Shortest path in a multiply-connected domain having curved ...

Accepted Manuscript
Shortest path in a multiply-connected domain having curved
boundaries
S. Bharath Ram, M. Ramanathan
PII: S0010-4485(12)00279-5
DOI: 10.1016/j.cad.2012.12.003
Reference: JCAD 2030
To appear in: Computer-Aided Design
Received date: 7 June 2012
Accepted date: 4 December 2012
Please cite this article as: Bharath Ram S, Ramanathan M. Shortest path in a
multiply-connected domain having curved boundaries. Computer-Aided Design (2012),
doi:10.1016/j.cad.2012.12.003
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Shortest path in a multiply-connected domain having
curved boundaries
S. Bharath Ram and M. Ramanathan, ∗
Department of Engineering Design,
Indian Institute of Technology Madras,
Chennai 600036, India
{bharathsram, emry01}@gmail.com
Abstract
Computing shortest path, overcoming obstacles in the plane, is a well-known geometric
problem. However, widely assumed obstacles are polygonal in nature. Very few
papers have focused on the curved obstacles, and in particular, for curved multiplyconnected
domains (domains having holes). Given a set of parametric curves forming a
multiply-connected domain (MCD), with one closed curve acting as an outer boundary
and several non-intersecting inner curves (loops) as representing holes and two distinct
points S and E lying on the outer boundary, this paper provides an algorithm to find
shortest interior path (SIP) between the two points in the domain. SIP consists of
portions of curves along with straight line segments that are tangential to the curve.
The algorithm initially computes point-curve tangents (PCTs) and bitangents (BTs)
using their respective constraints. A generalized region lemma is proposed, which is
then employed to remove PCTs/BTs that will not contribute to the potential path and
subsequently aiding in removing a few inner loops. The algorithm is designed to explore
all potential paths. Merging of paths is also proposed to avoid redundant computations.
Final SIP is chosen from potential paths using the length of each path. The algorithm
also has the potential to give all path of equal lengths contributing to shortest path
(within a tolerance level). As the algorithm computes all the potential paths on the
fly, there is no need to employ visibility graph to compute the shortest path. Curves
are represented using non-uniform rational B-splines (NURBS). The algorithm uses the
curves as such and does not discretise into point-sets or polygons. Extension to handle
curves with C 1 discontinuities and S and E not on the outer boundary has also been
described. Results of the implementation are provided and complexity of the algorithm
is also discussed. This paper follows up the one presented for simply-connected domain
(SCD) in [18].
1 Introduction
Computing Euclidean shortest path between two points avoiding obstacles in a domain is one
of the fundamental problems in geometry. A well-researched problem is the one that involve
∗ Corresponding author. Tel: +91-44-22574734
1

non-intersecting polygonal obstacles (e.g. [14, 19]). A prominent approach to compute the
shortest path for polygonal obstacles is via the visibility graph construction [2] as shortest
path is a subset of the visibility graph.
Shortest interior path (SIP) in a curved boundary can be thought of as a combination
of Euclidean and geodesic, where the SIP will consist of straight lines and arcs of curved
boundaries [18]. Approximating curved boundaries with polygons generates inaccuracies in
SIP and is not satisfactory in general [7]. Another approach to compute SIP in a curved
boundary is to sample points on the curved boundary [12] and then employ a typical shortest
path algorithm (such as Dijkstra [1]). The accuracy will depend on the sampling technique
used and is an approximate computation of the SIP. Algorithm for piecewise-linear boundaries
such [11] has also been extended to curved boundary using “funnels” in [13] assuming the
input (curved boundary) is already triangulated. Numerical approximation of shortest path
in curved obstacles using fast-sweeping of Eikonal equations [22] and grid-based approach has
been presented in [17].
Very few algorithms have handled the computation of SIP of curved boundary without
approximating with polygons or sample points, albeit theoretically. Computing visibility
graph based on greedy approach pseudo-triangulation for a set of non-intersecting convex
curved objects has been presented [16]. A similar approach for convex curves that intersect
transversely has been given in [5].
For arbitrarily shape curved boundary (not just convex), Bourgin and Howe [3] presented
an algorithm using string formulation on the intersection of the line connecting the start and
end points with the curved boundary. Based on the maximal distance of the curve from the
line, different cases are delineated to find the points on the curve that belong to the SIP.
Fabel [10] has presented an algorithm for computing the shortest arcs in a closed planar disk.
Both [3] and [10] are theoretical descriptions and no implementation results are provided.
An algorithm with implementation results for simply-connected arbitrarily shaped curved
boundary has been presented in [18] using tangent computation and region check. The major
idea in computing the SIP of curved boundaries [3, 10, 18] is to avoid computing visibility
graph through identifying potential straight lines/arcs of the curves using strategies such as
maximal distance [3] , region check [18] etc.
To the best of our knowledge, no known algorithm with implementation results is available
for arbitrarily shape curved boundary of a multiply-connected domain (MCD). In this paper,
the aim is to find the shortest interior path (SIP) of a MCD having curved boundaries,
extending the algorithm presented in [18], using the basic idea of tangent computation. All
the possible paths are directly identified during the computation effectively eliminating the
use of algorithms such as Dijkstra. SIP is then identified from the set of paths. The following
are the key differences from [18].
• In case of simply-connected domain (SCD) [18], the focus was on computing the SIP
itself, since SIP was unique. As there will be number of paths that could contribute
towards SIP in the MCD, in this paper, the focus is on eliminating a path that will not
play a role in the set of potential paths. Lemma 3 in the paper indicates the interaction
that a path can have and the cases for which the path will cease to exist.
• In [18], line connecting S and E was used to divide the boundary curve into two portions.
As such a division is not possible in multiply-connected domain (MCD), Lemma 4 in the
2

