Accepted Manuscript

Minimum area enclosure and alpha hull of a set of freeform planarclosed curves

A.V. Vishwanath, R. Arun Srivatsan, M. Ramanathan

PII: S0010-4485(12)00277-1DOI: 10.1016/j.cad.2012.12.001Reference: JCAD 2028

To appear in: Computer-Aided Design

Received date: 1 March 2012Accepted date: 3 December 2012

Please cite this article as: Vishwanath AV, Arun Srivatsan R, Ramanathan M. Minimum areaenclosure and alpha hull of a set of freeform planar closed curves. Computer-Aided Design(2012), doi:10.1016/j.cad.2012.12.001

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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-600036,India

Abstract

Of late, researchers appear to be intrigued with the question; Given a set of points, what isthe 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 non-convex polygons and other variants such as α-hull, poly hull, r-shape and s-shape etc. have beenproposed. In this paper, we extend the question of finding a minimum area enclosure (MAE) toa set of closed planar freeform curves, not resorting to sampling them. An algorithm to computeMAE 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 explorethe relation between alpha hull (using negative alpha) and the MAE.

Keywords: concave hull; alpha hull; enclosing curve; region occupied; convex hull; freeformcurves; minimum area;

1 Introduction

In the domain of point sets, convex hull [39, 14] has been the traditional answer when one asked tofind the region enclosed by the set. Quite a few algorithms exist for computing the convex hull ofa point set [39], both in R2 as well as in R3. Though convex hull has found numerous applications,ranging from interference checking [32] to shape matching [13] of geometric objects, one of thedisadvantages of the convex hull is that, at times it does not best represent the area occupied bythe input set. Hence, researchers started asking the question of what is the region occupied by theset of points and one of the earlier answer appears to be alpha hull (α-hull) [17] (whose discretecounterpart is α-shape [18]). α-hull for a point-set is a shape that generalizes the notion of convexhull. The alpha shape uses a real parameter α, variations of which leads to a family of shapes. Theoutput 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 bythe points by generating non-convex polygons. Figure 1(a) shows a concave hull for a set of pointswhereas the convex hull of the same set of points, shown in Figure 1(b) does not closely represent theset of points. A tighter enclosure can be achieved using concave hull (Figure 1(a)) than using convexhull (Figure 1(b)). For concave hull of set of points, a user-controlled parameter, called as tuningparameter is used to smooth the concave hull. Figure 1(c) shows a coarse concave hull whereas theone in Figure 1(d) is a smoother one.

∗Corresponding author. Email: emry01@gmail.com, Tel: +91-44-22574734, Fax: +91-44-22574732

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(a) Concave hull (b) Convex hull (c) Coarse concavehull

(d) Smoother con-cave hull

Figure 1: Concave hull vs. convex hull [1]

(a) A model of Piston andshaft 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 aprecise definition [35]. In particular, convex hull is the minimum perimeter as well as minimumarea convex enclosure of the set of points. However, for non-convex enclosures, such objectives oftenconflict each other i.e., minimizing area and perimeter is not possible simultaneously [2] and one hasto 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 beenproposed in [16]. A-shape, another variant of α-shape was introduced in [34]. Other parameter-basedhulls such as using exterior angle ω [48], depth-based digging from convex hull [4] have also beenproposed recently. Computation of minimum area polygon that passes through all the points in theset has been addressed in [24].

So far, the question of region occupation seems to be more on the set of points and not muchappear to have been done for other domains, for example, set of closed curves. An algorithm forcomputing the convex hull of a set of freeform curves has been provided in [21] which was thenextended to freeform surfaces [45]. Recently, algorithm’s efficiency in [21] was improved using biarcapproximation in [31]. For set of straight line and circular segments, an algorithm for computingconvex hull has been presented in [49].

Other prominent problems from the point-set that have now been extended to the domain ofgeometric modeling and CAD involving curves and surfaces as input are medial axis [42], Voronoicomputation [12, 27, 23], Delaunay graph for ellipses [22], visibility graph [41], minimum spanninghypershpere [38], and minimum enclosing ellipse [6].

