2005 Autumn Seminar
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Michael S.FloaterGéza Kós
Martin ReimersCAGD 22(2005) 623-631
Reporter: Zhang Xingwang
Mean Value Coordinates in 3D
2005 Autumn Seminar
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2005 Autumn Seminar
Overview
1. About the authors2. Motivation3. Introduction4. Mean value coordinates in 3D
6. Numerical examples7. Conclusions and future work
5. Convex polyhedra
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2005 Autumn Seminar
About the AuthorsMichael S.Floater:
Department of Informatics of the University of Oslo, Centre of Mathematics for Applications(CMA)
Geometric modeling, Numerical analysis, Approximation theory
Géza Kós Department of Analysis at Eötvös University in Budapest, Hung
ary Approximation theory, Surface and solid modeling, Surface rec
onstruction.
Martin Reimers Postdoc at CMA, University of Oslo Geometric modeling&splines, Approximation theory, Mesh bas
ed modeling, Computer graphics
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2005 Autumn Seminar
A point represented as a convex combination of its neighboring vertices
Motivation
Generalizing coordinates to convex polyhedra and the kernels of star-shaped polyhedra
Key: barycentric coordinates
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IntroductionMean value coordinates in 2D(Floater 2
003)Applications:
Convex combination maps between pairs of planar regions (Surazhsky and Gotsman, 2003).
Smoothly interpolating piecewise linear height data given on the boundary of a convex polygon.
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2005 Autumn Seminar
Mean Value Coordinates in
Some notations:
3 :a polyhedron R
3R
31 2, , , : verticesnv v v R
3 :
,
, 1, , .
kernel of , open set consisting of all
points in the interior Int( ) with property
that the only intersection between [ ] and
is i
i
K
v
v v
v i n
R
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2005 Autumn Seminar
Mean Value Coordinates in
Some notations:
3R
3 , , :If star-shapedK K R
1 2
1 1
, , , : , ,
( ) 1 ( )
barycentric coordinates:
non-negative functions such that
and
n
n n
i i ii i
K v K
v v v v
R
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2005 Autumn Seminar
Mean Value Coordinates in 3R
:a mesh of triangular facets T
, [ , , ] ,
[ , , , ]
each oriented triangle
a tetrahedron with a positive volume
i j k
i j k
v K v v v
v v v v
T
Tetrahedron
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2005 Autumn Seminar
Mean Value Coordinates in 3R
ˆ, , ,
[ , , ]
ˆ ˆ ˆ ˆ, ,
Project triangle onto the sphere,
a sphere triangle with vertices
ii i i i
i j k
i j k
v ve r v v v v e
rT v v v
T v v v
Spherical triangle
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2005 Autumn Seminar
Mean Value Coordinates in 3R
ˆ
( ) , ( )
0 ( ) ( ) ( )
outward normal to at any point
TS S T
n p S p S n p p v
n p p v p v
T
( ) ( ) ( )
, ,
, , ) , , ), , )( ) 0, ( ) 0, ( ) 0
, , ) , , ) , , )
where are the spherical barycentric coordinates of
vol( vol(vol(
vol( vol( vol(
i i j j k k
i j k
j k i ji ki j k
i j k i j k i j k
e e e e e e e
e
e e e e e ee e ee e e
e e e e e e e e e
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Mean Value Coordinates in
1 2
1
, , , : ,
( )
Theorem 1. The functions defined by
are barycentric coordinates which belong to
n
ii n
jj
K
wC K
w
R
3R, , ,
ˆ
,ˆ
( )
( ) 0, { , , }where
i T i j T j k T k
T
l T l
T
p v e e e
e l i j k
, ,1 1
10 ( ) 0
Reorganizing the sum
where i i
n n
i T i i i i i Ti v T i v Ti
e w v v wr
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Mean Value Coordinates in 3R
, 2
, :
[ , ] [ , ]
Theorem 2
where the angle between the line
segment and
Reasons: the integral of all unit normal over
any compact surface is zero
jk ij ij jk ki ki jki T
i jk
r srs rs
r s
s s
n n n n
e n
e en
e e
v v v v
See
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Convex Polyhedra
( ).In this case, Int
Coordinates:
well-defined, positive, and infinitely differentiable in Int( )
not well-defined at the boundary of
Extend coordinates continuously to the boudary.
K
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Convex Polyhedra
1 2, , , : ( )
Theorem 3:
If is convex and Int are a set of
continuous barycentic coordinates, then they have a unique
continuous extension to the boundary . The extended
coordinates are l
n
i
R
( )
, , , ) , , , ]
inear on each facet of and
Key: : convex, vol( : signed volume of [
i j ij
j k l j k l
v
v v v v v v v v
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Numerical Examples
{ : ( ) } 0.5,0.05,0.005Iso-surfaces for iv v c c
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Numerical Examples
{ : ( ) } 0.2,0.05,0.005Iso-surfaces for iv v c c
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2005 Autumn Seminar
Numerical Examples
{ : ( ) } 0.001,0.0001,0.00001Iso-surfaces for iv v c c
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Numerical Examples
{ : ( ) } 0.01,0.0005,0.0002Iso-surfaces for iv v c c
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Conclusions and Future WorkNatural extension of mean value coordinates to kernels of star-shaped polyhedra3D coordinates well-defined everywhere in a convex polyhedron, including the boundaryPolyhedron with multi-sided facets, first triangulate each facet. Depending on the choice of triangulation.Extend 3D coordinates to arbitrary points, even for arbitrary polyhedra.
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Any Questions?
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Thanks for you attention!