# Delaunay Triangulation

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## WHAT IS IT?

This model constructs a constructs a Delaunay triangulation from a set of 2-dimensional spatial coordinates. A point-set triangulation is a division of a 2-dimensional space into triangles with vertices established by a given set of points, and in a Delaunay triangulation these triangles are created such that the minimum angle of each triangle is maximized, which avoids sliver triangles (thin triangles with two highly acute angles). It also has the defining characteristic that the circumcircle of every triangle contains none of the (other) points in the set.

Delaunay triangulation has a number of computational uses, including the creation of a Voronoi tesselation (of Thiessen polygons) corresponding to the point set and other processes associated with interpolating a spatially varying scalar quantity such as elevation or rainfall. Finding the Delaunay triangulation is a first step for some methods of creating contour lines for topographic or isopleth maps.

## HOW IT WORKS

Each point in the point-set is associated with a mobile agent (a turtle of breed "point") and triangle sides are established as undirected links (of breed "edge"). Additional agents of breed "triangle" are created to establish connectivity and to facilitate the logic for determining whether a point is within the circumcircle of a given triangle (by comparing the distance of that point from the circumcenter of the triangle to the radius of the circumcircle).

The model uses the Bowyer-Watson algorithm for computing the Delaunay triangulation, which requires the creation of three additional points as the vertices of a "super triangle" that circumscribes the original set of points, or, as in this case, a pair of adjacent super triangles forming a super 'square'. The algorithmn considers each point in the data set incrementally (based on horizontal coordinate left-to-right), removing triangles that are interposed by the new point and replacing them with new conforming triangles by connecting the new point to the vertices of a "star-shaped polygonal hole" created by the union of the recently removed triangles. The super triangle vertices and all associated connections are then removed from consideration leaving the desired triangulation.

Algorithmic note: in order to ensure the creation of a convex hull, the "super" vertices should be considered at an infinite distance from the data set (a consideration that does not appear to be part of the original Bowyer or Watson literature)

## HOW TO USE IT

X1,Y1,X2,Y2,...: The user can enter a desired set of coordinates into the provided input boxes being sure that the values stay within the range listed as 'minimum x','maximum x','minimum y' and 'maximum y' above the viewscreen. Each ordered pair should also be unique. Screen size can be expanded using SETTINGS which will be reflected in the min-max monitors.

POINT1, POINT2, ...: Each spatial point can be toggled ON or OFF using the switch to have the point included in (ON) or excluded from (OFF) the point set, but note that each active point should have a unique location to avoid an error message.

VAL1, VAL2,...: These input boxes are provided to associate a scalar value with each spatial point. This value serves no function in this model, but is included under the assumption that the model might be expanded as part of a larger interpolation model.

SETUP: When pressed, displays all the active points with an identifying label

TRIANGULATE: When pressed, executes SETUP and then calculates and displayes the Delaunay triangulation of the active spatial points. Intermediate triangulations following the Bowyer-Watson algorithm are temporarily displayed but may not be visible.

PAUSE-BETWEEN-INCREMENTS?: When toggled ON, the algorithm will pause at specified places in the algorithm, namely when considering the removal of triangles violated by a newly introduced vertex. The triangles are outlined in yellow and the vertex is displayed with a larger circle. Edges deleted from consideration are subsequently drawn in red.

## THINGS TO TRY

Adjust the speed slider towards "slower" to watch a visualization of the Bowyer-Watson calculation.

## EXTENDING THE MODEL

Delaunay triangulation is a first step for some methods of creating contour lines for topographic or isopleth maps (see "Related Models"), and for creating Thiessen polygons.

## NETLOGO FEATURES

The model uses specialized breeds of turtle agents to represent both vertices and triangles, as well as a breed of undirected link agents to represent triangle edges, greatly simplifying the visualization of the triangulation.

## RELATED MODELS

The author plans to share a model entitled 'Contour20' that uses this triangulation in the creation of an isopleth map using contour lines.

## AUTHORSHIP

Created Dec 2017

Garrett Love, Ph.D.

Engineering Instuctor

NC School of Science and Math

This work supported by the National Science Foundation

STEM + C Award 1543228

Comp Hydro: Integrating data, computation, and visualization for model-based water literacy

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/4.0/.

## CREDITS AND REFERENCES

En.wikipedia.org. (2017). Bowyer–Watson algorithm. [online] Available at: https://en.wikipedia.org/wiki/Bowyer-Watson_algorithm [Accessed 20 Dec. 2017].

Bowyer, Adrian (1981). "Computing Dirichlet tessellations". Comput. J. 24 (2): 162–166.

Watson, David F. (1981). "Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes". Comput. J. 24 (2): 167–172.

C. Arens. The Bowyer-Watson Algorithm; An efficient Implementation in a Database Environment. Technical report, Delft University of Technology, January 2002.