paper, termed as generalized region lemma (GRL) (a generalized version of the region
check lemma in [18]) has been proposed.
• Exterior and interior regions are formulated based on Lemma 4 to eliminate PCTs/BTs/
inner curves that do not further play a role in path computation.
• Merging of paths (which was not required in SIP of SCD) to avoid redundant computation
has also been proposed.
• A tree data structure has been proposed and implemented for storing and manipulation
of identified paths.
• A methodology has been proposed and implemented for curves having C 1 discontinuities
(which was not handled in SCD).
In this paper, the curved boundary is assumed to be defined by a set of non-self-intersecting
closed parametric curves without discontinuities (an extension to curves with discontinuities is
also discussed with implementation results) and also they do not contain straight line portions.
The domain is assumed to be multiply-connected, with one outer boundary (bounding box
replacing the outer boundary has also been discussed) and at least one inner boundary. It is
assumed that, when traveled along the increasing direction of paramaterization of the curve,
its interior lies to its left.The curves are typically defined by Non-Uniform Rational B-Spline
(NURBS) [15]. The degree of the curves are assumed such that the inflection points on the
curve can be captured (i.e. minimum degree of the curve used is 3).
2 Definitions
Let the set of parametric curves be Q = {C0(r0), C1(r1), ... , Coc = Cn(rn)} for which SIP
has to be determined. Let S and E lie on the outer boundary curve Coc. All definitions from
Section 2 of [18] are directly applicable and hence not repeated here.
Definition 1 A path is a set of PCTs/BTs which are completely contained and curve portions
forming a contiguous segment.
Definition 2 An extended PCT/BT is the extension (along the direction of PCT/BT) of a
completely contained PCT/BT till its first intersection with one of the input curves.
By virtue of this definition, the extended PCT/BT is also completely contained.
Definition 3 SIP is the shortest among all available paths from S to E.
3 Preliminaries/Observations
Lemma 1 SIP consists of only concave portions of outer/inner loops and straight lines that
are completely contained in the MCD (adopted from [18]).
Lemma 2 The SIP preserve the direction of traversal (clockwise or counter-clockwise) in the
input curves [18].
3