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In this paper, we explore the computation of minimal area enclosure (MAE) of a set of curves.Moreover, we also extend the notion of α-hull, usually defined for a set of points, to a set of curves andcompare it with MAE. This paper is the manuscript version of the abstract, accepted for presentationtrack 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 computationalfluid dynamics (CFD) is to create a mesh of the CAD model for analysis. The CAD model mayconsists of overlapping edges/faces, non-manifold surfaces etc., making the meshing a very longand difficult task. Hence, surface wrappers are being used for the purpose of creating a usableCFD model from the CAD model [10]. For example, for an external flow analysis in CFD, onlythe ‘outer surfaces’, that is, a single continuous surface that wraps the complete model is requiredand no internal geometry and components are needed. However, the outer surface should representthe outer geometry accurately, including sharp corners. Meshing can then be performed on thewrapper, reducing the overall time for analysis considerably. The idea of MAE can be considered asone such wrapper, albeit in two dimensions. Other abstractions such as midsurface [44] or medialaxis transform [8, 43], aid in defeaturing [33], a process where not-so necessary features are removedfrom a model. However, these abstractions have traditionally proved difficult to automate. Figure2 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, convexhull and in applications such as profile milling. The construction of MAE for a set of disjoint curvesis based on the minimal spanning tree, which has perhaps been introduced here for a set of curvesfor the first time. It does given an indication about the connectivity of the curves in general (orhow it is arranged in the plane). Using MAE as a starting connectivity information, an algorithmto find the curves that lie in the convex hull has been developed (where again, the only known workfor arbitrary curves seems to be [21]). Assuming MAE as a rubber band with zero area and thenexpand them (inverse of compressing the band) along with capturing the change in the connectivitywhile 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 haveto be machined and hence the order of the profiles can be used to reduce the total length of tooltravel. MAE, depicted as MST for the disjoint curves can be used to determine the tool path andtravel across profiles.

In the case of pocket machining, modification of the contours are done using the connectionpoints for obtaining simplified tool path. The connection points are essentially minimum distancepoints and then MST is employed (please see Section 2.3.1, p22 in [28]). In general, as MST ofpoints 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 surround-ing trees , the bounding area will be greatly reduced when it is applied around the MAE. Otherareas of application for MAE include geographic information processing, image processing, patternrecognition, and feature detection [3]. Applications are also discussed in bit more detail in Section7.1.

In this paper, the input curves are parametric freeform curves represented using NURBS [40],and are closed C1-continuous (while implementation of MAE of intersecting curves, this conditionis relaxed, please see Section 7), non self-intersecting. We assume that as we travel along a curve inthe increasing direction of parameterization, its interior lies to the left. All the curves are assumedto be simply-connected (i.e. each one does not have holes in it) and no portions of any two curvesoverlap 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 curvesand then use an algorithm such as in [35]. However, this approach may result in a very coarselyapproximated MAE (depending on the sampling on each curve) of the input curves which mightimpede the accuracy of the results when used in applications. This has been shown for algorithms

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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 Booleanunion operation.

• For set of curves that do not intersect, antipodal points are identified and a minimum spanningtree approach has been proposed for obtaining MAE.

• Alpha hull, using a negative alpha, has been shown to be related to the external Voronoidiagram for the set of curves.

• For α = −∞, it is shown that the boundary of the alpha hull is a subset of MAE when curvesdo not intersect each other.

• It is also shown that for α = −∞ for curves that intersect each other, the boundary of thealpha hull is a superset of MAE.

Remainder of this paper is structured as follows: Section 2 presents some basic definitions em-ployed in this paper. Section 3 describes the approach for determining the MAE when only twocurves 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 MAEfor a set of curves is presented in Section 6. Results and discussion are presented in Section 7.Section 8 concludes the paper.

