Linear Size Meshes
Gary Miller, Todd Phillips, Don Sheehy
The Meshing Problem
Decompose a geometric domain into simplices.
Recover features.
Guarantee Quality.
The Meshing Problem
Decompose a geometric domain into simplices.
Recover features.
Guarantee Quality.
Meshes Represent Functions
Store f(v) for all mesh vertices v.
Approximate f(x) by interpolating from vertices
in simplex ...
Meshes Represent Functions
Store f(v) for all mesh vertices v.
Approximate f(x) by interpolating from vertices
in simplex ...
Meshes Represent Functions
Store f(v) for all mesh vertices v.
Approximate f(x) by interpolating from vertices
in simplex ...
Meshes for fluid flow simulation.
Quality

A Quality triangle is one with radius-edge ratio
bounded by a constant.
Quality

A Quality triangle is one with radius-edge ratio
bounded by a constant.
Output Size
•   Delaunay Triangulation of n input points may have
    O(n d/2 ) simplices.
Output Size
•   Delaunay Triangulation of n input points may have
    O(n d/2 ) simplices.

•   Quality meshes with m vert...
Output Size
•   Delaunay Triangulation of n input points may have
    O(n d/2 ) simplices.

•   Quality meshes with m vert...
Size Optimality

m: The size of our mesh.
mOPT: The size of the smallest possible mesh
achieving similar quality guarantee...
The Ruppert Bound
The Ruppert Bound




lfs(x) = distance to second nearest vertex
The Ruppert Bound




lfs(x) = distance to second nearest vertex

               Note: This bound is tight.
   [BernEppste...
This work

•   An algorithm for producing linear size delaunay
    meshes of point sets in Rd.
This work

•   An algorithm for producing linear size delaunay
    meshes of point sets in Rd.

•   A new method for analy...
The local feature size at x is
    lfs(x) = |x - SN(x)|.
The local feature size at x is
    lfs(x) = |x - SN(x)|.



 A point x is θ-medial w.r.t S if
  |x - NN(x)| ≥ θ |x - SN(x)...
The local feature size at x is
                            lfs(x) = |x - SN(x)|.



                         A point x is ...
Well Paced Points

A point x is θ-medial w.r.t S if
             |x - NNx| ≥ θ |x - SNx|.
An ordered set of points P is a ...
Linear Cost to balance a Quadtree.
           [Moore95, Weiser81]
The Main Result
Proof Idea

We want to prove that the amortized change in
mopt as we add a θ-medial point is constant.




lfs’ is the new...
Proof Idea



Key ideas to bound this integral.
•   Integrate over the entire space using polar
    coordinates.

•   Spli...
How do we get around the
  Ruppert lower bound?
Linear Size Delaunay Meshes
The Algorithm:
•   Pick a maximal well-paced extension of the
    bounding box.

•   Refine to...
Guarantees

•   Output size is O(n).

•   Bound on radius/longest edge. (No-large angles in R2)
Summary

•   An algorithm for producing linear size delaunay
    meshes of point sets in Rd.

•   A new method for analyzi...
Sneak Preview

In a follow-up paper, we show how the theory
of well-paced points leads to a proof of a
classic folk conjec...
Thank you.
Thank you.
  Questions?
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
Linear Size Meshes
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Linear Size Meshes

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An algorithm for producing a linear size superset of a point set that yields a linear size Delaunay triangulation in any dimension. This talk was presented at CCCG 2008.