It is to be noted that when both S and E lie on the same concave portion, then the SIP is
the portion of the curve C(t) between the two points [[3], Theorem 2.7, p220]. Also, straight
portions of SIP are tangential to the concave portions [18].
Definition 4 A BT is said to be consistent with respect to a path if it preserves the direction
of traversal of the path as stated in Lemma 2 .
It is also to be noted that, only one SIP is possible for a SCD, joining two distinct points,
lying on the boundary or interior [[21], Theorem 6.1, p147], which does not hold good for a
MCD.
Suppose that there is an SIP for a set curves constituting MCD. If an inner loop is added,
it should be observed that the entire SIP can change altogether. This suggests that an
incremental construction may not be possible and one has to consider all possible inner loops
at the same time. Moreover, local shortest path cannot be fine tuned to get the global path.
For example, a shortest path between two inner loops may not play a role at all in the global
shortest path between S and E (even-though those two curves may contribute to the final
path). Hence, all possible paths have to be detected to compute SIP.
Lemma 3 indicates the interaction that a path can have and the cases for which the path
can be eliminated, reducing the number of candidates and also computation of further BTs
for identifying SIP.
Lemma 3 (Path Elimination): Let there be a path. Let there be extended PCTs from S or
extended BTs consistent with the direction of traversal of the path. The path is eliminated
if it intersects any extended PCT from S or extended BT originating from the same concave
portion through which the path has passed.
Proof Let Fi represent the starting footpoint of BTi. Let η consist of the PCT, concave
portion between T and F1, and BT1 itself (refer to Figure 1(a)). Let the the bitangents BT2,
and BT3 arise out of the same concave portion as that of BT1. Let EBT2 and EBT3 represent
the intersection point of a path with their respective extended BTs.
Case 1: If η intersects any extended PCT from S at some point, say X, then the portion of
η till X can be replaced by the straight line portion SX which would make the path shorter
and hence η cannot be a potential SIP.
Case 2: Let the footpoint of a bitangent lies between T and F1 (such as F2 of BT2). Let
the path ζ = η + ζa and ζa intersects extended BT2 at EBT2. Then the cumulative portion
formed by concave portion of F2F1, BT1 and ζa can be replaced by straight line portion from
F2 to EBT2 to get a shorter path. Hence ζ can be removed.
Case 3a: Let the footpoint of a BT lies outside of T and F1 (such as F3 of BT3). Let the
path ζ = η + ζc. BT1 + ζc is longer than the path (F1F3 + F3 to EBT3) by the convex hull
property. Hence ζ can be removed.
Case 3b: Let the path ζ = η + ζb. The path ζ has self-intersection and hence it can be
eliminated.
Lemma 4 Let ζ be a path from S to E. Let G ∈ ζ be a point on a concave portion of one of
the input curves. Let SG and GE denote the portions of ζ before and after G respectively.
4

S
EBT3
BT3
T F2
F3
ζc
ζb
BT2
BT1
EBT2
ζa
(a) Barrier lemma (BTs from same concave region)
Figure 1: Barrier lemma and region lemma
S
ζ
G B
(b) Generalized region lemma
Consider SG, portions of an input curves and an extended BT from the same concave portion
in which G is present (or an extended PCT from S) as boundary components forming a closed
region, R. If R does not contain E, then for ζ to be a SIP, no portion of ζ shall be contained
in R.
Proof Proof is by contradiction. There exists a region R and some portion of GE is contained
in R. E does not belong to R and for ζ to reach E it has to cross the boundary of R. Thus
ζ has to intersect one of the boundary components of R. Considering each case separately,
Case 1: ζ cutting a portion of an input curve
Cutting one of the input curves would mean ζ is not completely contained in the region
specified by the set of curves.
Case 2: ζ cutting a PCT or BT or ζ cutting SG
This case can be proved by applying Lemma 3.
Lemma 4, termed as generalized region lemma (GRL) (a generalized version of the region
check lemma in [18]) helps in eliminating some of the inner loops/PCTs/BTs by forming
closed regions. Figure 1(b) shows a closed region R formed from SGBIS. This lemma holds
even if the roles of S and E are swapped.
4 Algorithm
Figure 2(a) shows a set of curves for which SIP has to be determined between S and E (line
SE is also shown). Let all curves except COC have clockwise orientation. Let S and E be the
distinct points on COC for which SIP has to be determined (do note that this condition is for
explanation purpose and can be relaxed, for example the outer boundary can be replaced with
a bounding box). Without loss of generality, let S be the point with lower parametric value
of the two. Let CT G be the curve at which a PCT becomes tangent from an originating curve
5
R
I
E

