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] while [not mouse-down?][] wait .5 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 @#$#@#$#@ GRAPHICS-WINDOW 429 54 996 622 -1 -1 13.0 1 14 1 1 1 0 0 0 1 -21 21 -21 21 0 0 1 ticks 30.0 SWITCH 5 53 108 86 point1 point1 0 1 -1000 SWITCH 5 84 108 117 point2 point2 0 1 -1000 SWITCH 5 115 108 148 point3 point3 0 1 -1000 SWITCH 5 145 108 178 point4 point4 0 1 -1000 SWITCH 5 176 108 209 point5 point5 0 1 -1000 SWITCH 5 206 108 239 point6 point6 0 1 -1000 SWITCH 5 236 108 269 point7 point7 0 1 -1000 SWITCH 5 267 108 300 point8 point8 0 1 -1000 SWITCH 5 297 108 330 point9 point9 0 1 -1000 SWITCH 5 328 108 361 point10 point10 0 1 -1000 SWITCH 5 372 108 405 point11 point11 1 1 -1000 SWITCH 5 402 108 435 point12 point12 1 1 -1000 SWITCH 5 432 108 465 point13 point13 1 1 -1000 SWITCH 5 462 108 495 point14 point14 1 1 -1000 SWITCH 5 493 108 526 point15 point15 1 1 -1000 SWITCH 5 524 108 557 point16 point16 1 1 -1000 SWITCH 5 555 108 588 point17 point17 1 1 -1000 SWITCH 5 586 108 619 point18 point18 1 1 -1000 SWITCH 5 617 108 650 point19 point19 1 1 -1000 SWITCH 5 648 108 681 point20 point20 1 1 -1000 INPUTBOX 114 53 164 113 x1 -7.0 1 0 Number INPUTBOX 276 53 326 113 x2 3.0 1 0 Number INPUTBOX 114 115 164 175 x3 12.0 1 0 Number INPUTBOX 276 116 326 176 x4 -10.0 1 0 Number INPUTBOX 114 177 164 237 x5 1.0 1 0 Number INPUTBOX 276 178 326 238 x6 13.0 1 0 Number INPUTBOX 114 238 164 298 x7 -1.0 1 0 Number INPUTBOX 162 53 212 113 y1 -11.0 1 0 Number INPUTBOX 210 53 260 113 val1 0.0 1 0 Number INPUTBOX 324 53 374 113 y2 -14.0 1 0 Number INPUTBOX 372 53 422 113 val2 0.0 1 0 Number INPUTBOX 162 115 212 175 y3 -14.0 1 0 Number INPUTBOX 210 115 260 175 val3 0.0 1 0 Number INPUTBOX 372 116 422 176 val4 0.0 1 0 Number INPUTBOX 210 177 260 237 val5 0.0 1 0 Number INPUTBOX 372 178 422 238 val6 0.0 1 0 Number INPUTBOX 210 238 260 298 val7 0.0 1 0 Number INPUTBOX 324 116 374 176 y4 2.0 1 0 Number INPUTBOX 162 177 212 237 y5 -2.0 1 0 Number INPUTBOX 324 178 374 238 y6 -1.0 1 0 Number INPUTBOX 162 238 212 298 y7 8.0 1 0 Number INPUTBOX 325 239 375 299 y8 9.0 1 0 Number INPUTBOX 162 299 212 359 y9 -18.0 1 0 Number INPUTBOX 114 299 164 359 x9 -8.0 1 0 Number INPUTBOX 114 372 164 432 x11 -19.0 1 0 Number INPUTBOX 114 433 164 493 x13 -19.0 1 0 Number INPUTBOX 114 494 164 554 x15 0.0 1 0 Number INPUTBOX 276 239 326 299 x8 10.0 1 0 Number INPUTBOX 276 300 326 360 x10 7.0 1 0 Number INPUTBOX 276 370 326 430 x12 19.0 1 0 Number INPUTBOX 276 431 326 491 x14 19.0 1 0 Number INPUTBOX 210 299 260 359 val9 0.0 1 0 Number INPUTBOX 211 372 261 432 val11 0.0 1 0 Number INPUTBOX 211 433 261 493 val13 0.0 1 0 Number INPUTBOX 114 556 164 616 x17 0.0 1 0 Number INPUTBOX 114 618 164 678 x19 0.0 1 0 Number INPUTBOX 275 618 325 678 x20 0.0 1 0 Number INPUTBOX 276 494 326 554 x16 0.0 1 0 Number INPUTBOX 276 556 326 616 x18 0.0 1 0 Number INPUTBOX 163 372 213 432 y11 -19.0 1 0 Number INPUTBOX 324 370 374 430 y12 19.0 1 0 Number INPUTBOX 163 433 213 493 y13 19.0 1 0 Number INPUTBOX 324 431 374 491 y14 -19.0 1 0 Number INPUTBOX 162 494 212 554 y15 0.0 1 0 Number INPUTBOX 211 494 261 554 val15 0.0 1 0 Number INPUTBOX 162 556 212 616 y17 0.0 1 0 Number INPUTBOX 373 239 423 299 val8 0.0 1 0 Number INPUTBOX 325 300 375 360 y10 9.0 1 0 Number INPUTBOX 373 300 423 360 val10 0.0 1 0 Number INPUTBOX 373 370 423 430 val12 0.0 1 0 Number INPUTBOX 373 431 423 491 val14 0.0 1 0 Number INPUTBOX 373 494 423 554 val16 0.0 1 0 Number INPUTBOX 373 556 423 616 val18 0.0 1 0 Number INPUTBOX 372 618 422 678 val20 0.0 1 0 Number INPUTBOX 210 556 260 616 val17 0.0 1 0 Number INPUTBOX 210 618 260 678 val19 0.0 1 0 Number INPUTBOX 162 618 212 678 y19 0.0 1 0 Number INPUTBOX 325 494 375 554 y16 0.0 1 0 Number INPUTBOX 324 556 374 616 y18 0.0 1 0 Number INPUTBOX 324 618 374 678 y20 0.0 1 0 Number MONITOR 429 643 499 688 mouse x ifelse-value (mouse-inside?) [mouse-xcor][\"\"] 1 1 11 MONITOR 500 643 565 688 mouse y ifelse-value mouse-inside? [mouse-ycor][\"\"] 1 1 11 BUTTON 6 10 72 43 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 75 10 173 43 NIL triangulate NIL 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 430 10 512 55 minimum x min-pxcor + 1 1 1 11 MONITOR 514 10 600 55 maximum x max-pxcor - 1 1 1 11 MONITOR 603 10 684 55 minimum y min-pycor + 1 1 1 11 MONITOR 684 10 769 55 maximum y max-pycor - 1 1 1 11 SWITCH 765 626 997 659 Pause-between-increments? Pause-between-increments? 0 1 -1000 TEXTBOX 721 663 1029 693 If 'Pause-between-increments' is \"On\", click inside the display screen to advance the Delaunay algorithm. 11 0.0 1 @#$#@#$#@ ## WHAT IS IT? This model 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. Click on the display screen to advance the algorithm. ## 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 has shared other models entitled 'Contour20' as well as a 3D version 'Contour30' 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. 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