Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.
-
Upload
job-roberts -
Category
Documents
-
view
218 -
download
0
Transcript of Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.
![Page 1: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/1.jpg)
Computing the shortest path on a polyhedral
surface
Presented by: Liu Gang2008.9.25
![Page 2: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/2.jpg)
![Page 3: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/3.jpg)
Overview of Presentation
Introduction Related works Dijkstra’s Algorithm Fast Marching Method Results
![Page 4: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/4.jpg)
Introduction
![Page 5: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/5.jpg)
Motivation Computing the shortest path between
two points s and t on a polyhedral surface S is a basic and natural problem
From the viewpoint of application: Robotics, geographic information
systems and navigation Establishing a surface distance metric
![Page 6: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/6.jpg)
Related worksExact globally shortest path algorithms Shair and Schorr (1986) On shortest paths in polyhedral spaces.
O(n3logn) Mount (1984) On finding shortest paths on convex polyhedral. O(n2logn) Mitchell, Mount and Papadimitriou (MMP)(1987) The discrete
geodesic problem. O(n2logn) Chen and Han(1990) shortest paths on a polyhedron. O(n2)
Approximate shortest path algorithms Sethian J.A (1996) A Fast Marching Level Set Method for Monotonically
Advancing Fronts. O(nlogn) Kimmel and Sethian(1998) Computing geodesic paths on manifolds.
O(nlogn) Xin Shi-Qing and Wang Guo-Jin(2007) Efficiently determining a locally
exact shortest path on polyhedral surface. O(nlogn)
![Page 7: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/7.jpg)
Exact globally shortest path algorithms Shair and Schorr (1986) On shortest paths in polyhedral spaces.
O(n3logn) Mount (1984) On finding shortest paths on convex polyhedral. O(n2logn) Mitchell, Mount and Papadimitriou (MMP)(1987) The discrete geodesic
problem. O(n2logn) Chen and Han(1990) shortest paths on a polyhedron. O(n2)
Approximate shortest path algorithms Sethian J.A (1996) A Fast Marching Level Set Method for Monotonically
Advancing Fronts. O(nlogn) Kimmel and Sethian(1998) Computing geodesic paths on manifolds.
O(nlogn) Xin Shi-Qing and Wang Guo-Jin(2007) Efficiently determining a locally
exact shortest path on polyhedral surface. O(nlogn)
Related works
Dijkstra's algorithm
Fast Marching Methods(FMM)
![Page 8: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/8.jpg)
Dijkstra's Algorithm
![Page 9: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/9.jpg)
Dijkstra's Shortest Path Algorithm
Find shortest path from s to t.
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
![Page 10: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/10.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
0
distance label
![Page 11: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/11.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
0
distance label
delmin
![Page 12: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/12.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
distance label
decrease key
X
X
X
![Page 13: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/13.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
(0,9)
14
0
distance label
X
X
X
delmin
![Page 14: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/14.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
(0,9)
14
0
X
X
X
![Page 15: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/15.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
(0,9)
14
0
X
X
X
decrease key
X 33
![Page 16: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/16.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
(0,9)
(0,14
)
0
X
X
X
X 33
delmin
![Page 17: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/17.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
(0,9)
(0,14
)
0
X
X
X
X 33
44X
X
32
![Page 18: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/18.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
(0,14
)
0
X
X
X
44X
delmin
X 33X
32
![Page 19: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/19.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14
)
0
X
X
X
44X
35X
59 X
24
X 33X
32
![Page 20: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/20.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
15
9
14
0
X
X
X
44X
35X
59 X
delmin
X 33X
(6,32)
![Page 21: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/21.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15
)
(0,9)
(0,14)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
X 33X
(6,32
)
![Page 22: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/22.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
delmin
X 33X
(6,32)
24
![Page 23: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/23.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15
)
(0,9)
(0,14)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
24
X 50
X 45
X 33X
(6,32)
![Page 24: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/24.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15
)
(0,9)
(0,14
)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
24
X 50
X(5, 45)
delmin
X 33X
(6,32)
![Page 25: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/25.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14
)
0
X
X
X
44X
35X
59 XX 51
X
(6,34)
24
X 50
X (5,45
)
X 33X
(6,32
)
![Page 26: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/26.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14
)
0
X
X
X
44X
35X
59 XX 51
X (3,34)
X 50
X (5,45)
delmin
X 33X
(6,32)
24
![Page 27: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/27.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
X
(5,50)
X(5, 45)
X 33X
(6,32
)
![Page 28: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/28.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
X
(5,50)
X(5, 45)
X 33X
(6,32
)
![Page 29: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/29.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
X
(5,50)
X(5, 45)
X 33X
(6,32
)
![Page 30: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/30.jpg)
Dijkstra's Shortest Path Algorithm
s
3
t
2
6
7
4
5
24
18
2
9
14
15 5
30
20
44
16
11
6
19
6
(0,15)
(0,9)
(0,14)
0
X
X
X
44X
35X
59 XX 51
X
(3,34)
X
(5,50)
X(5, 45)
X 33X
(6,32
)
![Page 31: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/31.jpg)
Summary Dijkstra’s algorithm is to construct a
tree of shortest paths from a start vertex to all the other vertices on the graph.