C D

kpN

BEq N k

AF

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 thecurve has been traditionally defined as the circle passing through p and a pair of additional pointson the curve infinitesimally close to p. The curvature of the curve at p is then defined to be thecurvature 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 changesdirection.

Definition 3 A point in a freeform curve is convex, if the outward normal at that point and thecurvature vector are in opposite directions. Otherwise, the point is called concave.

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C1(t)

C2(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 eachother whereas point q is concave.

Definition 4 A convex (concave) portion of a curve is a contiguous portion of the curve where allpoints are convex (concave).

When traveled in the increasing direction of parameterization, portion DC in Figure 3 is aconvex portion whereas F to E is concave. The portions can be identified by first computing pointsof 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-connectedi.e. will not have holes in it. Typically, MAE implies the boundary along with the underlying area,though while computing we do compute the boundary only (in a way similar to most of the algorithmsfor convex hull).

3 MAE of two curves

MAE of a single closed curve is the curve itself. When two curves are involved, the following caseshave 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 thetwo curves (say C1(t) and C2(r)) (Figure 4). To determine whether one curve is completely insidethe other, there are several approaches. We adopted the following approach: first it is required toidentify if the curves intersect or not. A quick check of this condition can be done using the convexhull of the control polygons of the two curves. One can use the “left” predicate (Chapter 1, [39]) todo this operation. If they do not intersect, the curves for sure do not intersect each other. However, ifthey intersect, the curves are then processed for intersection, which typically amounts to polynomialroot 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 P1 on the curve C1(t) and usingthe winding number [36], determine whether the point lies completely inside the curve C2(r). If so,

5

(a) A pair of curvesoutside each other

(b) Convex Hull of thecurves

(c) Pushing the con-vex Hull of the curves

(d) CorrespondingMAE

Figure 5: Rubber band analogy to get MAE for two curves that do not intersect.

the curve C1(t) lies inside the curve C2(r), and the MAE is now C2(r). If not, we take a point (sayP2) on C2(r) and do a similar check. It is to be noted that a point is outside the curve only whenthe 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 alsoassume that the curves contain no straight line segments. Imagine a rubber band that is tightlyenclosing them, such as the convex hull between the two (Figure 5(b). As one keeps pushing theconvex hull and still enclosing the curves (Figure 5(c)), it will reach a point where the area of theenclosure between the curves tends to become null.

Typically, in a rubber-band analogy, from the convex hull to polygonization (such as minimum-area polygon passing through all points), it is implicitly assumed that the rubber-band does notbreak (the analogy for α-hull, which breaks, comes from ice-cream scooping). In the case of disjointcurves, 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 thecurves 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 andhence infinitely many solutions are possible (it is similar to the fact that minimum area polygon fora set of points is not unique and NP-complete [24]). A minimum distance line has been chosen, asthis 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 twocurves. The minimal distance occurs when the points on the respective curves are antipodal to eachother (Lemma 1).

Lemma 1 The distance between two closed C1 non intersecting curves is minimum only when thenormals of the corresponding points are opposite to each other (i.e., antipodal.)

Proof Let us assume that we have found points P1 and P2 on the two curves that are minimal distantbut not antipodal. Construct a circle with these two points as the diameter. If this circle intersectsany of the curves at any point other than the minimal distant point, say P , then it implies thatdistance P1−P or P2−P is less than P1−P2 as diameter is the largest distance between two pointson a circle. This further implies that P1 − P2 cannot be the minimum distance points at the firstplace. Hence this directly means that the circle with P1 and P2 as diameter cannot intersect thecurves at any point other than P1 and P2, and hence is tangential at these two points. The fact that

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the circle is tangential to both the curves and the points are on diametric ends results in a conditionwhere the normals are in opposite directions. Hence the Lemma.

