Introduction Segment-Segment intersections Polygon … › CADCG › CADCG_11.pdf · 2020-06-10 ·...

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CAD & Computational Geometry

Course plan

Introduction Segment-Segment intersections Polygon Triangulation Intro to Voronoï Diagrams & Delaunay Triangulations Geometric Search Sweeping algorithm for Voronoï Diagrams

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Voronoi Diagrams

Voronoi Diagrams or Dirichlet Tessellations In dimension 2 or 3

Johann Peter Gustav Lejeune Dirichlet (1805-1859) ~ 1850.

For all dimensions :

Georgy Feodosevich Voronoy (1868-1908) ~ 1908

G.F. Voronoï (1908). "Nouvelles applications des paramètres continus à la théorie des formes quadratiques". Journal für die reine und angewandte Mathematik 134: 198–287.

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Voronoi Diagrams

Let P={ p0, p

1, … , p

n-1 } a point set in the plane

The Voronoi diagram Vor(P) is a paving of the plane into n cells V( p

i ).

Inside cell V( pi ), the closest point belonging to P is p

i .

Here, the we consider the euclidean distance :

A point q is inside the cell corresponding to point p

i iff

Dist p , q= p x−q x2 p y−q y

2

Dist q , piDist q , p j ∀ p j∈P , j≠i

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Voronoi Diagrams

What is the structure of the Voronoi diagram ? The bissector of points p and q separates the plane in two half-planes Let h(p,q) the open half plane containing p We have then

and V( p

i) is the intersection of the n-1 half-planes, a possibly open polygonal region,

limited by at most n-1 edges and n-1 vertices. The shape is necessarily convex.

p

q

r∈h( p ,q) iff Dist (r , p)<Dist (r , q)

h(p,q)

V pi= ∩j≠i h pi , p j

v(pi)

Vor(P) is either

a) a set formed by n-1 parallel lines

b) a connected set of line segments or half-lines.

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Voronoi Diagrams

Demonstration Case a) straightforward Case b) : Let us consider that an edge e that

bounds two Voronoi cells V(pi) and V(p

j) is a

line : it separates the plane in two parts. Let pk

a third point, not colinear with pip

j : necessarily,

the edge e' bounding V(pj) and V(p

k) is not

parallel to e and intersects it. The part of e which is in h(p

k,p

j) cannot be on the boundary

of V(pj) since it is closer to p

k than p

j .

Therefore, the edges of the Voronoi diagram are either half-lines or line segments, or only complete lines as in the case a)

n-1 parallel lines

pi

pj

pk

e

e'

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Voronoi Diagrams

There remains to prove that the Voronoi diagram is a connected graph.

Let us suppose that it is not the case. At least one Voronoi cell woud separate the plane in two parts. Since Voronoi cells are convex, its bundaries are necessarily made of at least some complete lines, which is a contradiction with the previous demonstration (either only complete lines; or only line segments and half-lines).

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Voronoi Diagrams

Complexity of the Voronoi diagram

What is the number of cells, edges and vertices ? There are n sites in P ; and in the worst case, a cell may have n-1

edges and vertices. The global complexity of Vor(P) could be quadratic. In fact, it is not the case:

Euler formula for a plane graph (including the unbounded face around the graph) : v – e + f = 2Here, we consider an additional vertex infinitely far away ; we have thereforen

v +1 – n

e + n = 2 → 2n

v – 2n

e + 2n = 2

Each edge has two vertices ; the sum of the valence of each vertex is twice the numberof edges. Also, the valence is at least 3.

v¥

2 ne≥3(nv+1)

2 nv−3(n v+1)+2 n≥2 → nv≤2 n−5

2 n−5+1−ne+n≥2 → ne≤3 n−6

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Voronoi Diagrams

The number of bisectors is quadratic, but only a small fraction (linear in n) are part of the Voronoi diagram.

Which ones ?

One defines CP(q) as the greatest circle of center q which does not

contain any point pi of P in its interior. It has at least one point of P on

its boundary.

Then : One point q is a vertex of Vor(P) iff C

P(q) has at least three points of P on its

boundary. A bissector of points p

i et p

j exists as an edge in Vor(P) iff there are points q on it

such that CP(q) has p

i and p

j on its boundary but not any other point of P.

q

CP(q)

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Voronoi Diagrams

Demonstration 3 points or more on the boundary → Center of the

circle is a vertex of the Voronoi diagram 2 points on the boundary → Center is on an edge

of the Voronoi diagram Only one point on the boundary → Center located

inside a cell of the Voronoi diagram

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Voronoi Diagrams How to compute the Voronoi diagram ?

For each cell V(pi), compute the intersections of all the half-planes

h(pi,p

j) with j≠i using the intersection of lines of chapter CG2

Complexity : in nlog n for each cell (one does not know in advance which intersections will be found in the final shape of the cell)

There are n cells → n²log n globally. However, the complexity of the diagram is only O(n)...