E
S
(a) Sample set of curves (b) PCTs that are
completely contained in
MCD
Figure 2: Sample set of curves and PCTs from S
(the footpoint denoted as T ). Let CF E denote the curve at which the extended PCT intersects
first, with the intersection point denoted as IF E. Also, let IOC be the first intersection of the
PCT with the outer curve COC. It is to be noted that the extended PCT can intersect the
outer curve first and in that case, CF E = COC, and IF E = IOC.
4.1 Processing PCTs
Initially, PCTs from S (using constraint equation in [18]) are used as the starting tangents.
All the PCTs that are not completely contained in the MCD are then removed (Figure 2(b)
shows only PCTs that are completely contained in the MCD). Considering the Figure 3(a),
for the PCT ST1, the curve C1 is CT G which has the footpoint T1. The extended PCT first
intersects at P1. Hence IF E = P1. Since P1 is at the outer curve COC, IF E = IOC = P1
and CF E = COC. Note that T1P1 is completely contained in the MCD. For the extended
PCT whose first intersection IF E = I3 with curve CF E = C3, CT G = COC. However, for the
extended PCT whose first intersection IF E = I4 with curve CF E = C3, CT G = C5. All the
extended portions of the PCTs are identified by dashed lines.
4.1.1 Exterior region identification
Among the PCTs that are completely contained in MCD, a PCT with its footpoint T or IF E
lying on COC, the closest in clockwise direction along the curve COC to the end point, E, is
selected. Though the search is done in clockwise direction, the closest point (for example,
P3 in Figure 3(a)) is in the counter-clockwise portion between S and E, when travelled from
S. Hence the PCT is named as CCRP CT (counter-clockwise region PCT). For a PCTs thus
selected, the following regions are described based on the conditions delineated as follows.
1. If CT G and CF E are both COC respectively, then two regions are identified. One region
consisting ST and portion of outer curve from T to S (that does not contain E) as
6
S

I4
C3
I3
P3
P1
C5
C1
T1
(a) PCTs Extended to get first intersection
S
I5
(b) Regions R1 and R2 for elimination
of PCTs/Inner curves
Figure 3: Processing PCTs using their extensions
boundaries. The other by T IF E and the portion of outer curve (that does not contain
E) from IF E to T as boundaries.
2. If CT G is not COC and IF E is on COC, then the region is defined by the boundary curves
SIF E and the portion of outer boundary (that does not contain E) from IF E to S (in
Figure 3(b), region R1 with boundary SI5S).
3. If CT G is COC and IF E not on COC, then the region is defined by the boundary curves
ST and the portion of outer curve (that does not contain E) from T to S . For e.g., in
Figure 3(b), region R2 defined by the closed boundary ST6S. It is to be noted that the
first intersection of ST6 is at I3 with the curve C3 (Figure 3(a)).
A similar search is done in the counter-clockwise direction and CLRP CT (clockwise region
PCT) is obtained. Also corresponding regions are identified.
4.1.2 Interior region identification
Interior regions are formed when footpoints of PCTs and/or extended PCTs first intersection
points are falling on the same curve, but not on COC. The curve portion between the mutually
farthest points on the curve and their respective PCT/extended PCT form a region. For
example, in Figure 4(a), several extended PCTs (a different starting point W for the PCTs
is used for illustrative purpose) have IF E’s on the same curve CSM. The interior region is
formed by W I1I2W , where I1 and I2 are the mutually farthest points along the curve CSM.
In this particular case, I1 and I2 are the first intersections of the extended PCTs of the curves
C1 and C2 respectively.
7
R1
T6
T5
R2
S

C1
I1
I2
CSM
(a) An example of interior
region elimination
C2
W
4.1.3 Region-based elimination
E E
(b) PCTs from S after elimination
Figure 4: Valid PCTs from S, E and interior region
S
(c) Termination list from E
Once the regions are identified, all PCTs and curves that are contained in the regions are
eliminated (Lemma 4). In order to get the PCTs within the regions, the values counterclockwise
angle with respect to the SE line or parameter values of IOC are used. Since the
PCTs are from the same starting point, the angle and the parameter of IOC correlate. Firstly
a range of parametric values is identified using the PCTs that contribute to the identification
of regions. Then tangents falling within or outside the range are eliminated accordingly. For
curve elimination within a region, typical intersection check followed by ray shooting is done.
Figure 4(b) and 4(c) show the remaining PCTs from S and E after employing region-based
elimination. Inner loops that are eliminated using exterior regions R1 and R2 are shown as
gray in Figure 3(b). Algorithm 1 describes the pseudocode for obtaining PCTs and processing
them.
4.2 Processing BTs
BTs are computed from each PCT in the starting list (which contains thus far processed
PCTs, Figure 4(b)). A similar region elimination strategy is followed for BTs with some
modifications to the region identification. Consider a path ζ, shown in Figure 5(a) from S
to a point T on a curve CT G, which is an incomplete potential SIP (i.e., the path has not
yet reached the end point E). At and near T , CT G is concave. Let φ (clockwise or counterclockwise)
be the direction induced by ζ on CT G. Let NIF be the closest inflection point to
T on CT G in the direction φ. Now T and NIF identify a concave portion of CT G.
To identify the next set of potential paths for ζ, the set of bitangents from concave portion
T NIF is computed. The bitangents that are not completely contained in MCD and those that
are not consistent with direction φ are removed. Let the starting foot point of each bitangent
be denoted by FST . The starting curve, CST , for all bitangents is CT G. Let the end foot point
and corresponding end curve be denoted by FEN and CEN. Each bitangent is extended from
8