3.2.1 Antipodal constraint of two curves in R2

Consider two planar closed C1-continuous curves C1(t) and C2(r). The antipodal constraint [38] is:

⟨C ′

1(t), C1(t)−C1(t) + C2(r)

2

⟩= 0,

⟨C ′

2(r), C2(r)−C1(t) + C2(r)

2

⟩= 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 thisset (which gives a set of t and corresponding r values), evaluate the points on the curves and then findthe 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 minimumdistance are then chosen. Assuming that there is only one set of points contributing to minimumdistance between the two curves, MAE is then the two curves and the line between the minimumdistance 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 calledminimum antipodal line (MAL) and the points as minimum antipodal points (MAP).

(a) (b)

C2(r)

C1(t)

C2(r)

C1(t)

Figure 6: Antipodal constraint for two freeform planar curves in R2.

3.3 Two intersecting curves

Consider two curves that intersect each other (Figure 7(a)). Imagine a rubber band that is tightlyenclosing them, such as the convex hull between the two (Figure 7(b). As one keeps pushing theconvex hull and still enclosing the curves (Figure 7(c)), it will reach a point where the enclosurecannot be moved beyond the points where the intersection of curves happen, which will then yieldminimum area enclosure (Figure 5(d)).

When two curves are intersecting, the minimum distance between them is zero. Do note that theantipodal condition need not be satisfied at the points of intersection. As the curves are intersecting,

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(a) A pair ofcurves thatintersect eachother

(b) Convex Hullof the curves

(c) Pushing theconvex hull ofthe curves

(d) Correspond-ing MAE

Figure 7: Rubber band analogy to get MAE for two curves that intersect.

(a) Two curves in-tersecting with eachother

(b) Remaining por-tion of the curve C1(t)

(c) Remaining por-tion of the curveC2(r)

(d) Remaining por-tions of the curvesC1(t) and C2(r),forming a MAE.

C1(t)

C2(r)

C1(t)

C2(r)

C1(t)

C2(r)

C1(t)

C2(r)

Figure 8: MAE for two intersecting free-form planar curves in R2.

points of intersections are then determined. The curves are then delineated between the points ofintersection. Portions of one curve that are lying completely inside the other are then removed. Thisagain amounts to checking if a curve is inside another. The resultant will be a set of curves formingthe MAE. Figure 8(a) shows two curves that are intersecting with each other. Figure 8(b) showsthe remaining portions of C1(t) after trimming the portion of C1(t) lying inside C2(r). Figure 8(c)shows the remaining portion of C2(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 unionbetween the curves. As Boolean union yields a unique result, so will be the MAE, for intersectingcurves.

It is to be noted 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 theelimination of portions of curves inside each other resulted in the formation of an inner loop asshown in Figure 9(b). The loops are eliminated using the fact that, for the curves parameterizedin the anticlockwise direction (interior of the curve is to the left), the loops will form a clockwisedirection when traveled in the increasing direction. This can be determined as follows: Let ab be aline completely contained in the loop. Using the tangent vectors Ta and Tb (Figure 9(b)), we can

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C2(r)

C1(t)

Figure 9: Two intersecting curves C1(t) and C2(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 extendthe approach to obtain the MAE for a set of curves. We also assume that there is only one MALbetween two curves. We first check for the curves that intersect and eliminate all the curves thatlie totally inside other curves, as they do not contribute to MAE. After this process, there will becurves that either intersect or lie completely outside.

For curves that intersect with each other, we first find the points of intersections between eachpair of curves. We apply the principle stated earlier to find the MAE for a pair of curves (Section 3.3).This will result in MAE for a pair of curves. We keep repeating this step till all the intersectingcurves are processed. This leaves us with clusters of curves (some obtained using Boolean). Forexample, for the set of curves in Figure 10(a), the processing of intersecting curves will results inFigure 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 othercurves (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 distancebetween MAP for the set of curves lying outside each other. Let δ1, δ2, .... be the minimum distances(computed using antipodal equation (1)) between curves in the MST. We assume that each δ’s aredifferent, 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. Itshould be noted that, a computed MAL between a pair of curves is not included while computingminimum 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 isthe 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 closedcomplements of discs (where these discs have radii −1/α) that contain all the points of S.

