning tree (MST) [5] is computed using the nodes as set of curves that are outside and the length as the distance between MAP for the set of curves lying outside each other. Let δ 1,δ 2,... be the minimum distances (computed using antipodal equation (1)) between curves in the MST. We assume that each δ s are different, so that MST returns an unique result and thereby giving an unique MAE. Figure 10(d) shows the MAE for the set of curves in Figure 10(a). Algorithm 1 indicates the MAE process. It should be noted that, a computed MAL between a pair of curves is not included while computing minimum distance between other curves. 5 Alpha Hull The alpha hull for a set of points S is defined in the following manner [17]. Definition 6 Let α be a sufficiently small but otherwise arbitrary positive real. The α-hull of S is the intersection of all closed discs with radius l/α that contain all the points of S. Definition 7 For arbitrary negative reals, the α-hull is defined as the intersection of all closed complements of discs (where these discs have radii 1/α) that contain all the points of S. 9

10 C 10 C 9 C 11 BU 3 C 3 C 11 C 7 C 8 C 1 C 2 C 6 C 5 C 4 BU 1 BU 2 (a) Initial set of curves C = {C 1,...,C 11}. S 1 = {C 1,C 2,C 3}, S 2 = {C 4,...,C 8}, S 3 = {C 9,C 10}. (b) After processing curves that intersect (using Boolean). BU 1 = {C 1 C2 C3} etc. C 11 BU 3 C 11 BU 3 BU 1 BU 2 BU 1 BU 2 (c) MALs for the set of curves in C, where C = {BU 1,BU 2,BU 3,C 11}. (d) MAE of the set of curves. Figure 10: Illustration of the algorithm. Definition 8 α-disc is a disc of radius 1 α for α > 0 and 1 α for α < 0 In this paper, our focus is on using the negative reals (i.e. with discs having radii 1/α) for the α-hull. From here, α-hull imply the hull with radii 1/α, unless otherwise mentioned. Figure 11 shows an α-hull for a set of points for a particular negative value for α. The inference obtained from the definition 7 is that, for a given set of points S, α-disc does not contain any point. The boundary of the α-hull occurs at all the places where the α-disc stays in contact with at least two points (called as neighboring points) simultaneously as shown in the Figure 12(b). Figure 12(a) shows an arbitrary disc of radius (-1/α) that does not form a part of the α-hull, whereas the disc in Figure 12(b) is constrained to pass through the points and hence is part of the α-hull boundary. Hereafter, α-disc will be synonymously used with constrained α-disc (Figure 12(b)), implying the disc touches the input set. Definition 9 The Voronoi cell of a point P i in S is the set of all points closer to P i than to P j, P j ǫs and i j. The Voronoi diagram is then the union of the Voronoi cells of all the points in the set. Note that the Voronoi diagram used here is the closest point Voronoi diagram, unless otherwise mentioned that will be the same hereafter. For completeness, Lemma 5 in [17] is restated here 10

11 Algorithm 1 MAE(C = C 1,...,C n ) Determine curves that intersect each other. Eliminate curves that lie completely inside. Find sets of curves that intersect. Let S i denote each set, and each S i will consist of curves from C. for each set S i of intersecting curves do Perform Boolean union. Represent each Boolean union as a single curve (say BU i ). Eliminate interior loops. end for Let C = {{BU i } {C {S i }}}. for each curve in in C do Compute MAP to all other curves in the set. end for if All curves in C are outside each other then Use curves as nodes and MAPs as distances, find the minimum spanning tree (MST). Return MST as MAE. else Return C as MAE. end if Figure 11: α-hull for a set of points. (Lemma 2) Lemma 2 The centers of the α-disc have to lie on the Voronoi diagram of the set of points for all α < 0. α-hull boundary occurs at places where the α-disc simultaneously stays in touch with at least two points. Since the α-disc is a circle of constant radius, the center of the disc at all times is equidistant from both the points (Figure 13(a)) and hence the centers of α-disc has to be on the Voronoi diagram (Figure 13(b) showing only the bisector). To the best of the knowledge of the authors, α-hull has been applied only for set of points so far. In this paper, we apply this to set of curves represented exactly (i.e. we do not approximate the curves using sample points). It is to be emphasized that we employ only the α-hull, as opposed to using its discrete variety called α-shapes [18]. We firstextend the concept of Voronoi of points to that of curves. Then the various combinations of the curves, its respective Voronoi diagrams and the α-disc traversal is explained. 11