IF E
FEN
FST
T5
R1
(a) Region R1 using
ST5FST FENIF ES as its boundary
ζ
S
Figure 5: Regions using BTs
R2
(b) Region R2 formed using BTs
FEN opposite to FST to get the first intersection point, IOC, with the outer boundary COC.
Similar to PCT case, the first point of intersection of the extended BT (i.e. BTs from FEN)
with input curves, is obtained as IF E on the curve CF E.
4.2.1 Region identification and elimination of BTs
As in the case of PCT, region boundaries are identified. Assuming, for a BT, IF E was on
the curve COC. An exterior region is formed by ζ (ST5), concave portion T5FST , bitangent
FST FEN, extended BT FENIF E and the portion of the outer curve between IF E and S that
does not contain E (region R1 in Figure 5(a)). Other cases for exterior region identification
are delegated similar to that of PCTs (another example of exterior region R2 is shown in
Figure 5(b)).
For the PCTs, starting or ending point is available to compute interior regions. In the
case of BTs, instead of a starting point, a concave portion (such as T5FST ) plays this role.
Identification of interior regions is then similar to that of the PCTs (i.e. finding farthest
tangents lying on the same curve). Using the interior and exterior regions, BTs and curves
falling in them are eliminated. This process is done recursively for all BTs further computed.
Along the recursion, a tree of paths is maintained.
4.2.2 Processing BTs from S and E
If S is in a convex portion, then no BT from the portion is completely contained. If S is
in a concave portion and CLRP CT as mentioned in 4.1.1 exists, then all BTs in clockwise
direction from the concave portion containing S have their starting foot point FST in the
region identified by CLRP CT . If CLRP CT does not exist, then BTs in clockwise direction
from the concave portion containing S are to be obtained and further processed according to
9

(a) Rejected path (in
blue)
(b) Rejected path (in
blue)
BT1
ζ2
ζ1
(c) ζ1 contains BT1
Figure 6: Rejected and merged paths (in blue)
(d) Merged paths
section 4.2. To do this, a dummy tangent line starting at S in the direction φ as clockwise
is added to the BT processing algorithm. Similar procedure is followed for counter-clockwise
direction and for the end point E.
4.3 Further elimination of paths
Sections 4.1 and 4.2 detailed how the PCTs/BTs can be eliminated based on the region
formulation using Lemma 4. However, there are further elimination of PCTs/BTs is possible.
This is done through the application of Lemma 3 which stated that a path cannot intersect
certain PCTs/BTs in specific ways. Thus a check is made to eliminate paths that are not
valid. Figures 6(a) and 6(b) show the path in blue that can be eliminated using Lemma 3.
4.4 Merging of paths
A tree structure is maintained to store the computed paths thus far. The tree starts with a
single node S, and the children for each node is the list of valid tangents obtained using the
node. The procedure processes every child recursively till E is reached via. the termination
list (which contains the PCTs processed from E) or no further valid tangent is available.
Along with every node, the length from S up to that node is also stored by computing the
Euclidean distance for tangents and line integral for curved portions. Thus the length of leaf
node when it reaches E will be the length of that particular path.
While the above procedures are followed recursively, a merging step is performed to avoid
redundant computations. Simply, paths are merged if a bitanget to be processed has been
already traversed. In this case all subtree of paths already computed is added to the current
path. The length corresponding to each node is updated accordingly. An example of merging
is shown in Figures 6(c) and 6(d). In Figure 6(c), the bitangent BT1 is already traversed by
the path ζ1 and hence the path ζ2 is merged when BT1 is reached. Algorithm 2 (in appendix)
gives the pseudocode for processing the bitangents.
10