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(a) Initial set of curves C ={C1, ..., C11}. S1 = {C1, C2, C3},S2 = {C4, ..., C8}, S3 = {C9, C10}.

(b) After processing curves that in-tersect (using Boolean). BU1 ={C1

⋃C2

⋃C3} etc.

(c) MALs for the set of curves in C′,

where C′ = {BU1, BU2, BU3, C11}.(d) MAE of the set of curves.

C9

C10

C4C5

C6

C7 C8

C11

C1

C2

C3

BU3

BU2

C11

BU1

BU3

BU2

C11

BU1

BU3

BU2

C11

BU1

Figure 10: Illustration of the algorithm.

Definition 8 α-disc is a disc of radius 1α 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 theα-hull. From here, α-hull imply the hull with radii −1/α, unless otherwise mentioned. Figure 11shows 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 doesnot contain any point. The boundary of the α-hull occurs at all the places where the α-disc staysin contact with at least two points (called as neighboring points) simultaneously as shown in theFigure 12(b). Figure 12(a) shows an arbitrary disc of radius (-1/α) that does not form a part ofthe α-hull, whereas the disc in Figure 12(b) is constrained to pass through the points and hence ispart 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 Pi in S is the set of all points closer to Pi than to Pj , ∀PjǫS and i 6= j. The Voronoi diagram is then the union of the Voronoi cells of all the points in theset.

Note that the Voronoi diagram used here is the closest point Voronoi diagram, unless otherwisementioned that will be the same hereafter. For completeness, Lemma 5 in [17] is restated here

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Algorithm 1 MAE(C = C1, ..., Cn)Determine curves that intersect each other.Eliminate curves that lie completely inside.Find sets of curves that intersect. Let Si denote each set, and each Si will consist of curves fromC.for each set Si of intersecting curves do

Perform Boolean union.Represent each Boolean union as a single curve (say BUi).Eliminate interior loops.

end forLet C ′ = {{BUi}

⋃{C − {Si}}}.for each curve in in C ′ do

Compute MAP to all other curves in the set.end forif 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.

elseReturn 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 allα < 0.

α-hull boundary occurs at places where the α-disc simultaneously stays in touch with at leasttwo points. Since the α-disc is a circle of constant radius, the center of the disc at all times isequidistant from both the points (Figure 13(a)) and hence the centers of α-disc has to be on theVoronoi 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 sofar. In this paper, we apply this to set of curves represented exactly (i.e. we do not approximatethe curves using sample points). It is to be emphasized that we employ only the α-hull, as opposedto using its discrete variety called α-shapes [18].

We first extend the concept of Voronoi of points to that of curves. Then the various combinationsof the curves, its respective Voronoi diagrams and the α-disc traversal is explained.

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(a) α-disc having noconstraint

(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 onDelaunay triangulation. It should be noted that theoretical foundation as well as algorithms forVoronoi cell/diagram of a set of freeform curves has been well-developed in the recent past (forexample, 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, weexplore 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 thatof points. Note that the interior of a closed curve lies to its left as we travel along the increasingdirection of parametrization.

Definition 10 Portions of Voronoi diagram that lie in the exterior are called external Voronoidiagram (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. Hencethe 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 trueeven when the curves begin to assume arbitrary convex shapes (Figure 14(c), where only a portionof 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. Itis to be noted that Definition 9, when employed for a set of closed curves, excludes the self-Voronoiportion [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 C1(r) in a concave portion, where apoint ‘r’ has two points r2 and r3 equidistant within C1(r) itself. A point such as p is equidistancebetween two different curves C0(t) and C1(r) and hence does not belong to self-Voronoi. Anotherfeature 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 thedisc curvature (Figure 16)[27]. Hence for a curve, when self-Voronoi is included, the α-hull may notretain most of the contour of the shape. Lemma 2, which was for point sets, is then extended forcurves (Lemma 3).

Lemma 3 The centers of the α-disc have to lie on the Voronoi diagram of the set of curves for allα < 0.

