Zufällige Anordnung von Kreisen in Tikz
Hallo,
ich suche ein Möglichkeit, wie ich eine vorgegebenen Anzahl von Kreisen in einem Viereck zufällig verteilen kann. Dabei soll es möglich sein einen Mindestabstand einzustellen, damit es nicht zu Überlappungen kommt. Ich habe das Paket poisson gefunden, das ziemlich genau das macht, was ich möchte. Leider verstehe ich noch nicht, wie ich da die Anzahl an Kreisen vorgeben kann.
Ich wäre über Verbesserungen oder auch Tipps zu anderen Ansätzen Dankbar.
Minimalbeispiel
Code:
\documentclass[12pt,listof=totoc,bibliography=totoc,pointlessnumbers]{scrartcl}
\usepackage[ngerman]{babel}
\usepackage[obeyspaces]{url}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{poisson}
\usepackage{tikz}
\usetikzlibrary{math}
\tikzset{plaettchenvoll/.style={line width = 0.2 mm,draw,fill=gray!80,circle,minimum width = 3.25mm},
}%\\
\begin{document}
\begin{tikzpicture}
\xdef\mylist{\poissonpointslist{5}{5}{0.7}{3}} % Sparse, fast
\draw (-0.5,-0.5)--(5.5,-0.5)--(5.5,5.5)--(-0.5,5.5)--cycle;
\foreach \x/\y[count=\i from 0]in \mylist {
\node[plaettchenvoll,fill=blue!\i] at(\x,\y) {};
}
\end{tikzpicture}
\end{document}
poisson.sty
Code:
\directlua{dofile("poisson.lua")}
\newcommand{\poissonpointslist}[4]{
\directlua{poisson_points_list(#1,#2,#3,#4)}
}
poisson.sty
Code:
-- This is a lua implementation of the algorithm described in
-- http://devmag.org.za/2009/05/03/poisson-disk-sampling/
--
-- The structure of the algorithm is exactly the same than in
-- the mentioned article. Its pseudo-code snippets were translated
-- to Lua.
--
-- One detail worths an explanation, though. The article uses a 2D matrix
-- called grid to store coordinates of points. In the article, it is
-- assumed that grid elements can be accesed via grid[point], being point
-- some structure with a pair of x and y integers, so grid[point] should
-- be equivalent to grid[x,y] or grid[x][y]. This grid is assumed to be
-- initially dimensioned and filled by nils.
--
-- In my implementation the grid is dynamic, and it is an associative array
-- indexed by string keys in the form grid["(x,y)"]. The function gridToString()
-- can be used to convert a Point to its string form, so the grid is indeed
-- accesed like this: grid[gridToString(p)] being p a Point with integer
-- coordinates (which in fact is found via imageToGrid, like in the article)
-- UTILITY FUNCTIONS (used in the article, without giving implementation)
-- =====================================================================
-- RandomQueue stores values and gives them in random order
RandomQueue = {}
function RandomQueue.new ()
return {last=-1}
end
function RandomQueue.push(q, item)
local last = q.last + 1
q.last = last
q[last] = item
end
function RandomQueue.pop(q)
if (RandomQueue.empty(q)) then
return nil
end
local index = math.random(0,q.last)
-- A random index is generated. The last element
-- is swaped with this random item, and the new
-- last item is popped.