Is it possible to be faster ? → yes ! Optimum : Ω(nlog n) Demonstration later on.

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Voronoi Diagrams

Line sweeping algorithme for the computation of a Voronoi Diagram

Also known as Fortune’s algorithm

Steven Fortune, A sweepline algorithm for Voronoi diagrams, Algorithmica 2 (1-4),pp. 153-174, 1987DOI: 10.1007/BF01840357

Algorithm in O(nlog n)

Known sweep line process :- an imaginary line l sweeps the plane- a status T is updated at each event- the event list F contains event in the order in which they appear (x-wise)- the Voronoi diagram is built step by step in relation with T- By duality, it is another way of computing the Delaunay triangulation...

l

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Voronoi Diagrams

The paradigm of sweep line algorithms implies an invariant. Here, it will be the portion of the Voronoi diagram that we know will not change for subsequent events.

It is not the portion of the Voronoi diagram on the left of the line l.

In fact, a part of Vor(P) located on the left of l still depends on some of the points p

i that are located on the right.

More precisely, the part that does not change is the set of points q such that the distance to the line l is greater or equal to the distance to the poinst p

j located on the left of l.

The limit of such a domain is a succession of parabolas, shaped somewhat like a “beach line”.

The “beach line” is y-monotone. The status T will be this beach line.

How does it change when events occur ?(What are in fact these events ?)

q

l

pj

l

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Voronoi Diagrams

Configurational changes in the beach line A new parabolic arc appears

In fact, only when the line l sweeps past a point pi. This is the first

kind of events (points of P)

Proof : cf Book , p 153

l l

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Voronoi Diagrams

Configuration changes in the beach line Existing parabolic arcs may disappear

This happens only when the beach line moves past a vertex of the Voronoi diagram, otherwise said, when the line l meets the extremal point of the circle going through three points p

i that correspond the

three consecutive arcs ai of the beach line.

This is the second type of events (limit circles)

l1

2

3

l1

3

l

1

3

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Voronoi Diagrams

Data structures The Voronoi diagram : e.g. a Doubly Connected Edge List. Ideally

one needs a bounding box so that the cells are closed, and the half edges all able to have a geometrical meaning (a direction and starting point)

s¥

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Voronoi Diagrams

Data structures Status T : The beach line – y-monotone – is a binary balanced tree.

The leaves store the parabolic arcs and are sorted by increasing y coodinate. For each leaf, there is a pointer to an event in the even list – the limit circle event that correspond to the disapearing of the corresponding parabolic arc. If it does not yet exist, store e.g. a null pointer.

The internal nodes inside the tree are the “breaks” in the beach line. Those are indentified with an ordered tuple (p

i, p

j) , where p

i is

the point corresponding to the inferior parabola, and pj that of the

superior parabola. At each interior node, one ca store a pointer to a half-edge of the Voronoi diagram

One may look for the are immediately to the left of a given point pk

inO(log n). Upon tree traversal, at each internal node, it is sufficient to compare the y coordinate of the point p

k with the y coordinate of

the “break”, which may be computed “on the fly” knowing the position of l (which goes through p

k).

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Voronoi Diagrams

Data structures Event list F : It is a priority queue for which the events are sorted

lexicographically by increasing x , then increasing y .

In this list, on can find all the points pi Î P, and the limit circle

events . The point used to classify the cicle events is the rightmost point of the circle. Moreover, a pointer to the arc inside the status T shall be stored.

Every point pi is known in advance, however circle events are not

and must be detected “on the fly”.

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Voronoi Diagrams

Determination of circle events As each event is processed, triplets of contiguous arcs appear or

disappear Those correspond potentially to a circle event (in the future)

It is clear that at each new triplet of arcs, a circle event should be inserted in the list F.

Two remarks: It is possible that, for a given triplet, the two edges that are concerned in the

Voronoi diagram are divergent. In this case, no need to insert a new event (it would be in the past !)

If the edges are convergent it is still possible that the corresponding triplet disappear before it is processed. This would be because of another event (e.g. a point event), and has to be considered as a “false alarm”, thus the corresponding circle event should be deleted from L.

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Voronoi Diagrams

Two remarks: It is possible that, for a given triplet, the two

edges that are concerned in the Voronoi diagram are divergent. In this case, no need to insert a new event (it would be in the past !)

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Voronoi Diagrams

Two remarks: If the edges are convergent it is still possible that

the corresponding triplet disappear before it is processed. This would be because of another event (e.g. a point event), and has to be considered as a “false alarm”, thus the corresponding circle event should be deleted from L.

It is easy if there is a link between the arcs in the status and the circle events (a pointer) stored along with the arc in the status.