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(a) Center of α-discequidistant from boththe points.

(b) Locus of α-disc ofvarying radius tracingthe voronoi

Figure 13: α-disc centers

Proof α-hull boundary occurs at places where the α-disc simultaneously stays in touch with twopoints on the curve. Since the α-disc is a circle of constant radius, the center of the disc at alltimes 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 (curvaturecondition, 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 ofthe set of curves.

When the input contains just two points, the α-hull between them is two points themselves forall α, except α = 0. However, when the input consists of two curves, α-hull between them is thecurves 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 anenclosure 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 area family of hulls, which depend on the user input parameter α, it is necessary to determine whichparticular α-hull that is similar to the MAE before the comparison is done. The corresponding αthat generates the MAE is used and its boundary lengths are compared. Prior to that, it is shownthat 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 provedfor completeness.

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(a) Voronoi of circles ofradius Zero

(b) Voronoi of circles ofradius ǫ

(c) A portion of bisector betweentwo curves that will contribute to theVoronoi cell of the set of curves forarbitrary convex shapes

Figure 14: Voronoi for curves

C0(t)

C1(r)

po

r

r1

t

r2 r3

Figure 15: Example of self-Voronoi

Lemma 4 The length of the boundary of the α-hull is inversely proportional to the area occupiedby the hull ∀ α < 0.

Proof From the proof of Lemma 3, it is clear that the alpha disc forms the boundary of the α-hullonly if it simultaneously touches two points, either of different curves or of the same curve. Considerthe case of α-hull of a single concave curve C(t) (The case of two separate curves intersecting/non-intersecting can be proved in a similar manner). Supposing the curve C(t) is arc length parameterizedwith 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

Lengthα = t1 + (1− t2) + 2rθ (2)

where θ= sin−1 C(t1)−C(t2)2∗r

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(a) (b)

C0(t)

C1(r)

a

C0(t)

C1(r)

b

Figure 16: [27] (a) The disk radius smaller than the radius of curvatures at the footpoints of thecurves. (b) The radius of the disk is greater than the radius of curvature of C1(r) at the footpointimplying 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 parametervalue 0 to t1, t2 to 1 and the arc of the α-disc. Let us consider the case when r decreases as wetravel 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 similarmanner). Hence the lemma.

From the Lemma 4, it is evident that the maximum length of the boundary occurs when theradius of the α-disc is equal to zero (or α = -∞). For curves that are intersecting each other, thefollowing lemma (Lemma 5) is applicable.

Lemma 5 For α = −∞, boundary of the α-hull is a superset of the boundary of MAE of the setof curves, when they intersect each other. Precisely, the MAE with its internal loops will constitutethe α-hull.

Proof For a set of intersecting curves, the MAE is a single enclosing curve that does not detectthe 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.

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(a) Input set of curves. (b) α-hull of the set of curveswith α = −∞

(c) MAE of the set of curves.

Figure 18: Generalization of α hull and MAE.

For set of curves that do not intersect, α-hull is related to the MAE in the following way (Lemma6).

Lemma 6 For α = −∞, boundary of the α-hull is a subset of the boundary of MAE of the set ofcurves, when they do not intersect each other and the curves lie outside each other. Precisely, theMAE 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 issame, as MALs’ have only zero area. MAE forms a continuous enclosure as opposed to α-hull whichgenerates disjoint sets. In this particular case, MAE involves the MAL between them. Thus theboundary 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 booleanunion, 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 minimumof all the minimum antipodal distances δ1, δ2, .... (MAP’s are computed for different curves but notwithin the same curve), then the α-hull is the set of input curves itself. This is similar to the factthat 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 6appear difficult when the set of curves contain both categories - i.e. some intersect and some curveslying outside. For the set of curves in Figure 18(a), α-hull (Figure 18(b)) has internal loops as well asdisjoint curves. MAE in Figure 18(c) does not contain the inner loops, where as the curves that donot intersect are connected using MAP and minimum spanning tree approach. One cannot concludethat whether the MAE is a superset/subset of the α-hull (with α = −∞).