4.5 Termination of a path
In the case of SCD, the algorithm terminates, when the latest footpoint of the PCT/BT
of the SIP algorithm thus far and the footpoint of the PCT from E lie on a same concave
portion. However, as the SIP is not unique in MCD, a similar check is used to compute only
the termination of a path. For completeness, the following corollary (Corollary 1) from [18]
has been modified for a path termination.
Corollary 1 Let CP1 be the latest footpoint of the PCT/BT, which will lie in a concave
portion of the curve of the SIP algorithm thus far. Let CP2 be the footpoint of the PCT from
E. If CP1 and CP2 lie on a same concave portion, then the path is terminated.
The algorithm terminates when Corollary 1 is satisfied for all the path. SIP is then
computed using the integration of each path and identifying the least of them. Algorithm 3
gives the pseudo-code for computing the shortest interior path, given two points S and E on
the set of curves CurveList (please note that the pseudocode does not include handling C 1
discontinuities).
5 Results and Discussion
Fig. 7 show the implementation results for a few test examples, with all potential SIPs (in
blue) and SIP (in black) and green indicating the boundaries (outer and inner curves) forming
MCD with different start and end points. Implementation in this paper have been carried out
using IRIT [8], a solid modeling kernel. The constraint equations for PCT and BT were solved
using the geometric constraint solver [9] in IRIT. The algorithm does not limit the number of
holes that can be present in the MCD. Also, local spiral-shapes such as the one in Fig. 7(p) are
supported. As the algorithm generates all potential SIPs (for eg., all blue paths in Fig. 7(a)),
it has the potential to give all path of approximately equal lengths (within a tolerance level,
specified by the user). Moreover, using generalized lemma (Lemma 4) eliminates inner loops
effectively reducing the number of curves involved in the computation of BTs and inflection
points. Fig. 7(o) shows an example where SIP did not pass through any of the inner loops.
For the input points S and E not on the outer boundary, a bounding box has been employed
to identify regions for eliminating curves and tangents (PCTs/BTs). Figure 8 shows a few
results with outer boundary replaced by a box.
5.1 Handling C 1 discontinuities
Points of C 1 discontinuity on the curves are identified [15]. From these points and the other
curves, PCTs are obtained and processed (as described in Section 4.1). Lines between C 1
discontinuity points are considered as pseudo-Bitangents (pBTs). Initially, pBTs that cuts
across the MCD in the neighbourhood of its C 1 discontinuity points are removed. Other
processing are done similar to that of BTs. Figure 9 shows a few test results for SIP with
curves having C 1 discontinuities.
11

(a) All potential SIPs (b) SIP (c) All potential SIPs (d) SIP
(e) All potential SIPs (f) SIP (g) All potential SIPs (h) SIP
(i) All potential SIPs (j) SIP (k) All potential SIPs (l) SIP
(m) All potential
SIPs
(n) SIP (o) SIP (p) SIP
Figure 7: Test results of the algorithm showing 12all
potential shortest paths (in blue) and SIP
(in black).

(a) All potential SIPs (b) SIP (c) All potential SIPs (d) SIP
(e) All potential SIPs (f) SIP (g) All potential SIPs (h) SIP
Figure 8: All potential shortest paths (in blue) and SIP (in black) for outer boundary as a
bounding box.
5.2 Discussion
To the best of the knowledge of the authors, not much work has been done on MCD. Ref.
2 in [3], which appear to be related to MCD is not accessible. [3, 10] seem to have handled
only SCD. However, both are theoretical in nature and no implementation result have been
provided. As the SIP is not unique in MCD, the algorithm has to be tuned to handle holes.
The MCD can be considered some what similar to the calculation of path from start to
end point through obstacles (i.e. a domain without the outer boundary in MCD). Most of
the algorithms related to SIP of obstacles handle polygonal boundaries, including a distance
function approach [20]. For curved obstacles, [16] proposed the computation of visibility
graph, assuming the obstacles as convex curves. For disk obstacles, shortest path based on
visibility graph and Dijkstra was proposed in [4]. Approach in [6] seems to be based on
Voronoi diagram and it is well known that computation of the diagram is very difficult and
not robust for arbitrary shaped curves. All the algorithms for curved obstacles appear to be
theoretical in nature and no implementation results have been provided (in [17], an approach
using grid-based approximation has been described used fast-sweeping method in [22]).
To use an algorithm such as Dijsktra for curved obstacles, visibility graph of the curved obstacles
can be used. However, to compute the visibility graph, the footpoints of the PCTs/BTs
are required. In our work, while traversal of the algorithm, all the potential paths are identified
and hence the application of visibility graph and Dijkstra need not be computed.
13

