globals [ ;ticks ;; model time counter. not essential to a* obstacle-color ;; color of obstacles--see information about this space-color ;; color of open space border-color ;; color of the world border ] breed [ bots bot ] ;; bots use a*nodes to search the space breed [ a*nodes a*node ] ;; each a*node occupies a place on the grid, contains the cost to get there breed [ maza mazum ] ;; a mazum is used to draw mazes. breed [ indicators indicator ] a*nodes-own [ owner ;; the bot that owns this node parent ;; the node that is the parent (that comes earlier in the path) of this node child ;; the node that is the child (comes later in the path) of this node (not standard a*) g-score ;; stores the cummulative cost of arriving at this point from the start h-score ;; stores the estimated cost of getting from this point to the goal/target f-score ;; the sum of g and h location ;; the patch that contains this node (duplicates "patch-here" when the node is on the path, but can be used in an -of expression, unlike patch-here) closed? ;; node are open (0), or closed (1). Nodes start out open ; target ;; copy of the PATCH that is the target patch ; start ;; copy of the PATCH that is the start patch on-path? ;; is this node on the final route (or route-so-far, if needed) of the path to the target ] bots-own [ start ;; the patch that the search starts from, may not be the current patch target ;; the patch that is the goal / target of the search owner ;; the owner of the search---always the bot itself... exists so it can be inherited by the a*nodes child ;; the child of this bot is the first a*node of the path.. the a*node on the start g-score ;; the bots own g-score is 0, unless the bot moves along its own path... then it might change... path-end ;; the a*node at the end of the path... is nobody until the goal/target is reached. ;; the path-end node is the only node that can construct the entire path back to the start done? ;; boolean that indicates the search is over--may mean that no path can be found current ;; the current a*node being examined or expanding the search o-current ;; the node that was previously current d-mode ;; directions mode, either 4 or 8 heuristic ;; heuristic-mode used by this bot ] maza-own [ maze-wall-color ;; color of maze walls--also color of unexplorred (not carved) paths maze-breadcrumb-color ;; color of bread-crumb trail, used to backtrack maze-path-color ;; final color of carved-out paths in the maze maze-border-color ;; color of maze-border maze-curviness ;; probability that maza will choose a non-straight ahead path maze-timeout ] to startup setup end to setup ca ;; ;; Initialize the color variables ;; set obstacle-color blue set border-color pink set space-color black let bread-crumb-color red ;; ;; define the start and target patches ;; (for this demo, they are fixed, but you they could be dynamic ;; let setup-start patch (min-pxcor + 2) ( min-pycor + 2 ) let setup-target patch (max-pxcor - 2) ( max-pycor - 2 ) ;; ;; Setup the search space, based on the selected senario ;; ;; ;; A completely blank field ;; if senario = "blank" [ fill-patches space-color do-terrain border ] ;; ;; A Random scattering of obstaces ;; if senario = "random" ;; random scattering of patches [ do-terrain ask patches [ if random 100 < density [ set pcolor obstacle-color ] ] border ;; make sure that there is at least a clearing around ;; the start and target patches--prevents needlessly creating ;; a large number of unpassable configurations clear-around setup-start clear-around setup-target ] ;; ;; A single path maze ;; if senario = "maze" [ senario-maze maze-new setup-start obstacle-color bread-crumb-color space-color border-color straightness 1 ] ;; ;; A maze with loops ;; if senario = "looped maze" [ senario-maze-looped density maze-new setup-start obstacle-color bread-crumb-color space-color border-color straightness 1 ] ;; ;; A random collection of circular regions that may overlap ;; if senario = "blobs" [ fill-patches space-color do-terrain ask patch (min-pxcor + world-width * .5) (min-pycor + world-height * .55) [ ask n-of (density * .8) (patches in-radius (world-width * .5)) [ ask patches in-radius-nowrap ((world-width * .05) + random (world-width * .05)) [ set pcolor obstacle-color ] ] ] ;ifelse terrain? ;[ border-3 ] ;[ border-2 ] border clear-around setup-start clear-around setup-target ] ;; ;; A pair of bars, crossing the center ;; if senario = "bars" [ fill-patches space-color do-terrain ;; draw horizontal and vertical bars, to within 5 units of edges ask patches with [ pxcor = 0 or pycor = 0 and abs pxcor + 5 < max-pxcor and abs pycor + 5 < max-pycor ] [ set pcolor obstacle-color ] ask patches with [ ( pxcor = int ( min-pxcor + world-width * .25 ) or pxcor = int ( max-pxcor - world-width * .25 ) ) and ( pycor > max-pycor - (world-height * .33) or pycor < min-pycor + (world-height * .33) ) ] [ set pcolor obstacle-color ] border ] ;; ;; Create the search bot with the start and target previously selected ;; create-bot setup-start setup-target extract-heuristic set-default-shape indicators "indicator" ask setup-start [ sprout-indicators 1 [ set color orange set size 4 set heading 180 ] ] ask setup-target [ sprout-indicators 1 [ set color orange set size 4 set heading 0 ] ] end to go ask bots [ go-bot ifelse is-turtle? path-end [ show-path-nodes get-path yellow ] [ ;if show-path-in-progress? ;[ ask a*nodes with [ owner = myself and color != magenta ] ; [ set color magenta ] ; show-path-nodes get-path white ; ] ] ] if show-path-in-progress? [ tick ] if not any? bots with [ done? = false ] [ ask bots [ ifelse is-turtle? path-end [ ask center-patch [ set plabel "Path Solved." ] ] [ ask center-patch [ set plabel "No Path." ] ] ] tick stop ] end to go-bot ;; ;; when path-end is not nobody, that means the target has been reached. ;; and path-end contains the a*node that lies on the target. ;; path-end can then be used to trace the path back to the start ;; ;; when done? is true, that means that no more locations ;; need to be searched. ;; ;; if path-end is nobody when done? is true, that means there is ;; NO path from the start to the target. ;; if path-end = nobody [ ;; ;; collect the nodes that are this bots nodes ;; (if netlogo had mutable lists, or mutable agentsets, this could be made faster!) ;; let my-nodes a*nodes with [ owner = myself ] ;; ;; are any of the nodes open? ;; ifelse any? my-nodes with [ closed? = 0 ] [ ;; ;; yes. do the path search ;; set current a*build-path ;; ;; having done that, was the target found? ;; if [location] of current = target [ set path-end current set done? true ] ] [ ;; ;; no more open nodes means no where left to search, so we are done. ;; set done? true ] ] end to fill-patches [ #color ] ask patches [ set pcolor #color ] end to maze-border ask patches with [ pxcor = min-pxcor or pxcor = max-pxcor or pycor = min-pycor or pycor = max-pycor ] [ set pcolor [ maze-border-color ] of myself ] end to maze-field ask patches with [ pxcor > min-pxcor and pxcor < max-pxcor and pycor > min-pycor and pycor < max-pycor ] [ set pcolor [ maze-wall-color ] of myself ] end to border ;; ;; colors the edge of the world to prevent searches and maze-making from leaking ;; ask patches with [ pxcor = min-pxcor or pxcor = max-pxcor or pycor = min-pycor or pycor = max-pycor ] [ set pcolor border-color ] end to border-2 ;; ;; creates open space along the border ;; ask patches with [ ( pxcor = ( min-pxcor + 1 ) or pxcor = ( max-pxcor - 1 ) ) or ( pycor = ( min-pycor + 1 ) or pycor = ( max-pycor - 1 ) )] [ set pcolor space-color ] end to border-3 ;; ;; creates a border of randomly "elevated" terrain just inside the border ;; ask patches with [ ( pxcor = ( min-pxcor + 1 ) or pxcor = ( max-pxcor - 1 ) ) or ( pycor = ( min-pycor + 1 ) or pycor = ( max-pycor - 1 ) )] [ set pcolor 5 + random-float 5 ] end to do-terrain ;; ;; puts elevation data in the world ;; black = minimum elevation ;; white = max-mum elevation ;; ;; when terrain? is on, path seeks best balance of shortness of path and lowness of elevation ;; if terrain? [ ask patches [ set pcolor 140 - ((distancexy-nowrap 0 0 ) / (.75 * world-width) * 140) ] repeat 3 [ diffuse pcolor .5 ] ask patches [ set pcolor scale-color gray pcolor 0 140 ] ] end to clear-around [ agent ] ;;; ;;; clears obstacles from under and around the given agent ;;; ask agent [ ask neighbors [ set pcolor space-color ] set pcolor space-color ] end to senario-maze [ #maze ] ;; takes a maze object, generates a maze using that object maze #maze end to senario-maze-looped [ #density #maze ] ;; capture color info from maze let #wall [maze-wall-color] of #maze let #path [maze-path-color] of #maze ;; make a standard maze using the maze object senario-maze #maze ;; punch holes in the maze walls at random to create loops ;; depends on the maze starting on an even patch, ;; so can assume that any patch with both coords odd ;; is definitely a wall (obstacle) patch ask n-of (2 * #density) (patches with [ pcolor = #wall and pxcor mod 2 != pycor mod 2 ]) [ set pcolor #path ] end to-report maze-new [ #start #wall #crumb #path #border #curve #timeout ] let me nobody ask #start [ sprout-maza 1 [ set me self set maze-wall-color #wall set maze-breadcrumb-color #crumb set maze-path-color #path set maze-border-color #border set maze-curviness #curve set maze-timeout #timeout ] ] report me end to maze [ #maze ] ;; given a maze turtle, create a single path maze in the field, ;; ask #maze [ maze-border maze-field let #timeout timer + maze-timeout set pcolor maze-breadcrumb-color ;; timeout prevents possibility that an infinite loop may occur ;; (which may not even be possible with this algorithm... ;; each call to maze-drawing carves out or backtracks the maze path ;; this is a reporter with special effects, returns TRUE when there ;; is no more maze to be carved. while [ maze-drawing and timer < #timeout ] [ ] ;; mazum die when they are through carving a maze die ] end to-report maze-drawing ;; a maze can make a maze drawing ;; draw a maze by first drawing a path with a marker color, then backtracking with the space color ;; allows drawing of mazes iterively, rather than recursively, and without using a stack! ;; essentially uses the patches themselves to remember the path and the path-so-far ;; ;; assume no path from here let path nobody ;; candidates for the next step are previously unexplored paths (ie, still colored as obstacles let candidates (patches at-points [ [ -2 0][0 -2 ] [ 2 0 ][ 0 2] ]) with [ pcolor = [maze-wall-color] of myself ] if-else any? candidates [ ;;if there are any unexplored paths, select one of them ;; which path is the straight-ahead path? let straight-patch patch-ahead 2 ; -at (dx * 2) (dy * 2) ;; either co straight (if curve test fails or straight is only choice) ;; or turn left or right (if possible) ifelse member? straight-patch candidates and (count candidates = 1 or random 100 < maze-curviness ) [ set path straight-patch] [ set path one-of candidates with [ self != straight-patch ] ] ;; point to it face-nowrap path ;; jump to it, in two steps, two draw a path from here to that patch ;; (note that jump is first, then color, leaving starting patch color that it was! repeat 2 [ jump 1 set pcolor maze-breadcrumb-color ] ;; report that the maze-drawing still has some exploring to do! report true ] [ ;; if we are here, then there are no unexplored paths in the vicinity. ;; so, candidates are previously carved (open) paths, that we have likely backed up to ;; or have just finished tracing (i.e. colored with the bread-crumb color set candidates (patches at-points [ [ -1 0][0 -1 ] [ 1 0 ][ 0 1] ]) with [ pcolor = [maze-breadcrumb-color] of myself ] ifelse any? candidates [ ;; there is a breadcrumb to retrace. trace it. set path one-of candidates face-nowrap path repeat 2 [ set pcolor maze-path-color jump 1 ] report true ] [ ;; we are finally back at the beginning ;; carve out the starting point, and we are done set pcolor maze-path-color report false ] ] ;; avoid coordinate creep ;; center maza in the patch ;; setxy pxcor pycor end to-report same-patch [ a b ] ;; ;; reports true if both agent a and b, whether turtles or patches, are on the same patch ;; report ([pxcor] of a = [pxcor] of b and [pycor] of a = [pycor] of b ) end to