breed [points point] ;data points to become Delaunay triangulated vertices points-own [value super?] ;value to be contoured, flag indicating "super" vertices of encompassing square breed [triangles triangle] ;triangle objects, defined by three unique vertex points triangles-own [Ux Uy rad2 ;circumcenter coordinates, circumcircle radius squared bad? ;flag for removal from triangulation vertices sides] ;vertex and side pointers undirected-link-breed [edges edge] ;triangle edges edges-own [keep?] ;retention flag for triangulation undirected-link-breed [isos iso] ;isopleth segments breed [tics tic] ;endpoints for isopleth segments globals [error? contour-list tic-list Pause-between-increments? numpoints triangulated? baseptsize minz maxz zrange] to setup ca zoom 100 set error? true set triangulated? false set Pause-between-increments? false;for debugging or viewing progress of Delaunay triangulation set numpoints 30 ;adjust if interface expanded to include more points set baseptsize .6 ;input data from interface, starting with index = 1 let xlist (list 0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30) let ylist (list 0 y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11 y12 y13 y14 y15 y16 y17 y18 y19 y20 y21 y22 y23 y24 y25 y26 y27 y28 y29 y30) let vallist (list 0 val1 val2 val3 val4 val5 val6 val7 val8 val9 val10 val11 val12 val13 val14 val15 val16 val17 val18 val19 val20 val21 val22 val23 val24 val25 val26 val27 val28 val29 val30) let onlist (list false point1 point2 point3 point4 point5 point6 point7 point8 point9 point10 point11 point12 point13 point14 point15 point16 point17 point18 point19 point20 point21 point22 point23 point24 point25 point26 point27 point28 point29 point30) ;error on out-of-bounds coordinates - note border let border ceiling (world-width / 4) if max xlist > max-pxcor - border or max ylist > max-pycor - border or min xlist < min-pxcor + border or min ylist < min-pycor + border [;beep ;not functional for NLweb user-message "One or more coordinate values are outside the screen range. Please check your inputs and try again" stop] let scalerange max (list (max xlist - min xlist) (max ylist - min ylist)) let scale (min (list world-width world-height) - 2 * border) / scalerange let xctr (min xlist + max xlist) / 2 let yctr (min ylist + max ylist) / 2 ;create data points create-points numpoints + 1 [ set shape "circle" set size baseptsize set color red set hidden? not item who onlist setxy item who xlist item who ylist if fit-to-frame? [setxy (xcor - xctr) * scale + (max-pxcor + min-pxcor) / 2 (ycor - yctr) * scale + (max-pycor + min-pycor) / 2] set value item who vallist set label value set label-color white set super? false face patch 0 0 0 ] mapvals ;error on too few data points if count points with [not hidden?] < 3 [user-message "You will need at least three active points to triangulate" stop] ;error on duplicated coordinates ask points with [not hidden?] [if distance min-one-of other points with [not hidden?] [distance myself] = 0 [;beep ;not functional for NLweb user-message "Two or more active points have identical coordinates. Please halt, revise your inputs and try again" stop]] ;create super-vertices at corners 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 set error? false end to update-vals let vallist (list 0 val1 val2 val3 val4 val5 val6 val7 val8 val9 val10 val11 val12 val13 val14 val15 val16 val17 val18 val19 val20) ask points with [who <= 20] [set value item who vallist set label value] set minz min (list min [value] of points with [not hidden?] alt-min-val) set maxz max (list max [value] of points with [not hidden?] alt-max-val) set zrange maxz - minz end to mapvals update-vals ifelse show3D? [ask points [set zcor min-pzcor + world-depth / 4 + (world-depth / 2) * (value - minz) / zrange]] [ask points [set zcor 0]] end to triangulate setup if error? [stop] ;create super-triangles 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?]) ;process left to right 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 [] ; initialize ask triangles with [bad?] ; for each new bad triangle [ if Pause-between-increments? ;for debugging or viewing progress of Delaunay triangulation [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] ;remove if duplicated - i.e. inside the polygonal hole [lput self edgelist]]] ;add if unique clean-up ; remove unused foreach edgelist [ed -> make-triangle [end1] of ed [end2] of ed p] ;create new triangles by connecting to vertices of the poly hole ] ; remove the super vertices and associated edges 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] set triangulated? true end to make-triangle [p1 p2 p3] ;define triangle by vertices 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 ; Adjustment to Bowyer-Watson to ensure convex hull - super vertices to "infinity" if [super?] of p1 [set Ax midx - 10000 * (midx - Ax) set Ay midx - 10000 * (midy - Ay)] if [super?] of p2 [set Bx midx - 10000 * (midx - Bx) set By midx - 10000 * (midy - By)] if [super?] of p3 [set Cx midx - 10000 * (midx - Cx) set Cy midx - 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 ;circumcenter coordinates set Uy (A2 * d2x + B2 * d3x + C2 * d1x) / D ;circumcenter coordinates set rad2 ((Ax - Ux) ^ 2 + (Ay - Uy) ^ 2) ;circumcircle radius squared set me who ] ask [vertices] of triangle me [create-edges-with other [vertices] of triangle me [set thickness .3 set hidden? false]] ;will not duplicate undirected links let vwho [who] of [vertices] of triangle me ;unordered list of vertex identifiers 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)] ;create list of sides end to-report inside-circum? [pt tri] ;inputs a point and a triangle 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 to contour if triangulated? = 0 [user-message "You will need a triangulation to generate a contour" stop] if not triangulated? [user-message "You will need a triangulation to generate a contour" stop] set tic-list [] ask tics [die] ask isos [die] update-vals ;determine interval to 3 sig figs let vrexp 3 - ceiling log zrange 10 let valint precision (zrange / num-intervals) vrexp set contour-list n-values (num-intervals + 1) [ n -> precision ((precision minz vrexp) + n * valint) vrexp ] ask triangles[ let vlist (sort-on [value] vertices) let maxv item 2 vlist let midv item 1 vlist let minv item 0 vlist let ranget [value] of maxv - [value] of minv ;total range let range1 [value] of midv - [value] of minv ;lower range let range2 [value] of maxv - [value] of midv ;upper range let dxt [xcor] of maxv - [xcor] of minv let dx1 [xcor] of midv - [xcor] of minv let dx2 [xcor] of maxv - [xcor] of midv let dyt [ycor] of maxv - [ycor] of minv let dy1 [ycor] of midv - [ycor] of minv let dy2 [ycor] of maxv - [ycor] of midv let dzt [zcor] of maxv - [zcor] of minv let dz1 [zcor] of midv - [zcor] of minv let dz2 [zcor] of maxv - [zcor] of midv let my-contours filter [i -> i > [value] of minv and i < [value] of maxv] contour-list ;find contours within triangle foreach my-contours [n -> let longwho 0 hatch-tics 1 [set color white set shape "dot" set hidden? true setxyz [xcor] of minv + (n - [value] of minv) * dxt / ranget ;linear interpolation along total side [ycor] of minv + (n - [value] of minv) * dyt / ranget ;linear interpolation along total side [zcor] of minv + (n - [value] of minv) * dzt / ranget ;linear interpolation along total side set longwho who ] hatch-tics 1 [set color sky set shape "dot" set hidden? true ifelse (n < [value] of midv) [setxyz [xcor] of minv + (n - [value] of minv) * dx1 / range1 ;linear interpolation along low side [ycor] of minv + (n - [value] of minv) * dy1 / range1 ;linear interpolation along low side [zcor] of minv + (n - [value] of minv) * dz1 / range1] ;linear interpolation along low side [setxyz [xcor] of midv + (n - [value] of midv) * dx2 / range2 ;linear interpolation along up side [ycor] of midv + (n - [value] of midv) * dy2 / range2 ;linear interpolation along up side [zcor] of midv + (n - [value] of midv) * dz2 / range2] ;linear interpolation along up side create-iso-with tic longwho [set color yellow set thickness .3 set label n set label-color yellow] ;connect tics with isoline segments ] ] ] foreach contour-list [n -> let these-isos isos with [label = n] if count these-isos > 1 [ask one-of these-isos [ask other these-isos [set label ""]]]] ;reduce to a single contour label end @#$#@#$#@ GRAPHICS-WINDOW 0 0 1004 1005 -1 -1 12.0 1 10 1 1 1 0 1 1 1 -41 41 -41 41 -20 20 1 1 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 0 1 -1000 SWITCH 5 402 108 435 point12 point12 0 1 -1000 SWITCH 5 432 108 465 point13 point13 0 1 -1000 SWITCH 5 462 108 495 point14 point14 0 1 -1000 SWITCH 5 493 108 526 point15 point15 0 1 -1000 SWITCH 5 524 108 557 point16 point16 0 1 -1000 SWITCH 5 555 108 588 point17 point17 0 1 -1000 SWITCH 5 586 108 619 point18 point18 0 1 -1000 SWITCH 5 617 108 650 point19 point19 0 1 -1000 SWITCH 5 648 108 681 point20 point20 0 1 -1000 INPUTBOX 114 53 164 113 x1 15.0 1 0 Number INPUTBOX 276 53 326 113 x2 14.3 1 0 Number INPUTBOX 114 115 164 175 x3 16.7 1 0 Number INPUTBOX 276 116 326 176 x4 11.5 1 0 Number INPUTBOX 114 177 164 237 x5 18.0 1 0 Number INPUTBOX 276 178 326 238 x6 9.7 1 0 Number INPUTBOX 114 238 164 298 x7 13.7 1 0 Number INPUTBOX 162 53 212 113 y1 0.8 1 0 Number INPUTBOX 210 53 260 113 val1 0.0 1 0 Number INPUTBOX 324 53 374 113 y2 9.3 1 0 Number INPUTBOX 372 53 422 113 val2 115.0 1 0 Number INPUTBOX 162 115 212 175 y3 9.3 1 0 Number INPUTBOX 210 115 260 175 val3 5.0 1 0 Number INPUTBOX 372 116 422 176 val4 120.0 1 0 Number INPUTBOX 210 177 260 237 val5 0.0 1 0 Number INPUTBOX 372 178 422 238 val6 83.0 1 0 Number INPUTBOX 210 238 260 298 val7 172.0 1 0 Number INPUTBOX 324 116 374 176 y4 9.4 1 0 Number INPUTBOX 162 177 212 237 y5 2.2 1 0 Number INPUTBOX 324 178 374 238 y6 11.6 1 0 Number INPUTBOX 162 238 212 298 y7 3.2 1 0 Number INPUTBOX 325 239 375 299 y8 10.7 1 0 Number INPUTBOX 162 299 212 359 y9 5.0 1 0 Number INPUTBOX 114 299 164 359 x9 17.2 1 0 Number INPUTBOX 114 372 164 432 x11 15.8 1 0 Number INPUTBOX 114 433 164 493 x13 14.2 1 0 Number INPUTBOX 114 494 164 554 x15 17.7 1 0 Number INPUTBOX 276 239 326 299 x8 11.5 1 0 Number INPUTBOX 276 300 326 360 x10 13.1 1 0 Number INPUTBOX 276 370 326 430 x12 10.8 1 0 Number INPUTBOX 276 431 326 491 x14 11.2 1 0 Number INPUTBOX 210 299 260 359 val9 23.0 1 0 Number INPUTBOX 211 372 261 432 val11 0.0 1 0 Number INPUTBOX 211 433 261 493 val13 300.0 1 0 Number INPUTBOX 114 556 164 616 x17 16.3 1 0 Number INPUTBOX 114 618 164 678 x19 12.3 1 0 Number INPUTBOX 275 618 325 678 x20 13.2 1 0 Number INPUTBOX 276 494 326 554 x16 7.0 1 0 Number INPUTBOX 276 556 326 616 x18 18.9 1 0 Number INPUTBOX 163 372 213 432 y11 5.0 1 0 Number INPUTBOX 324 370 374 430 y12 3.3 1 0 Number INPUTBOX 163 433 213 493 y13 5.0 1 0 Number INPUTBOX 324 431 374 491 y14 5.3 1 0 Number INPUTBOX 162 494 212 554 y15 7.0 1 0 Number INPUTBOX 211 494 261 554 val15 4.0 1 0 Number INPUTBOX 162 556 212 616 y17 7.4 1 0 Number INPUTBOX 373 239 423 299 val8 100.0 1 0 Number INPUTBOX 325 300 375 360 y10 5.4 1 0 Number INPUTBOX 373 300 423 360 val10 360.0 1 0 Number INPUTBOX 373 370 423 