## Comments and Questions

breed [points point] undirected-link-breed [edges edge] breed [triangles triangle] globals [numpoints] points-own [value super?] triangles-own [Ux Uy rad2 bad? vertices sides] edges-own [keep?] to setup ca set numpoints 20; adjust if interface expanded to include more points let xlist (list 0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20) let ylist (list 0 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 y16 y17 y18 y19 y20) let vallist (list 0 val1 val2 val3 val4 val5 val6 val7 val8 val9 val10 val11 val12 val13 val14 val15 val16 val17 val18 val19 val20) let onlist (list false point1 point2 point3 point4 point5 point6 point7 point8 point9 point10 point11 point12 point13 point14 point15 point16 point17 point18 point19 point20) if max xlist > max-pxcor - 1 or max ylist > max-pycor - 1 or min xlist < min-pxcor + 1 or min ylist < min-pycor + 1 [user-message "One or more coordinate values are outside the screen range. Please check your inputs and setup again" stop] create-points numpoints + 1 [ if who != 0 [set shape "circle" set size .6 set color red] setxy item who xlist item who ylist set value item who vallist set label who set label-color yellow set hidden? not item who onlist set super? false face patch 0 0 ] ask points with [not hidden?] [if distance min-one-of other points with [not hidden?] [distance myself] = 0 [user-message "Two or more active points have identical coordinates. Please halt, revise your inputs and setup again" stop]] create-points 1 [set hidden? true set super? true setxy (min-pxcor + 1) (min-pycor + 1)] ; who numpoints + 1 create-points 1 [set hidden? true set super? true setxy (min-pxcor + 1) (max-pycor - 1)] ; who numpoints + 2 create-points 1 [set hidden? true set super? true setxy (max-pxcor - 1) (max-pycor - 1)] ; who numpoints + 3 create-points 1 [set hidden? true set super? true setxy (max-pxcor - 1) (min-pycor + 1)] ; who numpoints + 4 end to triangulate setup ifelse count points with [not hidden?] < 3 [user-message "You will need at least three active points to triangulate"] [ make-triangle point (numpoints + 1) point (numpoints + 2) point (numpoints + 3) ; super-triangle 1 make-triangle point (numpoints + 3) point (numpoints + 4) point (numpoints + 1) ; super-triangle 2 let triset (sort-on [xcor] points with [not hidden?]) foreach triset ; for each point in the pointlist [p -> ask triangles [if inside-circum? p self [set bad? true]] ; for each triangle in the triangulation flag points within circumcircle let edgelist [] ask triangles with [bad?] ; for each new bad triangle [ if Pause-between-increments? [ask link-set sides [set color yellow] ask p [set size 2] user-message "Next step?" ask p [set size 1]] ask link-set sides [set edgelist ifelse-value member? self edgelist [remove self edgelist][lput self edgelist]]] clean-up foreach edgelist [ed -> make-triangle [end1] of ed [end2] of ed p];create new triangles ] ask triangles with [((member? point (numpoints + 1) vertices) or (member? point (numpoints + 2) vertices) or (member? point (numpoints + 3) vertices) or (member? point (numpoints + 4) vertices))][set bad? true] if Pause-between-increments? [ user-message "Complete triangulation?"] clean-up ask links with [color = red][die] ] end to make-triangle [p1 p2 p3] let me 0 create-triangles 1 [set hidden? true set bad? false set vertices turtle-set (list p1 p2 p3) let Ax [xcor] of p1 let Ay [ycor] of p1 let Bx [xcor] of p2 let By [ycor] of p2 let Cx [xcor] of p3 let Cy [ycor] of p3 let midx (max-pxcor + min-pxcor) / 2 let midy (max-pycor + min-pycor) / 2 if [super?] of p1 [set Ax midx - 10000 * (midx - Ax) set Ay midy - 10000 * (midy - Ay)] ; super vertices to "infinity" if [super?] of p2 [set Bx midx - 10000 * (midx - Bx) set By midy - 10000 * (midy - By)] if [super?] of p3 [set Cx midx - 10000 * (midx - Cx) set Cy midy - 10000 * (midy - Cy)] let d1x Bx - Ax let d1y Ay - By let d2x Cx - Bx let d2y By - Cy let d3x Ax - Cx let d3y Cy - Ay let A2 (Ax ^ 2 + Ay ^ 2) let B2 (Bx ^ 2 + By ^ 2) let C2 (Cx ^ 2 + Cy ^ 2) let D 2 * (Ax * d2y + Bx * d3y + Cx * d1y) set Ux (A2 * d2y + B2 * d3y + C2 * d1y) / D set Uy (A2 * d2x + B2 * d3x + C2 * d1x) / D set rad2 ((Ax - Ux) ^ 2 + (Ay - Uy) ^ 2) set me who ] ask [vertices] of triangle me [create-edges-with other [vertices] of triangle me [set hidden? false]] let vwho [who] of [vertices] of triangle me ask triangle me [set sides (list edge item 0 vwho item 1 vwho edge item 1 vwho item 2 vwho edge item 2 vwho item 0 vwho)] end to-report inside-circum? [pt tri] let r1sq ([xcor] of pt - [Ux] of tri ) ^ 2 + ([ycor] of pt - [Uy] of tri ) ^ 2 report r1sq < [rad2] of tri end to clean-up ask edges [set keep? false] ask triangles [ifelse bad? [die][ask link-set sides [set keep? true set color green]]];remove bad triangles ask edges with [not keep?][set color red] ;remove unused edges end

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## Attached files

File | Type | Description | Last updated | |
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Delaunay Triangulation.png | preview | Preview for 'Delaunay Triangulation' | about 1 month ago, by Garrett Love | Download |

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