highlight-path [ path-color ] ;; ;; recursive routine that colors the nodes, tracing back up through the node parents ;; until the start node is reached ;; set on-path? true set color path-color if color = yellow [ ifelse heading mod 90 = 0 [set shape "path" ] [ set shape "path-long" ] ] if is-turtle? parent [ set heading towards-nowrap parent ask parent [ highlight-path path-color ] ] end to-report get-path let n path-end if not is-turtle? n [ set n current ] if not is-turtle? n [ stop ] let p (list n ) while [ [location] of n != start ] [ set n [parent] of n set p fput n p ] report p end to show-path-nodes [ p hue ] ask (turtle-set p) [ set color hue if color = yellow [ ifelse heading mod 90 = 0 [set shape "path" ] [ set shape "path-long" ] ] ] ; ask a*nodes with [ member? self p ] [ set color hue ] display ;; foreach p ;; [ set color hue ;; if color = yellow [ ifelse heading mod 90 = 0 [set shape "path" ] [ set shape "path-long" ] ] ;; ] end to color-path-patches [ p ] ;; ;; non-recursive routine that, ;; given the path-end, increments the pcolor of the patches covered by the path, ;; tracing back to the start ;; foreach p [ if pcolor = 0 [ set pcolor 2 ] if pcolor < 9.5 [ set pcolor pcolor + .5 ] ] end to create-bot [ starting-patch target-patch #heuristic ] ;; ;; creates a search bot ;; and the first a*node for that bot. ;; so: a bot always has at least one a*node, and that first node is ;; the child of the bot ;; the first node, however, does not have any "parent"... ;; that is, its parent is always "nobody" ;; this is how a trace-back routine can know that it has reached ;; the begining of a path: when there is no more parents to trace-back to... ;; create-bots 1 [ set color gray set start starting-patch set target target-patch set owner self set path-end nobody set g-score 0 setxy [pxcor] of start [pycor] of start set shape "bot" set done? false set current nobody set o-current self set child nobody set heuristic #heuristic expand-into self start ask child [ set closed? 0 set shape "node" set parent nobody set on-path? true ] ] end to expand-into [ parent-agent location-agent ] ;; ;; causes the given parent agent to ;; expand into the given location (patch) ;; ;; this means that nodes are created in the given patch, and the parent ;; of these nodes is the given parent agent. ;; ask parent-agent [ hatch-a*nodes 1 [ set location location-agent setxy [pxcor] of location [pycor] of location ;; first thing--is this a dead end? if so, don't ;; bother to add it to any list... just die. let my-owner owner ifelse location = [ target ] of owner or any? neighbors with [ shade-of? pcolor space-color and not any? a*nodes-here with [ owner = my-owner ] ] [ set breed a*nodes set shape "node" set size 1.01 set color green set parent parent-agent set owner [owner] of parent-agent set [child] of parent-agent self set child nobody face parent ;; target is inherited set g-score calculate-g parent-agent set h-score calculate-h set f-score calculate-f set closed? -1 ;; new... neither open or closed set on-path? false ] [ die ] ] ] end to-report calculate-f ;; ;; calculates the f score for this s*node ;; ifelse location = [ target ] of owner [ report -999999 ] [ report g-score + h-score ] end to-report calculate-g [ candidate ] ;; ;; calculates the g score relative to the candidate for this s*node ;; let g [g-score] of candidate + distance-nowrap candidate if terrain? [ set g g + pcolor * 10] report g end to-report calculate-h let result 0 if [ heuristic ] of owner = 0 ;; euclidian distance to target [ set result distance-nowrap [ target ] of owner ] if [ heuristic ] of owner = 1 ;; manhattan distance [ let xdiff abs(pxcor - [pxcor] of [ target ] of owner) let ydiff abs(pycor - [pycor] of [ target ] of owner) set result ( xdiff + ydiff ) ] if [ heuristic ] of owner = 2 ;; diagonal distance [ let D 1 let D2 1.414214 let xdiff abs(pxcor - [pxcor] of [ target ] of owner) let ydiff abs(pycor - [pycor] of [ target ] of owner) let h_diagonal min (list xdiff ydiff) let h_straight xdiff + ydiff set result D2 * h_diagonal + D * ( h_straight - 2 * h_diagonal ) ] if [ heuristic ] of owner = 3 ;; diagonal distance + tie-breaker [ let D 1 let D2 1.414214 let xdiff abs(pxcor - [ [pxcor] of target ] of owner) let ydiff abs(pycor - [ [pycor] of target ] of owner) let h_diagonal min (list xdiff ydiff) let h_straight xdiff + ydiff set result D2 * h_diagonal + D * ( h_straight - 2 * h_diagonal ) ;; tie-breaker: nudge H up by a small amount let h-scale (1 + (16 / directions / world-width * world-height)) set result result * h-scale ] if [ heuristic ] of owner = 4 ;; euclidian distance to target with tie-breaker [ set result distance-nowrap [ target ] of owner let h-scale (1 + (16 / directions / world-width + world-height)) set result result * h-scale ] report result end to-report a*build-path let o-c o-current set current min-one-of a*nodes with [ owner = myself and closed? = 0 ] [ f-score ] ; + distance-nowrap o-c ] set o-current current let cc current if is-turtle? cc [ ask cc [ set closed? 1 set color magenta if not same-patch location [ target ] of owner [ let me owner let paths nobody ifelse directions = 8 [ set paths neighbors with [ shade-of? pcolor space-color ] ] [ set paths neighbors4 with [ shade-of? pcolor space-color ] ] let new-paths (paths with [ not any? a*nodes-here with [ owner = me ] ] ) if any? new-paths [ ask new-paths [ expand-into cc self ] ] set new-paths (a*nodes-on paths ) with [ owner = me and closed? < 1 ] ; if any? new-paths [ set new-paths min-one-of new-paths [ f-score ] ] ask new-paths [ ifelse closed? = 0 ;; already open [ ;; see if g from current is better than prior g let new-g calculate-g cc set f-score calculate-f if new-g < g-score [ ;; if it is, then change path to go from this point ;; set parent of this node to current set parent cc set shape "node" face parent set [child] of parent self set g-score new-g set f-score calculate-f ] ] [ ;; must be new (not yet open, not previously closed. set closed? 0 ;; open it set parent cc ] ] ] ] ] report current end to-report extract-heuristic report first choose-heuristic end to reset-bots ask a*nodes [ die ] ask bots [ die ] let setup-start patch (min-pxcor + 2) ( min-pycor + 2 ) let setup-target patch (max-pxcor - 2) ( max-pycor - 2 ) create-bot setup-start setup-target extract-heuristic end to-report center-patch report patch (int (min-pxcor + world-width * .5)) (int (min-pycor + world-height * .5)) end ;; 1) Add the starting square (or node) to the open list. ;; 2) Repeat the following: ;; a) Look for the lowest F cost square on the open list. We refer to this as the current square. ;; b) Switch it to the closed list. ;; c) For each of the 8 squares adjacent to this current square … ;; ;; * If it is not walkable or if it is on the closed list, ignore it. ;; Otherwise do the following. ;; * If it is on the open list already... ;; Check to see if this path to that square is better, ;; using G cost as the measure. ;; A lower G cost means that this is a better path. ;; If so, change the parent of the square to the current square, ;; and recalculate the G and F scores of the square. ;; If you are keeping your open list sorted by F score, ;; you may need to resort the list to account for the change. ;; * If it isn’t on the open list... ;; Add it to the open list. ;; Make the current square the parent of this square. ;; Record the F, G, and H costs of the square. ;; ;; d) Stop when you: ;; ;; * Add the target square to the closed list, in which case the path has been found (see note below), or ;; * Fail to find the target square, and the open list is empty. In this case, there is no path. ;; ;; 3) Save the path. Working backwards from the target square, go from each square to its parent square until you reach the starting square. That is your path. @#$#@#$#@ GRAPHICS-WINDOW 255 10 691 467 35 35 6.0 1 20 1 1 1 0 1 1 1 -35 35 -35 35 1 1 1 ticks CC-WINDOW 5 534 700 629 Command Center 0 BUTTON 10 235 65 268 NIL setup NIL 1 T OBSERVER NIL NIL NIL NIL BUTTON 70 255 125 288 go go T 1 T OBSERVER NIL NIL NIL NIL MONITOR 255 475 340 520 NIL count a*nodes 3 1 11 SLIDER 120 30 215 63 density density 0 100 14 1 1 NIL HORIZONTAL BUTTON 70 295 125 328 go 1x go NIL 1 T OBSERVER NIL NIL NIL NIL CHOOSER 10 30 102 75 senario senario "blank" "bars" "blobs" "maze" "looped maze" "random" 4 BUTTON 130 235 195 268 reset-bots reset-bots ask patch 0 0 [ set plabel \"\" ] NIL 1 T OBSERVER NIL NIL NIL NIL SLIDER 10 190 105 223 directions directions 4 8 4 4 1 NIL HORIZONTAL SWITCH 10 340 197 373 show-path-in-progress? show-path-in-progress? 