12 (a) α-disc having no constraint (b) Constrained α-disc Figure 12: α-discs based on positions. 5.1 Extension of Alpha Hull of a set of points to that of curves Traditionally, construction α-shape [17] or khi-shape [16] for a set of points has been based on Delaunay triangulation. It should be noted that theoretical foundation as well as algorithms for Voronoi cell/diagram of a set of freeform curves has been well-developed in the recent past (for example, please refer to [27]). However, Delaunay triangulation for a set of curves is not well known (under the condition that the curves are not discretized into set of points). Hence, in this paper, we explore α-hull of set of curves in terms of the Voronoi diagram of the set. It is to be noted that as the closed curves have well defined exterior and interior unlike that of points. Note that the interior of a closed curve lies to its left as we travel along the increasing direction of parametrization. Definition 10 Portions of Voronoi diagram that lie in the exterior are called external Voronoi diagram (EVD). A similar definition holds for interior Voronoi diagram (IVD). In the Figure 14(a), the set of points can be considered as set of circles of zero radius. Hence the Voronoi diagram of a set of circles of radius ǫ is same as that of the points (Figure 14(b)). However, it should be noted that the diagram is an external Voronoi diagram (EVD). This is true even when the curves begin to assume arbitrary convex shapes (Figure 14(c), where only a portion of the bisector between two curves is shown). When a curve possesses concave portions (Definition 4), there is a possibility of self-voronoi (which has equidistant points on the same curve) in EVD within a concave portion of the curve. It is to be noted that Definition 9, when employed for a set of closed curves, excludes the self-voronoi portion [7]. In general, the self-voronoi segments have to be considered as path of traversal of α- disc. This aspect of self-voronoi is a feature existing only for the curves and not for the point-set. For example, Figure 15 indicates a self-voronoi for the curve C 1 (r) in a concave portion, where a point r has two points r 2 and r 3 equidistant within C 1 (r) itself. A point such as p is equidistance between two different curves C 0 (t) and C 1 (r) and hence does not belong to self-voronoi. Another feature for the Voronoi of curves is that the points on the Voronoi has to satisfy curvature condition, which compares the radius of curvature of the curve at the point where the disc touches with the disc curvature (Figure 16)[27]. Hence for a curve, when self-voronoi is included, the α-hull may not retain most of the contour of the shape. Lemma 2, which was for point sets, is then extended for curves (Lemma 3). Lemma 3 The centers of the α-disc have to lie on the Voronoi diagram of the set of curves for all α < 0. 12

13 (a) Center of α-disc equidistant from both the points. (b) Locus of α-disc of varying radius tracing the voronoi Figure 13: α-disc centers Proof α-hull boundary occurs at places where the α-disc simultaneously stays in touch with two points on the curve. Since the α-disc is a circle of constant radius, the center of the disc at all times is equidistant from both the points (Figure 14(c)). Since the disc cannot pierce the curve, the curvature of the disc has to be larger than the curvatures at which the disc touches (curvature condition, Figure 16[27]). This implies that the centers of α-disc has to be on the Voronoi diagram. Hence the lemma. Corollary 1 The centers of the α-disc (for α < 0) have to lie on the external Voronoi diagram of the set of curves. When the input contains just two points, the α-hull between them is two points themselves for all α, except α = 0. However, when the input consists of two curves, α-hull between them is the curves themselves only for values of disc diameter less than or equal to the MAP between them (i.e. 2α > 1/MAP). Figures 5(c) and 7(c) are examples for α-hull for two curves, which produces an enclosure between them. 6 Relation between Alpha hull and MAE of curves The relation between the α-hull and MAE of curves is established in this section. Since α-hulls are a family of hulls, which depend on the user input parameter α, it is necessary to determine which particular α-hull that is similar to the MAE before the comparison is done. The corresponding α that generates the MAE is used and its boundary lengths are compared. Prior to that, it is shown that the length of the boundary of the α-hull increases as the area occupied by the α-disc decreases. Though the argument seems intuitive (which, however, may not be true for positive α s), it is proved for completeness. 13

14 (a) Voronoi of circles of radius Zero (b) Voronoi of circles of radius ǫ (c) A portion of bisector between two curves that will contribute to the Voronoi cell of the set of curves for arbitrary convex shapes Figure 14: Voronoi for curves r 2 r 3 r 1 r o p C 1 (r) t C 0 (t) Figure 15: Example of self-voronoi Lemma 4 The length of the boundary of the α-hull is inversely proportional to the area occupied by the hull α < 0. Proof From the proof of Lemma 3, it is clear that the alpha disc forms the boundary of the α-hull only if it simultaneously touches two points, either of different curves or of the same curve. Consider the case of α-hull of a single concave curve C(t) (The case of two separate curves intersecting/nonintersecting can be proved in a similar manner). Supposing the curve C(t) is arc length parameterized with the closed interval [0,1] and the α-disc of radius r touches the curve at parameters t1 and t2 (Figure 17(a)), then the length of the boundary of the α-hull is given by where θ= sin 1 C(t1) C(t2) 2 r Length α = t1+(1 t2)+2rθ (2) 14