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Linear Size Meshes

  1. 1. Linear Size Meshes Gary Miller, Todd Phillips, Don Sheehy
  2. 2. The Meshing Problem Decompose a geometric domain into simplices. Recover features. Guarantee Quality.
  3. 3. The Meshing Problem Decompose a geometric domain into simplices. Recover features. Guarantee Quality.
  4. 4. Meshes Represent Functions Store f(v) for all mesh vertices v. Approximate f(x) by interpolating from vertices in simplex containing x.
  5. 5. Meshes Represent Functions Store f(v) for all mesh vertices v. Approximate f(x) by interpolating from vertices in simplex containing x.
  6. 6. Meshes Represent Functions Store f(v) for all mesh vertices v. Approximate f(x) by interpolating from vertices in simplex containing x.
  7. 7. Meshes for fluid flow simulation.
  8. 8. Quality A Quality triangle is one with radius-edge ratio bounded by a constant.
  9. 9. Quality A Quality triangle is one with radius-edge ratio bounded by a constant.
  10. 10. Output Size • Delaunay Triangulation of n input points may have O(n d/2 ) simplices.
  11. 11. Output Size • Delaunay Triangulation of n input points may have O(n d/2 ) simplices. • Quality meshes with m vertices have O(m) simplices.
  12. 12. Output Size • Delaunay Triangulation of n input points may have O(n d/2 ) simplices. • Quality meshes with m vertices have O(m) simplices. The number of points we add matters.
  13. 13. Size Optimality m: The size of our mesh. mOPT: The size of the smallest possible mesh achieving similar quality guarantees. Size Optimal: m = O(mOPT)
  14. 14. The Ruppert Bound
  15. 15. The Ruppert Bound lfs(x) = distance to second nearest vertex
  16. 16. The Ruppert Bound lfs(x) = distance to second nearest vertex Note: This bound is tight. [BernEppsteinGilbert94, Ruppert95, Ungor04, HudsonMillerPhillips06]
  17. 17. This work • An algorithm for producing linear size delaunay meshes of point sets in Rd.
  18. 18. This work • An algorithm for producing linear size delaunay meshes of point sets in Rd. • A new method for analyzing the size of quality meshes in terms of input size.
  19. 19. The local feature size at x is lfs(x) = |x - SN(x)|.
  20. 20. The local feature size at x is lfs(x) = |x - SN(x)|. A point x is θ-medial w.r.t S if |x - NN(x)| ≥ θ |x - SN(x)|.
  21. 21. The local feature size at x is lfs(x) = |x - SN(x)|. A point x is θ-medial w.r.t S if |x - NN(x)| ≥ θ |x - SN(x)|. An ordered set of points P is a well-paced extension of a point set Q if pi is θ-medial w.r.t. Q U {p1,...,pi-1}.
  22. 22. Well Paced Points A point x is θ-medial w.r.t S if |x - NNx| ≥ θ |x - SNx|. An ordered set of points P is a well-paced extension of a point set Q if pi is θ-medial w.r.t. Q U {p1,...,pi-1}.
  23. 23. Linear Cost to balance a Quadtree. [Moore95, Weiser81]
  24. 24. The Main Result
  25. 25. Proof Idea We want to prove that the amortized change in mopt as we add a θ-medial point is constant. lfs’ is the new local feature size after adding one point.
  26. 26. Proof Idea Key ideas to bound this integral. • Integrate over the entire space using polar coordinates. • Split the integral into two parts: the region near the the new point and the region far from the new point. • Use every trick you learned in high school.
  27. 27. How do we get around the Ruppert lower bound?
  28. 28. Linear Size Delaunay Meshes The Algorithm: • Pick a maximal well-paced extension of the bounding box. • Refine to a quality mesh. • Remaining points form small clusters. Surround them with smaller bounding boxes. • Recursively mesh the smaller bounding boxes. • Return the Delaunay triangulation of the entire point set.
  29. 29. Guarantees • Output size is O(n). • Bound on radius/longest edge. (No-large angles in R2)
  30. 30. Summary • An algorithm for producing linear size delaunay meshes of point sets in Rd. • A new method for analyzing the size of quality meshes in terms of input size.
  31. 31. Sneak Preview In a follow-up paper, we show how the theory of well-paced points leads to a proof of a classic folk conjecture. • Suppose we are meshing a domain with an odd shaped boundary. • We enclose the domain in a bounding box and mesh the whole thing. • We throw away the “extra.” • The amount we throw away is at most a constant fraction.
  32. 32. Thank you.
  33. 33. Thank you. Questions?

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