430 val12 0.0 1 0 Number INPUTBOX 373 431 423 491 val14 195.0 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 82.0 1 0 Number INPUTBOX 210 556 260 616 val17 55.0 1 0 Number INPUTBOX 210 618 260 678 val19 210.0 1 0 Number INPUTBOX 162 618 212 678 y19 7.9 1 0 Number INPUTBOX 325 494 375 554 y16 6.8 1 0 Number INPUTBOX 324 556 374 616 y18 7.0 1 0 Number INPUTBOX 324 618 374 678 y20 10.6 1 0 Number BUTTON 342 10 423 43 NIL contour NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 7 10 82 43 display setup NIL 1 T OBSERVER NIL NIL NIL NIL 1 BUTTON 85 10 183 43 NIL triangulate NIL 1 T OBSERVER NIL NIL NIL NIL 1 SLIDER 442 10 589 43 num-intervals num-intervals 3 12 6.0 1 1 NIL HORIZONTAL MONITOR 599 10 782 55 contours contour-list 1 1 11 BUTTON 442 195 574 228 Toggle triangles ask edges [set hidden? not hidden?] NIL 1 T OBSERVER NIL NIL NIL NIL 1 MONITOR 440 134 522 179 minimum x min-pxcor + ceiling (world-width / 4) 1 1 11 MONITOR 522 133 608 178 maximum x max-pxcor - 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This model constructs an isopleth (or "contour") map of a value that varies in 2-dimensional space, interpolating from a set of up to 30 values and associated coordinates. ## HOW IT WORKS The model first divides the 2-dimensional space into triangles with vertices at each coordinate pair in the "point set". 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). This construction, known as a Delaunay Triangulation, also has the defining characteristic that the circumcircle of every triangle contains none of the (other) points in the set. The model uses the Bowyer-Watson algorithm for computing the Delaunay triangulation, establishing four points in a "super square" around the original point set, and then incrementally considering each point within the square, removing and adding connections to maintain a viable triangulation. The added vertices are then removed from consideration leaving the desired triangulation. The sides of each triangular element are then divided into increments using linear interpolation that compares the values at each vertex to the distances between. Increments of similar value along each side of a triangle are then connected with segments to create lines of equal value, i.e. isopleths or contour lines. ## 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' in the monitors below 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. These are values that will be interpolated to create the contour map. DISPLAY: When pressed, just displays all the active points with an identifying label TRIANGULATE: When pressed, displays all the active points with an identifying label 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 unless the model is adjusted to a slower speed. CONTOUR: When pressed, calculates a series of intervals based on the value of the NUM-INTERVALS slider and displays isopleth values for each interval boundary in the 'contours' monitor (one more than the number of intervals). It then constructs and displays isopleths in the triangulation with labels. Only functions if a triangulation has been already computed, but will recalculate new contours if adjustments have been made to VAL# values. NUM-INTERVALS: Establishes the number of intervals to be visualized with contour lines, spaced at equal intervals between the minimum and maximum active values in the point set OR the range values specified by ALT-MIN-VAL and ALT-MAX-VAL. Note that the interval is rounded to 3 significant digits. ALT-MIN-VAL: Establishes an alternative value for the lowest value contour line, only if it is less than the smallest of the current active data points (as displayed in the 'Min value' monitor). ALT-MAX-VAL: Establishes an