15 C 1 (r) C 1 (r) b a C 0 (t) (a) C 0 (t) (b) Figure 16: [27] (a) The disk radius smaller than the radius of curvatures at the footpoints of the curves. (b) The radius of the disk is greater than the radius of curvature of C 1 (r) at the footpoint implying the violation of the constraint. (a) (b) Figure 17: Comparison of the length of the boundary of α-hull for varying values of α Equation (2) is obtained by simple addition of the length of portions of the curve from parameter value 0 to t1, t2 to 1 and the arc of the α-disc. Let us consider the case when r decreases as we travel along the self-voronoi (Figure 17(b)). The value of t1 increases and the value of t2 decreases, increasing the effective length of the boundary of the α-hull (other cases can be handled in a similar manner). Hence the lemma. From the Lemma 4, it is evident that the maximum length of the boundary occurs when the radius of the α-disc is equal to zero (or α = - ). For curves that are intersecting each other, the following lemma (Lemma 5) is applicable. Lemma 5 For α =, boundary of the α-hull is a superset of the boundary of MAE of the set of curves, when they intersect each other. Precisely, the MAE with its internal loops will constitute the α-hull. Proof For a set of intersecting curves, the MAE is a single enclosing curve that does not detect the interior region. The area occupied by the MAE is more than that of the corresponding α-hull. Applying Lemma 4, the length of the boundary of the alpha hull is greater. Hence the Lemma. 15

16 (a) Input set of curves. (b) α-hull of the set of curves with α = (c) MAE of the set of curves. Figure 18: Generalization of α hull and MAE. 6). For set of curves that do not intersect, α-hull is related to the MAE in the following way (Lemma Lemma 6 For α =, boundary of the α-hull is a subset of the boundary of MAE of the set of curves, when they do not intersect each other and the curves lie outside each other. Precisely, the MAE without its minimum antipodal lines will constitute the α-hull. Proof In the case of curves independent of each other, the area of both MAE and the α-hull is same, as MALs have only zero area. MAE forms a continuous enclosure as opposed to α-hull which generates disjoint sets. In this particular case, MAE involves the MAL between them. Thus the boundary of the MAE is a superset of the corresponding α-hull. Hence the Lemma Corollary 2 For the curves that are intersecting, when there is no interior loop in the boolean union, the MAE is equal to α-hull for α =. When all the curves do not intersect, and if the α-disc has radius that is less than the minimum of all the minimum antipodal distances δ 1,δ 2,... (MAP s are computed for different curves but not within the same curve), then the α-hull is the set of input curves itself. This is similar to the fact that when the radius of the α-disc for point set is less than the minimum length of Delaunay edges, the α-hull will be the set of points itself. It is to be noted that generalization of lemmas 5 and 6 appear difficult when the set of curves contain both categories - i.e. some intersect and some curves lying outside. For the set of curves in Figure 18(a), α-hull (Figure 18(b)) has internal loops as well as disjoint curves. MAE in Figure 18(c) does not contain the inner loops, where as the curves that do not intersect are connected using MAP and minimum spanning tree approach. One cannot conclude that whether the MAE is a superset/subset of the α-hull (with α = ). 7 Implementation results of MAE Figures shows the implementation results of the algorithm. All the implementation have been carried out using IRIT [19] geometric kernel and its constraint solver [20]. Figure 19(a) shows two curves for which cmae was sought. The antipodal line has points on the curve which are convex (Figure 19(b)). Figure 20 shows the results for set of curves (the top row shows the input curves 16

17 (a) Two curves (b) Antipodal line falls in the convex region of both curves Figure 19: MAE for two free-form planar curves in R 2. and their respective MAE is shown in the bottom row). Figure 21 shows the model, convex hull, α-hull (with α = ) and MAE respectively for three objects. The maximum degree of the curve used is 4. MAE has also been computed for curves in Figures 21(a) and 21(e), which contains straight line portions as computation of MAP does not play a role. The results also indicate that the MAE approximates the domain better than the convex hull. (a) Test set of curves 1 (b) Test set of curves 2 (c) Test set of curves 3 (d) Test set of curves 4 (e) MAE for set 1 (f) MAE for set 2 (g) MAE for set 3 (h) MAE for set 4 Figure 20: MAE for a set of free-form planar curves in R 2. 17