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Voronoi Diagrams

Global algorithm

Voronoi(P)Input : a set of points P={p

0,p

1 … , p

n-1}

Output : a DCEL of the V.D. of P in its bounding box (V){ Initialize an event list F with the points in P ( ‘point’ events) Initialize en empty status T Initializer an empty DCEL V While F contains events { Take the first event from F (lexicographic order) If it is a “point” event then ProcessPointEvt(pi) Else ProcessCircleEvt(f) // f is the leaf in the status corresponding to the arc that will be deleted } The internal nodes of T are half edges of the V.D. they should be limited by the bounding box so that “outer” cells are bounded and allow traversal Walk the DCEL and create faces (cells)}

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Voronoi Diagrams

ProcessPointEvt(pi)

{ If T is empty, insert p

i into T

Else { Look for the arc a immediately on the left to p

i ; it is associated to another point p

j

If the leaf containing a has a pointer toward a circle event in F, delete that event. Replace the leaf in T with a subtree having 3 leaves that correspond to the 3 new triplets In this subtree : The middle leaf corresponds to the new arc a

i associated to p

i ;

The two others are copies of the old arc a (cut in two) Insert two new internal nodes (p

i,p

j) and (p

j,p

i)

Re-balance the tree. Create two new half edges in the DCEL V, for the edge separating V(p

i) et V(p

j)

Check that the triplets of consecutive arcs that are above or below the new arc ai (from p

i ) correspond to

converging edges : if yes then add a corresponding circle event in the list F. }}

a,pj

a,pj

a,pj

a',pi

(pj,p

i)

(pi,p

j)

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Voronoi Diagrams

ProcessCircleEvt(c){ Delete the leaf c (that contain the arc a) from T. Update the tuples in the internal nodes of T Delete every circle event associated to a (they can be found by walking T from c) Add the center of the circle as a new vertex into the DCEL of the V.D. Create two new half edges that correspond to the new triplets Check that the new triplets correspond to converging edges : if yes then add a corresponding circle event in the list F. }

a',pj

a",pk

a,pi

(pj,p

i)

(pi,p

k)

a',pj

a",pk

(pj,p

k)

c......

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Voronoi Diagrams

Some degenerate cases : At the beginning of the algorithm : points having the same x

coordinate

Concyclic points (4 or more)

Point event exactly on the right of a break in the beack line (same y coordinate)

Points corresponding to three successive arcs that are also co-linear

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Voronoi Diagrams

Some othe types of Voronoï diagrams

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Voronoi Diagrams

Order p Voronoi diagrams Aims to build regions closest to p of the sites Order 1 diagram is the regular Voronoi diagram With n sites, it is possible to build a diagram of maximum

order n – 1 This diagram of order n – 1 is also called the Maximum

Distance Diagram (MDD)

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Voronoi Diagrams

Order 2 diagram Order n – 1 diagram

Cell where p

1 and p

2 are

the closest

p1

p2 p

3

p4

p5

p6

p7

p8

Cell where p

4 and p

5 are

the closest

Cell closestto p

2 – p

9

(furthestawayfrom p

1)p

1

p2 p

3

p4

p5

p6

p7

p8

p9

Cell where p

3 and p

8 are

the closest

Cell furthestawayfrom p

4

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Voronoi Diagrams

Properties of the MDD

Let P = {p1 … p

n–1} a set of points in the plane

A point pi of P is associated to a cell of the

MDD of P only if it belongs to theconvex hull of P.

The proof is trivial : a point p inside the convex hull of P, is never the furthest away fromany other point q : on a straight line joining p and q, there necessarily exist points on the convex hullsuch that the distance to q is greater.

q

p

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Voronoi Diagrams

Properties of the MDD The cells are infinite, hence there are no

cycles in the MDD, therefore its graph is a tree.

Proof : from a point q ; located in a cellassociated to p

i, one draw a line (qp

i).

For every point of the half line from q goingopposite to p

i, p

i is the most far away

point from P.

There are O(n) cells, vertices; edges(the proof is similar to the one for regularVoronoï diagrams)In particular,

qp

i

ne≤2n−3

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Voronoi Diagrams

Properties of the MDD The center c of any circumscribed circle to P

(three points or more) is located on a vertex of the MDD Or exactly on the middle of a segment joining

the two sites associated to an edge of the MDD(two points).

cc

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Voronoi Diagrams

Computation of the MDD

- Computation of the convex hull of P. The vertices located on the convex hull are ordered in a cycle s

0 … s

h–1 (seen in a previous

chapter)

- One « mixes » randomly the vertices that hare located on the convex hull. Lets call p

0 … p

h–1 this ordering.