alternative value for the highest value contour line, only if it is greater than the largest of the current active data points (as displayed in the 'Max value' monitor). TOGGLE TRIANGLES: When pressed, renders the triangulation lines invisible (or visible), allowing for better view of the contour lines. TOGGLE POINTS: When pressed, renders the data point markers and labels invisible (or visible), allowing for better view of triangulation and/or contour lines. FIT-TO-FRAME?: If ON when DISPLAY or TRIANGULATE is pressed, the data points will be scaled in the x and y dimensions and recentered in the 3D View SHOW3D?: If OFF when DISPLAY or TRIANGULATE is pressed, the data points will be plotted with a z-coordinate value of 0. If ON, each point will be plotted with a z-coordinate scaled to the associated value. ## THINGS TO TRY Adjust the speed slider towards "slower" to watch a visualization of the Bowyer-Watson calculation. Gather data (eg. precipitation, temperature, elevation) from an online resource or classroom experiment and create a contour plot. Note that you may need to scale units, particularly the spatial coordinates. Turn many of the points in a data set "Off" and generate a contour with limited data. Turn some of the points back "On" and regenerate the contour. Consider how the addition of data changes the contour and the estimated values in other spatial locations. ## NETLOGO FEATURES Each point in the point-set is associated with a mobile agent (a turtle of breed "point") and triangle sides and isolines are established as undirected links (of breed "edge" and "iso"). 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 makes extensive use of anonymous procedures ("->" syntax) to deal with the sets of different agent types. ## AUTHORSHIP Created Jan 2018 Garrett Love, Ph.D. Engineering Instructor 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. 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Circle -7500403 true true 70 85 38 Circle -7500403 true true 130 25 38 Circle -7500403 true true 96 51 108 Circle -16777216 true false 113 68 74 Polygon -10899396 true false 189 233 219 188 249 173 279 188 234 218 Polygon -10899396 true false 180 255 150 210 105 210 75 240 135 240 house false 0 Rectangle -7500403 true true 45 120 255 285 Rectangle -16777216 true false 120 210 180 285 Polygon -7500403 true true 15 120 150 15 285 120 Line -16777216 false 30 120 270 120 leaf false 0 Polygon -7500403 true true 150 210 135 195 120 210 60 210 30 195 60 180 60 165 15 135 30 120 15 105 40 104 45 90 60 90 90 105 105 120 120 120 105 60 120 60 135 30 150 15 165 30 180 60 195 60 180 120 195 120 210 105 240 90 255 90 263 104 285 105 270 120 285 135 240 165 240 180 270 195 240 210 180 210 165 195 Polygon -7500403 true true 135 195 135 240 120 255 105 255 105 285 135 285 165 240 165 195 line true 0 Line -7500403 true 150 0 150 300 line half true 0 Line -7500403 true 150 0 150 150 pentagon false 0 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166 210 172 213 173 238 167 251 160 248 154 265 169 264 178 247 186 240 198 260 200 271 217 271 219 262 207 258 195 230 192 198 210 184 227 164 242 144 259 145 284 151 277 141 293 140 299 134 297 127 273 119 270 105 Polygon -7500403 true true -1 195 14 180 36 166 40 153 53 140 82 131 134 133 159 126 188 115 227 108 236 102 238 98 268 86 269 92 281 87 269 103 269 113 x false 0 Polygon -7500403 true true 270 75 225 30 30 225 75 270 Polygon -7500403 true true 30 75 75 30 270 225 225 270 @#$#@#$#@ NetLogo 3D 6.0.4 @#$#@#$#@ @#$#@#$#@ @#$#@#$#@ @#$#@#$#@ @#$#@#$#@ default 0.0 -0.2 0 0.0 1.0 0.0 1 1.0 0.0 0.2 0 0.0 1.0 link direction true 0 Line -7500403 true 150 150 90 180 Line -7500403 true 150 150 210 180 @#$#@#$#@ 0 @#$#@#$#@