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