Characteristics: 1. Monotone property: Every vertex is processed exactly once 2. When reach the destination, trace backward to find the shortest path.
![Page 32: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/32.jpg)
PreliminariesLet S be a triangulated
polyhedral surface in R3, defined by a set
of faces, edges and vertices.
Assume that the Surface S has n faces
and x0, x are two points on the
surface.
Face sequence: F is defined by a list of adjacent faces f1, f2,…, fm+1 such that fi and fi+1 share a common edge ei . Then we call the list of edges E=(e1,e2,…,em) an edge sequence.
![Page 33: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/33.jpg)
Fast Marching Method(FMM) R. Kimmel
ProfessorDepartment of Computer
ScienceTechnion-Israel Institute of
TechnologyIEEE Transactions on Image Processing
J. A. Sethian
ProfessorDepartment of Mathematics
University of California, Berkeley
Norbert Wiener Prize in Applied Mathematics
![Page 34: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/34.jpg)
Forest fire
![Page 35: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/35.jpg)
Eikonal equation
Let be a minimal geodesic between and .
The derivative
is the fire front propagation direction.
In arclength parametrization .
Fermat’s principle:
Propagation direction = direction of steepest increase of
.
Geodesic is perpendicular to the level sets of on .
![Page 36: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/36.jpg)
Eikonal equation
Eikonal equation (from Greek εικων)
Hyperbolic PDE with boundary
condition
Minimal geodesics are
characteristics.
Describes propagation of waves
in medium.
![Page 37: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/37.jpg)
Fast marching algorithm
Initialize and mark it as black.
Initialize for other vertices and mark them as
green.
Initialize queue of red vertices .
Repeat
Mark green neighbors of black vertices as red (add to
)
For each red vertex
For each triangle sharing the vertex
Update from the triangle.
Mark with minimum value of as black (remove
from )
Until there are no more green vertices.
Return distance map .
![Page 38: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/38.jpg)
Update step difference
Dijkstra’s update
Vertex updated from
adjacent vertex
Distance computed
from
Path restricted to graph
edges
Fast marching update
Vertex updated from
triangle
Distance computed
from and
Path can pass on mesh faces
![Page 39: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/39.jpg)
Fast Marching Method cont.
![Page 40: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/40.jpg)
propagationUsing the intrinsic variable of the triangulation
Acute triangulation
The update procedure is given as follows
![Page 41: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/41.jpg)
Fast Marching Method cont.
Acute triangulation guarantee that the consistent solution approximating the Viscosity solution of the Eikonal equation has the monotone property
![Page 42: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/42.jpg)
Obtuse triangulation
Inconsistent solution if the mesh contains obtuse
triangles
Remeshing is costly
![Page 43: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/43.jpg)
Obtuse triangulation cont.
Solution: split obtuse triangles by adding virtual connections to non-adjacent vertices
![Page 44: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/44.jpg)
Obtuse triangulation cont.
Done as a pre-processing step in
![Page 45: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/45.jpg)
Results (Good)
![Page 46: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/46.jpg)
Results (Bad)
![Page 47: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/47.jpg)
Why?
![Page 48: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/48.jpg)
More Flags for Keeping Track of an Advancing Wavefront
From Vertex : the shortest path to the vertex v goes via its adjacent vertex v’.
From Edge: the shortest path to the vertex v goes across the edge v1v2; p is the access point.
![Page 49: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/49.jpg)
More Flags for Keeping Track of an Advancing Wavefront
(a) From Left Part: edge v1v3 receives wavefront coming from theleft part of edge v1v2.
(b) From Right Part: Edge v3v2
receives wavefront coming fromthe right part of edge v1v2.
![Page 50: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/50.jpg)
![Page 51: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/51.jpg)
![Page 52: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/52.jpg)
![Page 53: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/53.jpg)
![Page 54: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/54.jpg)
Backtracing Paths
![Page 55: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/55.jpg)
![Page 56: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/56.jpg)
![Page 57: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/57.jpg)
![Page 58: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/58.jpg)
Result
![Page 59: Computing the shortest path on a polyhedral surface Presented by: Liu Gang 2008.9.25.](https://reader035.fdocuments.net/reader035/viewer/2022062516/56649e445503460f94b381f8/html5/thumbnails/59.jpg)
Questions will be welcome !