18 (a) Model of a flange (b) Convex hull of the flange (c) Alpha hull (d) MAE (e) Model of a knuckle joint (f) Convex hull of the joint (g) Alpha hull (h) MAE Figure 21: MAE for mechanical joints. 7.1 Discussion Comparison with sampling and point-set based approaches MAE counterpart is the minimum area polygon problem in the case of a set of points. However, this has been shown to be NP-Complete [25] and hence no polynomial time algorithm to compute it is not available. In our testing, we have used the best known approximation algorithm for computing minimum area polygon [37]. Recent papers on concave hulls, the ω-hull [48] and digging of convex hull [4] have also been used for comparison. It should be noted that it is our implementation of these algorithms. For the object in Figure 22(a), 483 points were generated from the set of curves. Minimum area polygon (in red) for the set of points (in green) is shown in Figure 22(b). Clearly, the result is way off the target (please see Figure 20(h)). Figure 22(c) shows a closer result (in red) where algorithm [4] was employed with a threshold value of 4. However, this threshold value was obtained after lot of experimentation. Result for other threshold values are shown in Figures 22(d) and 22(e). Note that the hull seems to be better approximated with threshold 4 than with 10 and hence one cannot say that a larger value will lead to better approximation of MAE. Results for ω values of 120 and 90 are shown in Figures 22(f) and 22(g). Better result seems to prevail with ω = 120, yet again, obtained using experimentation, even though the result does not seem to be close to MAE. Also, for a ω of 90, for a sampling density of 1124, the result is shown in Figure 22(h), which is different from Figure 22(g) This reiterates that, when curves are approximated using set of sample points, not only one has 18

19 (a) Test object (b) Minimum area polygon (c) Using [4] with threshold = 4. (d) Using [4] with threshold = 1. (e) Using [4] with threshold = 10. (f) Using [48] with ω = 120 (g) Using [48] with ω = 90 (h) Using [48] with ω = 90, samples = 1124 Figure 22: Comparison of our result with minimum area polygon and concave hull algorithms of point-sets. to determine an appropriate sampling, the parameter (such as threshold in [4] and ω in [48]) also has to be identified, which is mostly by experimentation, not so feasible tasks in general Running time Permute and reject strategy [9], which computes all the permutation to return minimum area polygon took fewhoursfor even aset of ninepoints. Theapproximation algorithm [37]runsinO(n 4 ). Table1 shows the running time (in Intel core i3-2330m Processor Clock rate-2.20ghz, RAM-2 GB Windows 7 (64 bit)) using Matlab-7.8.0(R2009a). MAE algorithm has been implemented using IRIT [19], written in C, and takes less than a minute for most of the test cases (one cannot strictly compare the timing as they have been implemented using different tools). Even though [48] takes less time, the final hull seems to be not close to MAE of the set of input curves, evident from Figure such as 22(f) Limitations Algorithm 1 returns an unique MAE only under the assumptions that MAP between two curves is unique and no distances computed using MAP are same across different curves as MST will then not be unique. We have also assumed that, for the computation of MAP, the curves neither have straight line portions or overlapping ones. Currently, the MAE is represented as a set of piecewise 19

20 Paper ref Time (in min.) for Sample size = 483 Time (in min.) for sample size = 1124 [37] 20 - [4] 2 (for all thresholds) 11 (for threshold = 4) [48] < 1 < 1 Table 1: Running time of different algorithms. parametric curves. However, it is best to unify them into a single curve representation as it could prove useful in various applications. For example, queries such as whether a point is inside a MAE can be easily addressed if there is a good representation for the MAE Applications (a) Race Car model (b) MAE (c) Convex hull Figure 23: Model of a race car, its convex hull and MAE. One of the major challenges in analysis such as computational fluid dynamics (CFD) is to create a mesh of the CAD model for analysis. Surface wrappers are being used for purpose creating a usable CFD model from the CAD model [10]. For example, in external flow analysis in CFD, only outer surface is required and hence MAE can be considered as once such wrapper, albeit in two dimensions. Figure 23(a) shows a top view of race car model and its MAE (Figure 23(b)) as the outer wrapper, which is better than its convex hull (Figure 23(c)), in terms of closer wrapping. It appears that current available wrapper requires trial and manual intervention and does not always return optimal surface [30]. MAE can aid in automating the process of creating wrappers once the modeling phase of the design in done. The MAE (or MST) of a set of curves, can be extremely useful in the path planning of profile milling operation. If the disjoint curves were to represent each of the profiles to be machined (Figure 24(a)), then the MST of the profiles (Figure 24(b)) can be used to derive the tool path. Figure 24(c) shows a rough sketch of the path (in blue) around the profiles, where the starting of the path can be anywhere. In the case of pocket machining, modification of the contours are done using 20

