Generalized Barycentric Coordinates€¦ · 22 Generalized Barycentric Coordinates – SPM, Bilbao...
Transcript of Generalized Barycentric Coordinates€¦ · 22 Generalized Barycentric Coordinates – SPM, Bilbao...
Kai Hormann
Faculty of InformaticsUniversità della Svizzera italiana, Lugano
School of Computer Science and EngineeringNanyang Technological University, Singapore
Generalized Barycentric Coordinates
Generalized Barycentric Coordinates
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Coordinates
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Coordinates
coordinates of Bilbao
43° 15′ 25″ N, 2° 55′ 25″ W
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Latitude and longitude
1693 world map by Louis de Courcillon, abbé de Dangeau (1643 – 1723)
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Latitude and longitude
c.194 BC world map by Eratosthenes (c. 276 BC – c.194 BC)[19th century reconstruction]
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Cartesian coordinates
René Descartes(1596 – 1650)
Appendix “La Géométrie”1637
x
y
1 2 3–3 –2 –1
1
2
3
–3
–2
–1
(0,0)
(1,–2)
(–3,1)
(2,2)
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Cartesian coordinates
René Descartes(1596 – 1650)
(0,0)
x
y
1 2 3–3 –2 –1
1
2
3
–3
–2
–1
(1,–2)
(–3,1)
(2,2)
point (2, 2) with
x-coordinate: 2 y-coordinate: 2
mathematically:
(2, 2) = (0, 0) + 2 · (1, 0)+ 2 · (0, 1)
in general:
(x, y) = (0, 0)+ x · (1, 0)+ y · (0, 1)
x- and y-coordinatesw.r.t. base points
(0,0), (1,0), (0,1)
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Cartesian coordinates
link between geometry and algebra
essential for Newton and Leibniz to develop calculus
x
y
1 2 3–3 –2 –1
1
2
3
–3
–2
–1
x2 + y2 = 4 y = x2 − 2
x
y
1 2 3–3 –2 –1
1
2
3
–3
–2
–1
Generalized Barycentric Coordinates
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Barycentric coordinates
“Der barycentrische Calcul”1827
August Ferdinand Möbius(1790 – 1868)
(1,0,0)
(0,1,0)
(0,0,1)(0.25,–0.25,1)
(0.25,0.25,0.5)
(0.5,0.5,0)
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Barycentric coordinates
August Ferdinand Möbius(1790 – 1868)
(1,0,0)
(0.5,0.5,0)
(0,1,0)
(0,0,1)(0.25,–0.25,1)
(0.25,0.25,0.5)
point (a, b, c) with 3 coordinates w.r.t. base points A, B, C
mathematically:
(a, b, c) = a · A+ b · B+ c · C
whereA = (1, 0, 0)B = (0, 1, 0)C = (0, 0, 1)
anda + b + c = 1
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Law of the lever
Archimedes(c. 287 BC – c. 212 BC)
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Law of the lever
Archimedes(c. 287 BC – c. 212 BC)
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system of masses at positions
position of the system’s barycentre :
Barycentric coordinates
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system of masses at positions
position of the system’s barycentre :
are the barycentric coordinates of
not unique
at leastpoints
needed tospan
Barycentric coordinates
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Theorem [Möbius, 1827]
The barycentric coordinates of with respect to are unique up to a common factor
example:
Barycentric coordinates
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Barycentric coordinates for triangles
normalized barycentric coordinates
properties
partition of unity
reproduction
positivity
Lagrange property
application
linear interpolation of data
Generalized Barycentric Coordinates
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Arbitrary polygons
barycentric coordinates
normalized coordinates
properties
partition of unity
reproduction
for all
linear precision
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Convex polygons
Theorem: If all , then
positivity
Lagrange property
linear along boundary
application
interpolation of data given at the vertices
inside the convex hull of the
direct and efficient evaluation
[Floater, H. & Kós 2006]
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Wachspress coordinates
mean value coordinates
discrete harmonic coordinates
Examples
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Theorem: Common form
Three-point coordinates[Floater, H. & Kós 2006]
Wachspress mean value discrete harmonic
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Non-convex polygons
poles, if , because
Wachspress mean value discrete harmonic
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Colour interpolation
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Vector fields
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Smooth shading
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Rendering of quadrilateral elements
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Transfinite interpolation
mean value coordinates radial basis functions
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Multi-sided Bézier patches
[Loop & DeRose 1986][Smith & Schaefer 2015]
[Salvi & Varády 2018]
[Varády et al. 2016]
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Mesh animation
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Image warping
original image warped imagemask
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Mesh warping
animation for Ratatouille [Joshi et al. 2007]
[H. & Sukumar 2008][Ju et al. 2005]
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Closed-form coordinates
Wachspress [Wachspress 1975]
discrete harmonic [Pinkall & Polthier 1993]
mean value [Floater 2003]
positive mean value [Lipman et al. 2007]
metric [Malsch et al. 2005]
Gordon–Wixom [Belyaev 2006]
positive Gordon–Wixom [Manson et al. 2011]
Poisson [Li & Hu 2013]
power [Budninsky et al. 2016]
blended [Anisimov et al. 2017]
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Computational coordinates
harmonic coordinates [Joshi et al. 2007]
define normalized coordinate as the solution of Laplace’s equation
subject to
maximum entropy coordinates [H. & Sukumar 2008]
maximize the Shannon–Jaynes entropy
subject to
local barycentric coordinates [Zhang et al. 2014]
minimize the sum of total variation
subject to
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Comparison
mean value blended harmonic max entropy local
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If you want to know more …
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Summary
Wachspress coordinates
inside convex polygon (exterior angles not too small)
mean value coordinates
arbitrary polygons, well-defined in R2, but can be negative
harmonic coordinates
inside arbitrary polygons, but no closed form
holy-grail coordinates
arbitrary polygons, closed form, shape similar to harmonic coordinates
Kai Hormann
Faculty of InformaticsUniversità della Svizzera italiana, Lugano
School of Computer Science and EngineeringNanyang Techonological University, Singapore
Generalized Barycentric Coordinates