(a) All potential SIPs (b) SIP (c) All potential SIPs (d) SIP
(e) SIP (f) SIP
Figure 9: All potential shortest paths (in blue) and SIP (in black) for curves having C 1
discontinuities.
5.2.1 Complexity of the algorithm
In this paper, as the computation of SIP involves PCTs/BTs, the complexity is analyzed
based on the number of PCTs/BTs that may be involved (the analysis does not take into
C 1 discontinuity points). Let n be the number of concave portions from the outer boundary
and h be the number of inner loops. Between two concave regions there can be at most
four completely contained BTs (this can be shown by employing the convex hull between
two curves). Thus maximum number of BTs for the outer boundary will be 4(nC2) (nC2 =
n(n − 1)/2). For h inner loops, number of BTs is 4(hC2) and between inner and outer
boundary, it will be 4nh. From the starting point S, there can be at most two completely
contained PCTs for a concave portion. This implies that at most 2n PCTs from S and 2n
PCTs from E will be available when n concave portions are considered. Similarly 2h from S
and 2h from E will be the number of PCTs for the inner loops. Let T = 4(nC2 + nh + n + h)
be the total number of tangents (PCTs + BTs).
Tangent lists are searched for the closest tangents (to E or S) as in discussed in section
4.1.1 and for mutually farthest tangents as in section 4.1.2. No tangent is processed twice
because of merging criteria followed in section 4.4. Thus the complexity of the algorithm is
O(T ) in the worst case.
In general, not all BTs will be used for our computation (as inner loops may be eliminated)
14

(a) All BTs in the given
shape (568 BTs)
(b) Completely contained
BTs (183 BTs)
Figure 10: Number of bitangents
though it cannot be quantified exactly how many tangents will be removed as it depends on
the shape of the curves. The number of BTs that will be used will be a substantial reduction
of the total number of BTs, when we compute on the fly over any algorithm that might use
all the completely contained tangents. Consider the input curves in Figure 7(a). The curves
have 568 BTs (Figure10(a)) and 183 completely contained BTs (Figure10(b)). By including
the PCTs, approximately there are 200 completely contained tangents. These are the input
for graph based Dijkstra’s algorithm. However, our algorithm has been found to use far
lesser number of tangents as shown in the Table 1. The table shows the number of tangents
selected/number contained tangents at each stage along the recursion. Blank entry in the
table means all paths have terminated.
Table 1: Number of tangents selected/Number contained tangents at each stage
Figure PCTs from E PCTs from S BTs BTs BTs BTs BTs BTs BTs
7(a) 4/11 5/12 2/8 1/10 1/6 1/12
7(c) 3/3 4/10 1/12 2/4 1/10 1/16 2/3 2/12 1/6
7(e) 3/14 6/10 3/11 2/12 1/15 2/16 1/12 1/12
It should be noted that application of Lemma 4 leads to inner loops and regions get
rejected at each stage, BTs are computed and used only in a closed band around the shortest
path as could be seen from figures 7(a) and 7(e). Thus the number of computations are also
greatly reduced.
5.2.2 Using discretization of the curve to compute SIP
Discretizing the curve into set of points or polylines and then employing Dijkstra’s algorithm
could be another approach to compute SIP. However, to compute a path, it is imperative that
the footpoints of the tangent (PCTs/BTs) were part of sample points or vertices of polylines.
However, there is no effective sampling strategy that can precisely capture all footpoints of
tangents and bitangents and hence more inaccuracies are introduced. As one has to anyway
15