7 Implementation results of MAE

Figures 19-21 shows the implementation results of the algorithm. All the implementation have beencarried out using IRIT [19] geometric kernel and its constraint solver [20]. Figure 19(a) shows twocurves 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

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(a) Two curves (b) Antipodal linefalls in the convexregion of both curves

Figure 19: MAE for two free-form planar curves in R2.

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 curveused is 4.

MAE has also been computed for curves in Figures 21(a) and 21(e), which contains straightline portions as computation of MAP does not play a role. The results also indicate that the MAEapproximates 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 R2.

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(a) Model of a flange (b) Convex hull of theflange

(c) Alpha hull (d) MAE

(e) Model of a knucklejoint

(f) Convex hull of thejoint

(g) Alpha hull (h) MAE

Figure 21: MAE for mechanical joints.

7.1 Discussion

7.1.1 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, thishas been shown to be NP-Complete [25] and hence no polynomial time algorithm to compute it isnot available. In our testing, we have used the best known approximation algorithm for computingminimum area polygon [37]. Recent papers on concave hulls, the ω-hull [48] and digging of convexhull [4] have also been used for comparison. It should be noted that it is our implementation ofthese algorithms.

For the object in Figure 22(a), 483 points were generated from the set of curves. Minimum areapolygon (in red) for the set of points (in green) is shown in Figure 22(b). Clearly, the result is wayoff 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 lotof experimentation. Result for other threshold values are shown in Figures 22(d) and 22(e). Notethat the hull seems to be better approximated with threshold 4 than with 10 and hence one cannotsay 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 toprevail with ω = 120 , yet again, obtained using experimentation, even though the result does notseem to be close to MAE. Also, for a ω of 90, for a sampling density of 1124, the result is shown inFigure 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

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(a) Test object (b) Minimumarea 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 ofpoint-sets.

to determine an appropriate sampling, the parameter (such as threshold in [4] and ω in [48]) alsohas to be identified, which is mostly by experimentation, not so feasible tasks in general.

7.1.2 Running time

Permute and reject strategy [9], which computes all the permutation to return minimum area polygontook few hours for even a set of nine points. The approximation algorithm [37] runs in O(n4). Table 1shows the running time (in Intel core i3-2330M Processor Clock rate-2.20GHz, RAM-2 GB Windows7 (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 comparethe 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 as22(f).

7.1.3 Limitations

Algorithm 1 returns an unique MAE only under the assumptions that MAP between two curves isunique and no distances computed using MAP are same across different curves as MST will thennot be unique. We have also assumed that, for the computation of MAP, the curves neither havestraight line portions or overlapping ones. Currently, the MAE is represented as a set of piecewise

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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 couldprove useful in various applications. For example, queries such as whether a point is inside a MAEcan be easily addressed if there is a good representation for the MAE.

7.1.4 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 createa mesh of the CAD model for analysis. Surface wrappers are being used for purpose creating ausable 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 twodimensions. Figure 23(a) shows a top view of race car model and its MAE (Figure 23(b)) as theouter wrapper, which is better than its convex hull (Figure 23(c)), in terms of closer wrapping. Itappears that current available wrapper requires trial and manual intervention and does not alwaysreturn optimal surface [30]. MAE can aid in automating the process of creating wrappers once themodeling phase of the design in done.

The MAE (or MST) of a set of curves, can be extremely useful in the path planning of profilemilling operation. If the disjoint curves were to represent each of the profiles to be machined (Figure24(a)), then the MST of the profiles (Figure 24(b)) can be used to derive the tool path. Figure24(c) shows a rough sketch of the path (in blue) around the profiles, where the starting of the pathcan be anywhere. In the case of pocket machining, modification of the contours are done using

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(a) Profiles on a foam glass - Outer box and thedisjoint curves as profiles

(b) MAE (MST) of the profiles to be cut. Antipodallines are shown in red.