0 1 -1000 SWITCH 10 80 113 113 terrain? terrain? 0 1 -1000 MONITOR 345 475 430 520 nodes in path count a*nodes with [ color = yellow ] 3 1 11 MONITOR 435 475 520 520 path-length ifelse-value (any? bots) [[ [g-score] of current ] of one-of bots] [\"\"] 3 1 11 CHOOSER 10 140 205 185 choose-heuristic choose-heuristic [0 "euclician distance"] [1 "manhattan distance"] [2 "diagonal distance"] [3 "diag-dist + tie-breaker"] [4 "euclid-dist + tie-breaker"] 4 TEXTBOX 20 395 195 480 After selecting senario and bot parameters, click SETUP, then GO. To see how different heuristics and number of directions affect solution, select new heuristic or directions, click RESET, then glick GO. 11 0.0 1 TEXTBOX 10 125 160 143 Search bot parameters 11 0.0 1 TEXTBOX 10 10 160 28 Senario parameters 11 0.0 1 SLIDER 120 70 215 103 straightness straightness 0 100 89 1 1 NIL HORIZONTAL @#$#@#$#@ == NOTES == === "Shortest" Paths === This model does find the shortest paths, as far as the g-score is concerned. Since the bot can only move one patch per step, in either 4 or eight directions, its "shortest" paths are not the same as would be obtained if steps in any direction were allowed, or if the graph was not made of every patch, connected to its adjactent pathes. For example, if we could process the map so that only "important" patches were included in the graph (in the set of connected patches searched), then both search efficiency and path-length would improve. In this situation, since shortest paths tend to skirt around obstacles, "important" patches are patches that border an obstacle. Further, they are the patches at the "corners" of obstacles. That is, space patches with exactly one obstacle patch among their neighbors. Now, once the patches are identified, how are they connected? Clearly, every patch is connected to every other patch, so long as no obstacle lies on the line connecting the two patches. The line is clear if it does not pass through any obstacle patch. Note that this is not the same as not passing through the *center* of the obstacle patch! Last, the score of the connection is the distance bewteen the patches. == CREDIT AND ACKNOWLEDGEMENTS == Thanks to Amit J. Patel at http://theory.stanford.edu/~amitp/GameProgramming/Heuristics.html for great expanations and insight into the A* algorithm, and the role of the heuristic cost function. ;; A framework for using the famous path-finding algorithm Copyright © 2009 James P. Steiner @#$#@#$#@ default true 0 Polygon -7500403 true true 150 5 40 250 150 205 260 250 bot true 6 Circle -13840069 true true 30 30 240 indicator true 2 Polygon -955883 false true 150 75 210 15 285 90 225 150 285 210 210 285 150 225 90 285 15 210 75 150 15 90 90 15 Polygon -955883 false true 105 150 60 105 105 60 150 105 195 60 240 105 195 150 240 195 195 240 150 195 105 240 60 195 Polygon -955883 false true 135 150 105 120 120 105 150 135 180 105 195 120 165 150 195 180 180 195 150 165 120 195 105 180 node true 6 Circle -13840069 true true 75 75 150 Polygon -13840069 true true 75 150 150 -150 225 150 path true 6 Rectangle -13840069 true true 75 -75 225 225 Polygon -16777216 true false 75 195 105 195 150 135 195 195 225 195 150 90 Polygon -16777216 true false 75 60 105 60 150 0 195 60 225 60 150 -45 path-long true 6 Rectangle -16777216 true false 75 -195 225 225 Polygon -13840069 true true 75 0 75 -195 225 -195 225 -60 150 -165 75 -60 105 -60 150 -120 195 -60 225 -60 225 60 150 -45 75 60 105 60 150 0 195 60 225 60 225 195 150 90 75 195 105 195 150 135 195 195 225 195 225 225 75 225 @#$#@#$#@ NetLogo 4.0.4 @#$#@#$#@ @#$#@#$#@ @#$#@#$#@ @#$#@#$#@ @#$#@#$#@ 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 @#$#@#$#@