michael s.floater g éza kós martin reimers cagd 22(2005) 623-631 reporter: zhang xingwang

2005 Autumn Seminar

©2005 zxw1

Michael S.FloaterGéza Kós

Martin ReimersCAGD 22(2005) 623-631

Reporter: Zhang Xingwang

Mean Value Coordinates in 3D

2005 Autumn Seminar

©2005 zxw2

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

2005 Autumn Seminar

©2005 zxw3

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

2005 Autumn Seminar

©2005 zxw4

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

2005 Autumn Seminar

©2005 zxw5

2005 Autumn Seminar

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.

2005 Autumn Seminar

©2005 zxw6

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

2005 Autumn Seminar

©2005 zxw7

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

2005 Autumn Seminar

©2005 zxw8

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

2005 Autumn Seminar

©2005 zxw9

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

2005 Autumn Seminar

©2005 zxw10

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

2005 Autumn Seminar

©2005 zxw11

2005 Autumn Seminar

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

2005 Autumn Seminar

©2005 zxw12

2005 Autumn Seminar

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

2005 Autumn Seminar

©2005 zxw13

2005 Autumn Seminar

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

2005 Autumn Seminar

©2005 zxw14

2005 Autumn Seminar

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

2005 Autumn Seminar

©2005 zxw15

2005 Autumn Seminar

Numerical Examples

{ : ( ) } 0.5,0.05,0.005Iso-surfaces for iv v c c

2005 Autumn Seminar

©2005 zxw16

2005 Autumn Seminar

Numerical Examples

{ : ( ) } 0.2,0.05,0.005Iso-surfaces for iv v c c

2005 Autumn Seminar

©2005 zxw17

2005 Autumn Seminar

Numerical Examples

{ : ( ) } 0.001,0.0001,0.00001Iso-surfaces for iv v c c

2005 Autumn Seminar

©2005 zxw18

2005 Autumn Seminar

Numerical Examples

{ : ( ) } 0.01,0.0005,0.0002Iso-surfaces for iv v c c

2005 Autumn Seminar

©2005 zxw19

2005 Autumn Seminar

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.

2005 Autumn Seminar

©2005 zxw20

Any Questions?

2005 Autumn Seminar

©2005 zxw21

Thanks for you attention!