1. Tao JuA general construction of coordinatesSlide 1 A general, geometric construction of coordinates in any dimensions Tao Ju Washington University in St. Louis

2. Tao JuA general construction of coordinatesSlide 2 Coordinates Homogeneous coordinates – Given points – Express a new point as affine combination of – are called homogeneous coordinates – Barycentric if all

3. Tao JuA general construction of coordinatesSlide 3 Applications Value interpolation – Color/Texture interpolation Mapping – Shell texture – “Cage-based” deformation [Hormann 06] [Porumbescu 05] [Ju 05]

4. Tao JuA general construction of coordinatesSlide 4 How to find coordinates?

5. Tao JuA general construction of coordinatesSlide 5 Coordinates In A Polytope Points form vertices of a closed polytope – x lies inside the polytope Example: A 2D triangle – Unique (barycentric): – Can be extended to any N-D simplex A general polytope – Non-unique – The triangle-trick can not be applied.

6. Tao JuA general construction of coordinatesSlide 6 Previous Work 2D Polygons – Wachspress [Wachspress 75][Loop 89][Meyer 02][Malsch 04] Barycentric within convex shapes – Discrete harmonic [Desbrun 02][Floater 06] Homogeneous within convex shapes – Mean value [Floater 03][Hormann 06] Barycentric within convex shapes, homogeneous within non-convex shapes 3D Polyhedrons and Beyond – Wachspress [Warren 96][Ju 05a] – Discrete harmonic [Meyer 02] – Mean value [Floater 05][Ju 05b]

7. Tao JuA general construction of coordinatesSlide 7 Previous Work A general construction in 2D [Floater 06] – Complete: a single scheme that can construct all possible homogeneous coordinates in a convex polygon – Homogeneous coordinates parameterized by variable p Wachspress: p=-1 Mean value: p=0 Discrete harmonic: p=1 What about 3D and beyond? (this talk) – Tao Ju, Peter Liepa, Joe Warren, CAGD 2007

8. Tao JuA general construction of coordinatesSlide 8 The Short Answer Yes, such general construction exists in N-D – Complete: A single scheme constructing all possible homogeneous coordinates Applicable to any convex simplicial polytope – 2D polygons, 3D triangular polyhedrons, etc. – Geometric: The coordinates are parameterized by an auxiliary shape Wachspress: Polar dual Mean value: Unit sphere Discrete harmonic: Original polytope

9. Tao JuA general construction of coordinatesSlide 9 The Longer Answer… We focus on an equivalent problem of finding weights such that – Yields homogeneous coordinates by normalization

10. Tao JuA general construction of coordinatesSlide 10 2D Mean Value Coordinates We start with a geometric construction of 2D MVC 1. Place a unit circle at. An edge projects to an arc on the circle. 2. Write the integral of outward unit normal of each arc,, using the two vectors: 3. The integral of outward unit normal over the whole circle is zero. So the following weights are homogeneous: v1v1 v2v2 x

11. Tao JuA general construction of coordinatesSlide 11 2D Mean Value Coordinates To obtain : – Apply Stoke’s Theorem v1v1 v2v2 x -n 2 T -n 1 T

12. Tao JuA general construction of coordinatesSlide 12 2D Mean Value Coordinates We start with a geometric construction of 2D MVC 1. Place a unit circle at. An edge projects to an arc on the circle. 2. Write the integral of outward unit normal of each arc,, using the two vectors: 3. The integral of outward unit normal over the whole circle is zero. So the following weights are homogeneous: v1v1 v2v2 x

13. Tao JuA general construction of coordinatesSlide 13 Our General Construction Instead of a circle, pick any closed curve 1. Project each edge of the polygon onto a curve segment on. 2. Write the integral of outward unit normal of each arc,, using the two vectors: 3. The integral of outward unit normal over any closed curve is zero (Stoke’s Theorem). So the following weights are homogeneous: v1v1 v2v2 x

14. Tao JuA general construction of coordinatesSlide 14 Our General Construction To obtain : – Apply Stoke’s Theorem v1v1 x d1d1 d2d2 v2v2 v1v1 x d1d1 -d 2 v2v2

15. Tao JuA general construction of coordinatesSlide 15 Examples Some interesting result in known coordinates – We call the generating curve Wachspress (G is the polar dual) Mean value (G is the unit circle) Discrete harmonic (G is the original polygon)

16. Tao JuA general construction of coordinatesSlide 16 General Construction in 3D Pick any closed generating surface 1. Project each triangle of the polyhedron onto a surface patch on. 2. Write the integral of outward unit normal of each patch,, using three vectors: 3. The integral of outward unit normal over any closed surface is zero. So the following weights are homogeneous: v2v2 v3v3 v1v1 x rTrT

17. Tao JuA general construction of coordinatesSlide 17 -n 3 T d 1,2 General Construction in 3D To obtain : – Apply Stoke’s Theorem v2v2 v3v3 v1v1 x rTrT

18. Tao JuA general construction of coordinatesSlide 18 Examples Wachspress (G: polar dual) Mean value (G: unit sphere) Discrete harmonic (G: the polyhedron) Voronoi (G: Voronoi cell)

19. Tao JuA general construction of coordinatesSlide 19 An Equivalent Form – 2D Same as in [Floater 06] – Implies that our construction reproduces all homogeneous coordinates in a convex polygon vivi v i-1 v i+1 x BiBi vivi x A i-1 AiAi v i-1 where

20. Tao JuA general construction of coordinatesSlide 20 An Equivalent Form – 3D 3D extension of [Floater 06] – We showed that our construction yields all homogeneous coordinates in a convex triangular polyhedron. where vivi v j-1 v j+1 vjvj x A j-1 AjAj vivi v j-1 v j+1 vjvj x BjBj CjCj

21. Tao JuA general construction of coordinatesSlide 21 Summary Homogenous coordinates construction in any dimensions – Complete: Reproduces all homogeneous coordinates in a convex simplicial polytope – Geometric: Each set of homogeneous coordinates is constructed from a closed generating shape (curve/surface/hyper-surface) Polar dual: Wachspress coordinates Unit sphere: Mean value coordinates Original polytope: Discrete harmonic coordinates Voronoi cell: Voronoi coordinates

22. Tao JuA general construction of coordinatesSlide 22 Open Questions What about continuous shapes? – General constructions known [Schaefer 07][Belyaev 06] – What is the link between continuous and discrete constructions? Discrete coordinates equivalent to continuous coordinates when applied to a piece- wise linear shape What about non-convex shapes? – Are there coordinates well-defined for non-convex shapes, besides MVC? What about non-simplicial polytopes? – Are there closed-form MVC for non-triangular meshes that agree with the continuous definition of MVC? What about on a sphere? – Initial attempt by [Langer 06], yet limited to within a hemisphere.