- One takes the vertices one by one : k=0... h–4 ( ph–1

… p3 )

- The preceding vertex is stored (clockwise order) and the successor (anticlockwise) : to p

h-k-1 correspond s

j, and we associate s

j–1 mod (h-k)

and sj+1 mod (h-k)

- One takes off ph-k-1

(hence sj) from the cycle (while updating the

cycle so that it remains ordered … ). Therefore, ph-k-1

cannot be the predecessor or successor of points canceled later on

- Finally, one obtains a structure that allows ; starting from 4 points of the convex hull, to rebuild the convex hull while maintaining the ordering at every moment.

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Voronoi Diagrams


- Associated to a point pj, one keeps a pointer to the most anti-

clockwise edge of the cell associated to pj.

pj

Cell of pj

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Voronoi Diagrams


- Incrémental approach : from the MDD of pi–1

, one builds the MDD for p

i

pi

pred(pi)

succ(pi)

Cellule de succ(p

i)

Cellule de pred(p

i)

pi+1 pred(p

i+1)succ(p

i)

Cellule de pi

Cellule de succ(p

i)

Cellule de pred(p

i)

succ(pi+1

)

pi

pred(pi)

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Voronoi Diagrams


- The cell for pi comes “in between” that of pred(p

i) and succ(p

i)

- These two cells are separated by a half line that is part of the bissector of pred(p

i) and succ(p

i)

- succ(pi) has in fact a pointer on this edge.

- The bissector of pi and succ(p

i) induces a new edge (a half line, in

red)

succ(pi)

pi succ(p

i)

pi

Cellule de succ(p

i)

pred(pi)pred(p

i)

Cellulede pred(p

i)

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Voronoi Diagrams


- One has to traverse the cell of succ(pi) in the clockwise order to

know which edge is intersected by the new half edge.

- From the other side of this edge, there is the cell corresponding to another point p

j

- One traverse this cell again, to find the intersection with the bissector of p

i-p

j

succ(pi)

pi

Cellule de succ(p

i)

pi

pj

pi

pk

Cellule de pj

Cell.de p

k

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Voronoi Diagrams

Computation of the MDD - One repeats this set of operations until we meet pred(p

i), which wil

give the other half-line that bounds the cell of pi.

Do not forget to update the pointers between the existing points pi

and the corresponding edges (most anticlockwise) ...

pi

pk

pi pred(p

i)

pi

Cellule de pi

Cell.de p

k

Cell.de pred(p

i)

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Voronoi Diagrams

Computation of the MDD The computation may be initialized with a MDD of the 4 first vertices

p0...p

3 for which the MDD is trivial :

p1

p2p

3

p0

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Voronoi Diagrams

Complexity of the algorithm Initial convex hull : O(nlog n), contains the n points in the worst case.

After the insertion of pi, the corresponding cell has k edges.

Hence, to build this cell from the MDD obtained after the insertion of pi–1

, the algorithm has walked in k neighboring cells.

These k cells are separated by a subtree that has at most 2k–3 distinct edges, and in the worst case, one will walk these edges two times, hence a total of 4k–6 tests (linear in k)

pi

The MDD at step i has at most 2i–3 edges. Each edge separates two cells. So, in average, each cell has less than (4i–6)/i edges, which is smaller than 4.

Each point pi has the same probability to be inserted at

the end of the process, so globally the complexity to build the MDD is in O(n).

Globally, the complexity is therefore in O(nlog n) with a storage in O(n).

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Voronoi Diagrams

Application of the MDD Metrology ( in relation with the classical V.D.)

Dertermine the two min-max circles having the same center that approximate a cloud of samples (coming e.g. from a laser scanner operating on a machined cylindrical part, to asses the quality of the machining)

There are multiple cases : The interior circle (max size) goes through 3 or more samples – its center is

therefore on a vertex of the classical V.D. The exterior circle is then coincident with only one point (the one corresponding to a cell of the MDD)

The exterior circle (min size) goes trough three or more points – its center is therefore on a vertex of the MDD. The internal circle is then coincident with only one sample (that corresponding the a cell of the V.D.)

Finally; the external circle goes through two points, as well as the interior circle. The center is therefore at the intersection of an edge of the MDD, and one of the V.D.

To compute these circles and find the optimum, one must therefore compute the arrangements between a V.D. and the MDD.

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Voronoi Diagrams

For each vertex of the MDD, find the point of P that is the closest(gives 4 points)

For each vertex of the V.D., find the point of P the most far away(gives 4 points)

For each intersection between the V.D. and the MDD, determine the 4 points in P

For each of these groups of candidates, find the one which gives the smallest ring.

This may be done in O(n²) in the worst case

There may be O(n²) additional sites due to the intersection of the two graphs.

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Voronoi Diagrams

In the present case; 13 combinations have to be checked

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Voronoi Diagrams

Voronoi diagrams in other norms (Lp,p≠2 ; polygonal ...)

Uniform L1 norm “local” polygonal norms

Introduction Segment-Segment intersections Polygon … › CADCG › CADCG_11.pdf · 2020-06-10 ·...

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