(c) Rough sketch of the tool path (in blue) for profilecutting.

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 bycomputing Voronoi diagram first (please see Section 2.3.1, p22 in [28]) and then applying MST. Theproposed MAE can directly be applied without computing Voronoi diagram (as connection pointsare minimum distance points) in such an application.

A good relation exists between negative α-hull and MAE, as discussed in Section 6. This factcan be used to develop an algorithm to compute negative α-hull for a set of curves, a topic which hasnot been explored for a set of curves. The negative α-hull [17] will give a tighter boundary (in thesense that it gives boundaries partially wrapping the initial curves) than the convex hull (straightboundaries), and it can definitely be used for purposes such as collision detection, for a cutter ofradius α. Also, when convex hull is used, one has to have the predicates ‘left or right’ for collisiondetection (let us say, between a cutter of specific diameter, and the convex hull). A negative α-hullwill only consist of set of α-disc and hence the collision detecting check is reduced to a comparisonof distance measure using the centers of α-discs and the cutter. It should be noted that the negativeα-hull led to its discrete variety, the α-shape [18], a prominent one for reconstructing a shape froma set of points.

MAE can also be used to identify branch points in a Voronoi diagram for the set of curves. Asone of the crucial question in the Voronoi computation of curves, a very prominent problem in therecent past, we believe that this MAE will give the connectivity information for locating the branchpoint computation. As MAE is actually a minimum spanning tree (MST) for the set of curves, andMST has shown to be the subset of Delaunay triangulation (at least for the point set [39]), which

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is the dual of Voronoi diagram, we can work with MAE as the starting point. This approach willreduce 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 forbranching 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 necessaryportions from two curves are identified, to generate Voronoi digram, local computation of bisectoris sufficient. This is quite different from the current trend, which typically generates the bisector tocompute branch points [27], a process seemingly more expensive.

7.1.5 Future work

The algorithm can be extended for C0-continuous curves and possibly for open curves. Anotherpotential application for using the theories that have been developed in this paper could be onblending 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 acurve should not contain straight line portions that will result in infinite solution while computingMAP is also being looked at currently.

8 Conclusion

In this paper, an algorithm for computing MAE of freeform curves has been presented. The curvesused are planar, closed, and C1-continuous. MAE was initially shown for two curves, consideringthe various configuration between them and then extended to a set of curves. Alpha hull is thendescribed for the set of curves, an extension of the ones used for a set of points. A comparisonbetween alpha hull and MAE for the input set has also been presented. Possible applications, wherethe MAE could be used, have also been highlighted.

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A. V. Vishwanath is currently working in General Electric, Bangalore. He did his M.S. in the Departmentof Engineering Design, Indian Institute of Technology Madras, Chennai, India. His current research interestsinclude computational geometry and geometric modeling.

R. Arun Srivatsan is currently a project officer at the Department of Engineering Design, Indian Instituteof Technology Madras, Chennai, India. He earned his M. Tech. and B. Tech. degrees from the same place.His current research interests include computational geometry, kinematics and dynamics of mechanisms, androbot manipulation.

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M. Ramanathan is currently an Assistant Professor in the Department of Engineering Design, IndianInstitute of Technology, Madras. He earned a B.E. degree in Mechanical Engineering from Thiagarajarcollege of Engineering, Madurai, India. He received his M. Sc. (Engg.) and Ph.D. from the Departmentof Mechanical Engineering at the Indian Institute of Science, Bangalore, India. He did his post-doctoralresearch at Technion, Haifa, Israel and then at Purdue university, USA. His current research interests includegeometric modeling, computational geometry, shape and image search, mesh model analysis, and BioCAD.

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Research Highlights

• This paper addresses region occupation to a set of closed planar freeform curves.

• Minimum area enclosure (MAE) has been proposed and implemented using curves exactly.

• MAE’s relation to α-hull of a set of curves has also been explored.

• Comparison with discretization-based approach has been perfomed.

• Applications of MAE are also discussed.

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