随机分布、随机旋转且不重叠的文本

Question

我正在尝试生成一个包含随机分布和随机旋转的文本的页面，并且代码这里做得很好。

但即使在空间充足的情况下（我可以看到很多空白），它也会导致文本重叠（我使用的是天城文字母 --- 有超过 48 个）。

避免重叠的一种方法是继续生成页面，希望随机性对我们有利并生成没有重叠的页面。

但这要求的随机性太高了。我尝试了近百次才放弃。:)

如何获得随机分布、随机旋转且不重叠？

Answer 1

这个答案几乎完全取自这个很好的答案. 它需要 lualatex。

\documentclass[tikz,border=3.14mm]{standalone}
\usepackage{filecontents}
% poisson.sty and poisson.lua are from https://tex.stackexchange.com/a/185423/121799
\begin{filecontents*}{poisson.sty}
\directlua{dofile("poisson.lua")}
\newcommand{\poissonpointslist}[4]{
    \directlua{poisson_points_list(#1,#2,#3,#4)}
}
\newcommand{\poissonpointslistordered}[4]{
    \directlua{poisson_points_list_ordered(#1,#2,#3,#4)}
}
\end{filecontents*}
\begin{filecontents*}{poisson.lua}
-- 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
local 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
local Point = {}

function Point.new(x,y)
    return {x=x, y=y}
end

-- Determines if a point is inside the rendered rectangle
local 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
local function gridToString(gridPoint)
    return string.format("(%d,%d)", gridPoint.x, gridPoint.y)
end

-- Computes the distance between two points
local 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
local 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
local function imageToGrid(point, cellSize)
    local gridX = math.floor(point.x/cellSize)
    local gridY = math.floor(point.y/cellSize)
    return Point.new(gridX, gridY)
end

local 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

-- 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
local 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

local 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 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


-- 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 ipairs(data) do
        table.insert(str, string.format("%f/%f", v.x, v.y))
    end
    tex.print(table.concat(str, ", "))
end

-- Function similar to the above, but the returned list is "ordered"
-- so that the points are more or less in the left-to-right, 
-- top-to-down order
function poisson_points_list_ordered(width, height, mindist, add_points)
    local cellSize = mindist/math.sqrt(2)
    local function compare_coords(a,b)
        aa = imageToGrid(a, cellSize);
        bb = imageToGrid(b, cellSize);
        if (aa.y == bb.y) then  -- If they are around the same row
            return a.x<b.x;     -- the x coord orders them
        else                    -- if not
            return a.y>b.y;     -- the y coord orders them
        end
    end

    local data = generate_poisson(width, height, mindist, add_points)
    table.sort(data, compare_coords);
    local str = {}
    for k,v in ipairs(data) do
        table.insert(str, string.format("%f/%f", v.x, v.y))
    end
    tex.print(table.concat(str, ", "))
end
\end{filecontents*}

\usepackage{tikz}
\usetikzlibrary{backgrounds}
\usepackage{poisson}

\begin{document}
\xdef\LstTxt{"duck","koala","marmot","honey","tiger","lion","bear","pineapple",
"pizza","sun","Mercure","Venus","Earth","Mars","Jupiter","Saturn","Uranus"}
\xdef\MaxWidth{0}
\foreach \X [count=\Y] in \LstTxt
{\pgfmathsetmacro{\mywidth}{max(\MaxWidth,width(\X))}
\xdef\MaxWidth{\mywidth}
\xdef\NumEntries{\Y}}
\pgfmathsetmacro{\GraceDist}{2*\MaxWidth*1pt/1cm}
\edef\mylist{\poissonpointslist{14}{14}{\GraceDist}{\NumEntries}}  
% 14 x 14 is the size of the area to fill
% \GraceDist is the minimum distance between centers
\typeout{\mylist}
\begin{tikzpicture}[framed,gridded,radius=0.5cm]
    \draw[red](-1,-1) rectangle (15,15);
    \foreach \x/\y [count=\n starting from 0] in \mylist 
    {\pgfmathsetmacro{\RndOne}{255*rnd}
    \pgfmathsetmacro{\RndTwo}{255*rnd}
    \pgfmathsetmacro{\RndThree}{255*rnd}
    \pgfmathsetmacro{\Scale}{0.5+1.5*rnd}
    \pgfmathsetmacro{\Rot}{360*rnd}
    \pgfmathsetmacro{\String}{{\LstTxt}[\n]}
    \definecolor{mycolor}{RGB}{\RndOne,\RndTwo,\RndThree}
    \draw[color=mycolor] (\x,\y) node[scale=\Scale,rotate=\Rot] {\String};}
\end{tikzpicture}
\end{document}