local last = q.last
item = q[index]
q[index] = q[last]
q[last] = nil
q.last = last -1
return item
end
function RandomQueue.empty(q)
return q.last==-1
end
function RandomQueue.print(q)
-- For debugging. Not used
local t = {}
for i=0, q.last do
table.insert(t, string.format("(%f,%f)", q[i].x, q[i].y))
end
print (string.format("RandomQueue %d elem: %s", q.last+1, table.concat(t, " ")))
end
-- Point stores a coordinate pair
Point = {}
function Point.new(x,y)
return {x=x, y=y}
end
-- Determines if a point is inside the rendered rectangle
function inRectangle(point, width, height)
return (point.x>0 and point.y>0 and point.x<width and point.y<height)
end
-- Converts a point to a string representation, to be used as index in the grid
function gridToString(gridPoint)
return string.format("(%d,%d)", gridPoint.x, gridPoint.y)
end
-- Computes the distance between two points
function distance(p1, p2)
return math.sqrt(math.pow(p2.x-p1.x,2) + math.pow(p2.y-p1.y,2))
end
-- Prints the grid. For debugging. Not used
function printGrid(grid)
print "==========="
for k,v in pairs(grid) do
print (string.format("%s: %f, %f", k, v.x, v.y))
end
end
-- THE FUNCTIONS GIVEN IN THE ARTICLE
-- This is the lua implementation of the pseudocode in the article
function generate_poisson(width, height, min_dist, new_points_count)
local cellSize = min_dist/math.sqrt(2)
local grid = {} -- Point.new(math.ceil(width/cellSize), math.ceil(height/cellSize))}
local processList = RandomQueue.new()
local samplePoints = {}; -- Empty list
-- Generate the first point
local firstPoint = Point.new(math.random()*width, math.random()*height)
-- print (string.format("newPoint: [%f, %f]", firstPoint.x, firstPoint.y))
RandomQueue.push(processList, firstPoint)
table.insert(samplePoints, firstPoint)
grid[gridToString(imageToGrid(firstPoint, cellSize))] = firstPoint
-- Generate other points from points in queue
while (not RandomQueue.empty(processList)) do
-- RandomQueue.print(processList)
-- printGrid(grid)
local point = RandomQueue.pop(processList)
for i=0,new_points_count do
local newPoint = generateRandomPointAround(point, min_dist)
-- print (string.format("newPoint: [%f, %f]", newPoint.x, newPoint.y))
-- Check the point is in the region and not too close
-- to other points
if inRectangle(newPoint, width, height) and
not inNeighbourhood(grid, newPoint, min_dist, cellSize) then
-- In this case, the point is accepted
RandomQueue.push(processList, newPoint)
table.insert(samplePoints, newPoint)
grid[gridToString(imageToGrid(newPoint, cellSize))] = newPoint;
end
end
end
return samplePoints
end
function imageToGrid(point, cellSize)
local gridX = math.floor(point.x/cellSize)
local gridY = math.floor(point.y/cellSize)
return Point.new(gridX, gridY)
end
function generateRandomPointAround(point, mindist)
local r1 = math.random()
local r2 = math.random()
local radius = mindist * (r1+1)
local angle = 2 * math.pi * r2
newX = point.x + radius * math.cos(angle)
newY = point.y + radius * math.sin(angle)
return Point.new(newX, newY)
end
function inNeighbourhood(grid, point, mindist, cellSize)
local gridPoint = imageToGrid(point, cellSize)
cellsAroundPoint = squareAroundPoint(grid, gridPoint, 5)
for k,cell in pairs(cellsAroundPoint) do
if not (cell==nil) then
local d = distance(cell, point)
if distance(cell, point) < mindist then
return true
end
end
end
return false
end
-- This one is not given in the article. It returns the
-- values of several cells around the give gridPoint
-- We are using string indexes for the grid, but if we
-- try to access to a key which is not stored, lua gives
-- nil instead of an exception, so it works as expected
-- because we get nils for cells which have no dot inside
function squareAroundPoint(grid, gridPoint, n)
local extreme = math.floor(n/2)
local result = {}
for i=-extreme,extreme do
for j=-extreme,extreme do
ii = i + gridPoint.x
jj = j + gridPoint.y
data = grid[gridToString(Point.new(ii,jj))]
if data == nil then
repr = "nil"
else
repr = string.format("(%f,%f)", data.x, data.y)
end
table.insert(result, data)
end
end
return result
end
-- Initialize random seed
math.randomseed(os.time())
-- Function to generate the list of dots in a tikz's foreach compatible syntax
function poisson_points_list(width, height, mindist, add_points)
local data = generate_poisson(width, height, mindist, add_points)
local str = {}
for k,v in pairs(data) do
table.insert(str, string.format("%f/%f", v.x, v.y))
end
tex.print(table.concat(str, ", "))
end