21 (a) Profiles on a foam glass - Outer box and the disjoint curves as profiles (b) MAE (MST) of the profiles to be cut. Antipodal lines are shown in red. (c)roughsketchof thetool path(inblue) for profile cutting. Figure 24: Profiles to be cut, MAE of the profiles, tool path. the connection points for obtaining simplified tool path. The connection points are obtained by computing Voronoi diagram first (please see Section 2.3.1, p22 in [28]) and then applying MST. The proposed MAE can directly be applied without computing Voronoi diagram (as connection points are minimum distance points) in such an application. A good relation exists between negative α-hull and MAE, as discussed in Section 6. This fact can be used to develop an algorithm to compute negative α-hull for a set of curves, a topic which has not been explored for a set of curves. The negative α-hull [17] will give a tighter boundary (in the sense that it gives boundaries partially wrapping the initial curves) than the convex hull (straight boundaries), and it can definitely be used for purposes such as collision detection, for a cutter of radius α. Also, when convex hull is used, one has to have the predicates left or right for collision detection (let us say, between a cutter of specific diameter, and the convex hull). A negative α-hull will only consist of set of α-disc and hence the collision detecting check is reduced to a comparison of distance measure using the centers of α-discs and the cutter. It should be noted that the negative α-hull led to its discrete variety, the α-shape [18], a prominent one for reconstructing a shape from a set of points. MAE can also be used to identify branch points in a Voronoi diagram for the set of curves. As one of the crucial question in the Voronoi computation of curves, a very prominent problem in the recent past, we believe that this MAE will give the connectivity information for locating the branch point computation. As MAE is actually a minimum spanning tree (MST) for the set of curves, and MST has shown to be the subset of Delaunay triangulation (at least for the point set [39]), which 21

22 is the dual of Voronoi diagram, we can work with MAE as the starting point. This approach will reduce the operation on bisectors (will not require processing such as left-left or curvature checks (please see [27] for more details) or tracing approaches [42], where at each normal point, a check for branching is done. Alternate constraint equations such as the ones in for constrained circles in [38] along with MST could be used to track branch points. Once the branch points and the necessary portions from two curves are identified, to generate Voronoi digram, local computation of bisector is sufficient. This is quite different from the current trend, which typically generates the bisector to compute branch points [27], a process seemingly more expensive Future work The algorithm can be extended for C 0 -continuous curves and possibly for open curves. Another potential application for using the theories that have been developed in this paper could be on blending of curves (see Chapter 14 in [29]). The application areas are being explored at present. Various properties that the computed MAE satisfy is also being explored. The restriction that a curve should not contain straight line portions that will result in infinite solution while computing MAP is also being looked at currently. 8 Conclusion In this paper, an algorithm for computing MAE of freeform curves has been presented. The curves used are planar, closed, and C 1 -continuous. MAE was initially shown for two curves, considering the various configuration between them and then extended to a set of curves. Alpha hull is then described for the set of curves, an extension of the ones used for a set of points. A comparison between alpha hull and MAE for the input set has also been presented. Possible applications, where the MAE could be used, have also been highlighted. References [1] [2] latingeo/noticias doc/108polyhulls-galton.pdf. [3] [4] A new concave hull algorithm and concaveness measure for n-dimensional data sets. Information Science and Engineering, 33(9): , [5] A. V. Aho, J. E. Hopcroft, and J. D. Ullman. Data Structures and Algorithms. Addison-Wesley, [6] D. Albocher and G. Elber. On the computation of the minimal ellipse enclosing a set of planar curves. In Shape Modeling International, pages , [7] H. Alt and O. Schwarzkopf. The voronoi diagram of curved objects. 11th Symposium on Compuational Geometry, pages 89 97, [8] C. G. Armstrong. Modeling requirements for finite element analysis. Computer-Aided Design, 26: , July [9] T. Auer and M. Held. Rpg - heuristics for the generation of random polygons. In Proc. 8th Canad. Conf. Comput. Geom, pages 38 44,

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