(4)– 4 – Nice Triangulations • θ

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Nice Triangulations

Given a point set S in the plane produce a nice triangulation T of S, i.e. each triangle ∆ of T is “nicely shaped.”

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Nice Triangulations

Different definitions of “nicely shaped” triangle ∆:

• smallest angle is large

• the largest angle is small

• all angles are acute

• ratio of radius of circumcirle and radius of incircle is small

• ratio of longest edge and shortest edge is small

• ratio of longest edge and corresponding altitude is small

• . . .

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Nice Triangulations

Different definitions of “nicely shaped” triangle ∆:

• smallest angle is large

• the largest angle is small

• all angles are acute

• ratio of radius of circumcirle and radius of incircle is small

• ratio of longest edge and shortest edge is small

• ratio of longest edge and corresponding altitude is small

• . . .

All this notions are closely related and the respective “small” and “large”

can be appropritely parameterized and related to each other.

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Nice Triangulations

• θ(∆): smallest angle of ∆

• R(∆): ratio of longest edge and shortest edge

• A(∆): ratio of longest edge and corresponding altitude (“aspect ratio”) Relations:

1

sin θ(∆) ≤ A(∆) ≤ 2

sin θ(∆) and R(∆) < A(∆)

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Nice Triangulations

• θ(∆): smallest angle of ∆

• R(∆): ratio of longest edge and shortest edge

• A(∆): ratio of longest edge and corresponding altitude (“aspect ratio”) Relations:

1

sin θ(∆) ≤ A(∆) ≤ 2

sin θ(∆) and R(∆) < A(∆) T triangulation

• θ(T ) = min_∆∈T θ(∆)

• R(T ) = max_∆∈T R(∆)

• A(T ) = max_∆∈T A(∆)

Looking for triangulations T of S so that min-angle θ(T ) is large, of R(T ) is small, of A(T ) is small.

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Nice Triangulations

Theorem:

Among all triangulations T of S the Delaunay triangulation maximizes the minimum angle, i.e. θ(DT (S)) = max{θ(T )|T triangulation of S}.

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Nice Triangulations

Theorem:

Among all triangulations T of S the Delaunay triangulation maximizes the minimum angle, i.e. θ(DT (S)) = max{θ(T )|T triangulation of S}.

Proof idea:

Flip algorithm makes sorted vector of all triangle angles in the triangulation lexicographically increase.

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Nice Triangulations

Problem:

For some point sets S, there are no nice triangulations, i.e. θ(T ) is small for all triangulations T of S (and A(T ) and R(T ) are large).

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Nice Triangulations

Problem:

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Nice Triangulations

Problem:

Idea: Add points to S (so-called “Steiner points”) so that a nice triangulation on the larger set is possible

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Nice Triangulations

Idea: Add points to S (so-called “Steiner points”) so that a nice triangulation on the larger set is possible.

This is always possible!!!

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Nice Triangulations

1.Draw sufficiently fine grid, so that points in S are separated by two layers of boxes.

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Nice Triangulations

2. For each point in S warp the closest grid point to its position.

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Nice Triangulations

2. For each point in S warp the closest grid point to its position.

3. Triangulate each quadrilateral.

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Nice Triangulations

2. For each point in S warp to closest grid point to its position.

3. Triangulate each quadrilateral.

Way too many new vertices!!!

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Quadtrees

From T. Mchedlidze, KIT

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Quadtrees

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Quadtrees

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Example

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Example

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Example

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Example

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Example

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Example

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Quadtree Properties

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Quadtree Properties

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Quadtree Properties

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Quadtree Properties

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Finding Neighbors

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Finding Neighbors

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Balanced Subtrees

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Balanced Quadtrees

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Balanced Quadtrees

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Balancing Quadtrees

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Balancing Quadtrees

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Balanicng Quadtrees

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Balancing Quadtrees

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Quadtrees for Nice Triangulations

Bern, Eppstein, Gilbert: Provably Good Mesh Generation (1994)

p p

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In each quadrilateral incident to p choose diagonal that yields

triangles with better aspect ratio.

p p p

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Lemma: For each triangle ∆ incident to an orange edge the aspect ratio is at most 4, i.e. A(∆) ≤ 4.

p p p

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Point needs one layer of empty boxes around its own box.

Need non-interference between layers of different points.

p p p

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Splitting Crowded Boxes

Box b is crowded if at least one of the following holds:

• b contains more than one point of S

• b has side length `, contains a single point p ∈ S, but some other point of S is closer than 2√

2` to p.

• b contains one point of S but one of the 8 neighbors around b has a split side.

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Splitting Crowded Boxes

Box b is crowded if at least one of the following holds:

• b contains more than one point of S

• b has side length `, contains a single point p ∈ S, but some other point of S is closer than 2√

2` to p.

• b contains one point of S but one of the 8 neighbors around b has a split side.

p

q

made impossible

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Nice triangulation for S:

0. Put a sufficiently large square box Q around S and make the root of a quadtree.

1. while there is a crowded box, split it and ensure balance

2. for each p ∈ S move the closest corner of its containing leaf box to p and triangulate the incident quadrilaterals with aspect ratio at most 4

3. triangulate each empty leaf box into at most 8 isosceles right triangles (aspect ratio √ 2)

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Nice triangulation for S:

0. Put a sufficiently large square box Q around S and make the root of a quadtree.

1. while there is a crowded box, split it and ensure balance

2. for each p ∈ S move the closes corner of its containing leaf box to p and traingulate the incident quadrilaterals with aspect ratio at most 4

3. triangulate each empty leaf box into at most 8 isosceles right triangles (aspect ratio √ 2)

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Results

Lemma: This algorithm produces a triangulation T for S with aspect ration A(T ) at most 4.

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Results

Lemma: There is a constant c independent of S so that the size of T is at most

c · X

∆∈DT (S)

log R(∆)

where DT (S) is the Delaunay triangulation of S, and R(∆) is the ratio of longest and shortest side of triangle ∆.

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Results

Lemma: There is a constant c independent of S so that the size of T is at most

c · X

∆∈DT (S)

log R(∆)

where DT (S) is the Delaunay triangulation of S, and R(∆) is the ratio of longest and shortest side of triangle ∆.

Theorem: There is a constant d independent of S so that for any triangulation T ⁰ that has S in its vertex set we have

|T | ≤ d · |T ⁰| log A(T ⁰)

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Outlook “Nice Triangulations” (Meshing)

Drawbacks of this result:

• bad constants

• not anisotropic (rotating the coordinate systems changes triangulations)

Viable alternativ algorithms via refining Delaunay triangulations.

Generalization to meshing problems when edges are given as input are possible, but quadtree based approaches have similar shortcomings.

Quadtree based approaches do generalize to higher dimensions.

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Outlook Quadtrees

• Quadtrees find many applications in Computer Graphics, Image Processing, Geographic Information Systerm, etc.

They are useful whenver different scales are to be represented

• There is a variant “compressed quadtree” that uses just space linear in the size of the input.

• “skip quadtrees” allow searches and also updates in logarithmic expected time.

• Quadtrees readily generalize to higher dimensions (“octtree”).

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