compute the tangency points to use an algorithm like Dijkstra, which will be of O(T 2 ), the
complexity of our algorithm is O(T ). Also, it is worth noting that the algorithm directly
computes all the potential path on the fly.
6 Conclusion
In this paper, an algorithm to compute SIP of curved boundaries, forming a multiplyconnected
domain, extending the algorithm in [18] has been presented. It is shown that all
potential paths can be computed, eliminating many PCTs/BTs/inner loops using a generalized
region lemma and then specifically finding exterior and interior regions using PCTs/BTs.
Merging of paths that will avoid redundant computation is also done. It is also shown that the
algorithm can be applied to any number of holes and also to curves having C 1 discontinuity
points. Test results on a few cases indicate that the algorithm is very amenable to implementation.
SIP can be computed without employing visibility graph and Dijkstra. Further work
involves focussing on applications based on [18] and this paper.
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7 Appendix: Pseudocode of the algorithms
Algorithm 1 GetP CT s(X,CurveList)
1: Get all PCTs from X to members in CurveList
2: Eliminate PCTs that are not completely contained in the MCD and store it in P CT List
3: Eliminate curves and PCTs using exterior and interior region identification.
4: return T angentList,CurveList
Algorithm 2 GetBT s(T gtLine,CurveList,P resentP ath,T erminationList)
1: [SearchCurve,SearchP ar,Dir] = T gtLine.[EndCurve, EndP ar, Dir]
2: if T gtLine not eliminated according to section 4.3 then
3: if T gtLine is merged with a processed tangent,P rocessedT angent then
4: Add the subtree of paths under P rocessedT angent and update path lengths
5: else
6: NextInflection= Identify next inflection point
7: for Each Curve in CurveList do
8: BT List= Get all BTs between SearchCurve and Curve
9: Identify BTs having StartP ar between SearchP ar and NextInflection
10: Eliminate BTs that are not consistent with the direction Dir according to lemma2
11: Eliminate BTs that are not completely contained in the MCD
12: end for
13: (NextT gtList,CurveList) = Remaining BTs and Curves after elimination using exterior
and interior region identification.
14: for Each T gtLine in NextT gtList do
15: P resentP ath += Concave portion + T gtLine
16: T ermCheck = Check for termination with tangents from T erminationList
17: if T ermCheck == false then
18: (NextBT List, CurveList) = GetBT s(T gtLine,CurveList,P resentP ath,
T erminationList)
19: else
20: P resentP ath += Concave portion + Corres. tgt. in T erminationList
21: Add P resentP ath to ShortestP athList
22: end if
23: end for
24: end if
25: end if
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Algorithm 3 ShortestInteriorP ath(S,E,CurveList)
1: if S and E are distinct then
2: if Straight line path SE not available then
3: (T erminationList, CurveList) = GetP CT s(E,CurveList)
4: Add dummy tangent T erminationList if required according to section 4.2.2
5: (NextT gtList,CurveList) = GetP CT s(S,CurveList)
6: Add dummy tangent to NextT gtList if required according to section 4.2.2
7: for Each T angentLine in NextT gtList do
8: P resentP ath=T angentLine
9: T ermCheck = Check for termination with tangents from T erminationList
10: if T ermCheck == false then
11: (NextBT List, CurveList) = GetBT s(T angentLine,
CurveList,P resentP ath,T erminationList)
12: else
13: P resentP ath += Concave portion + Corres. tgt. in T erminationList
14: Add P resentP ath to ShortestP athList
15: end if
16: end for
17: To get SIP, Compute all path lengths and find the minimum.
18: else
19: Add SE to SIP.
20: end if
21: return SIP
22: else
23: return The start and the end points are same.
24: end if
19

S. Bharath Ram is currently a research scholar in the Department of Engineering Design, Indian Institute
of Technology Madras, Chennai, India. He earned a BE degree in Mechanical Engineering from College
of Engineering, Guindy, Anna University, India. His current research interests include geometric modeling,
differential and computational geometry.
M. Ramanathan is currently an Assistant Professor in the Department of Engineering Design, Indian
Institute of Technology, Madras. He earned a B.E. degree in Mechanical Engineering from Thiagarajar
college of Engineering, Madurai, India. He received his M. Sc. (Engg.) and Ph.D. from the Department
of Mechanical Engineering at the Indian Institute of Science, Bangalore, India. He did his post-doctoral
research at Technion, Haifa, Israel and then at Purdue university, USA. His current research interests include
geometric modeling, computational geometry, shape and image search, mesh model analysis, and BioCAD.
1

Research Highlights
• This paper computes shortest path for multiply-connected curved boundaries.
• The algorithm does not discretize the boundaries into points or polylines.
• No need to employ visibility graph to compute the shortest path.
• The algorithm can handle C 1 discontinuities.
• Start and end points need not be on the outer boundary.