Smooth Convex Hull

Question

我已經開始研究凸包算法，並且想知道我可以使用什麼方法來平滑多邊形邊緣。船體輪廓不平滑。我想要做的是通過頂點的線條變得更平滑，以免它們成角度。

我曾嘗試推行貝濟耶（只實現了形狀是不一樣的船體形狀）和B樣條（同樣的形狀是不一樣的，其實我不能做的B輪廓封閉的形狀）。

我失敗了，希望有人能提供指導。

Answer 1

（注意！不是解決）

我試圖找到確切的解決方案，在極座標Lagrange polynomial，但找出來，那somtimes「平滑曲線」位於凸多邊形內部。通過在theta in [0:2 * pi]區間外添加額外的可移動不可見點，基本上可以解決一階導數匹配條件（在起點）。但上述問題在我的腦海裏無法解決。

這裏是的Lua腳本與我attemptions（使用qhull，rbox（從qhull工具鏈）和gnuplot的公用事業）：

function using() 
    return error('using: ' .. arg[0] .. ' <number of points>') 
end 

function points_from_file(infile) 
    local points = {} 
    local infile = io.open(infile, 'r') 
    local d = infile:read('*number') 
    if d ~= 2 then 
     error('dimensions is not two') 
    end 
    local n = infile:read('*number') 
    while true do 
     local x, y = infile:read('*number', '*number') 
     if not x and not y then 
      break 
     end 
     if not x or not y then 
      error('wrong format of input file: line does not contain two coordinates') 
     end 
     table.insert(points, {x, y}) 
    end 
    infile:close() 
    if n ~= #points then 
     error('second line not contain real count of points') 
    end 
    return points 
end 

if not arg then 
    error("script should use as standalone") 
end 
if #arg ~= 1 then 
    using() 
end 
local n = tonumber(arg[1]) 
if not n then 
    using() 
end 
local bounding_box = math.sqrt(math.pi)/2.0 
local fnp = os.tmpname() 
local fnchp = os.tmpname() 
os.execute('rbox ' .. n .. ' B' .. bounding_box .. ' D2 n t | tee ' .. fnp .. ' | qhull p | tee ' .. fnchp .. ' > nul') -- Windows specific part is "> nul" 
local sp = points_from_file(fnp) -- source points 
os.remove(fnp) 
local chp = points_from_file(fnchp) -- convex hull points 
os.remove(fnchp) 
local m = #chp 
if m < 3 then 
    io.stderr:write('convex hull consist of less than three points') 
    return 
end 
local pole = {0.0, 0.0} -- offset of polar origin relative to cartesian origin 
for _, point in ipairs(chp) do 
    pole[1] = pole[1] + point[1] 
    pole[2] = pole[2] + point[2] 
end 
pole[1] = pole[1]/m 
pole[2] = pole[2]/m 
print("pole = ", pole[1], pole[2]) 
local chcc = {} 
for _, point in ipairs(chp) do 
    table.insert(chcc, {point[1] - pole[1], point[2] - pole[2]}) 
end 
local theta_min = 2.0 * math.pi -- angle between abscissa ort of cartesian and ort of polar coordinates 
local rho_mean = 0.0 
local rho_max = 0.0 
local chpc = {} -- {theta, rho} pairs 
for _, point in ipairs(chcc) do 
    local rho = math.sqrt(point[1] * point[1] + point[2] * point[2]) 
    local theta = math.atan2(point[2], point[1]) 
    if theta < 0.0 then -- [-pi:pi] -> [0:2 * pi] 
     theta = theta + 2.0 * math.pi 
    end 
    table.insert(chpc, {theta, rho}) 
    if theta_min > theta then 
     theta_min = theta 
    end 
    rho_mean = rho_mean + rho 
    if rho_max < rho then 
     rho_max = rho 
    end 
end 
theta_min = -theta_min 
rho_mean = rho_mean/m 
rho_max = rho_max/rho_mean 
for pos, point in ipairs(chpc) do 
    local theta = (point[1] + theta_min)/math.pi -- [0:2 * pi] -> [0:2] 
    local rho = point[2]/rho_mean 
    table.remove(chpc, pos) 
    table.insert(chpc, pos, {theta, rho}) 
end 
table.sort(chpc, function (lhs, rhs) return lhs[1] < rhs[1] end) 
-- table.insert(chpc, {chpc[#chpc][1] - 2.0 * math.pi, chpc[#chpc][2]}) 
table.insert(chpc, {2.0, chpc[1][2]}) 
-- table.sort(chpc, function (lhs, rhs) return lhs[1] < rhs[1] end) 

local solution = {} 
solution.x = {} 
solution.y = {} 
for _, point in ipairs(chpc) do 
    table.insert(solution.x, point[1]) 
    table.insert(solution.y, point[2]) 
end 
solution.c = {} 
for i, xi in ipairs(solution.x) do 
    local c = solution.y[i] 
    for j, xj in ipairs(solution.x) do 
     if i ~= j then 
      c = c/(xi - xj) 
     end 
    end 
    solution.c[i] = c 
end 
function solution:monomial(i, x) 
    local y = self.c[i] 
    for j, xj in ipairs(solution.x) do 
     if xj == x then 
      if i == j then 
       return self.y[i] 
      else 
       return 0.0 
      end 
     end 
     if i ~= j then 
      y = y * (x - xj) 
     end 
    end 
    return y 
end 
function solution:polynomial(x) 
    local y = self:monomial(1, x) 
    for i = 2, #solution.y do 
     y = y + self:monomial(i, x) 
    end 
    return y 
end 

local gnuplot = io.popen('gnuplot', 'w') 

gnuplot:write('reset;\n') 
gnuplot:write('set terminal wxt 1;\n') 
gnuplot:write(string.format('set xrange [%f:%f];\n', -bounding_box, bounding_box)) 
gnuplot:write(string.format('set yrange [%f:%f];\n', -bounding_box, bounding_box)) 
gnuplot:write('set size square;\n') 
gnuplot:write(string.format('set xtics %f;\n', 0.1)) 
gnuplot:write(string.format('set ytics %f;\n', 0.1)) 
gnuplot:write('set grid xtics ytics;\n') 
gnuplot:write('plot "-" using 1:2 notitle with points, "-" using 1:2:3:4 notitle with vectors;\n') 
for _, point in ipairs(sp) do 
    gnuplot:write(string.format('%f %f\n', point[1], point[2])) 
end 
gnuplot:write('e\n') 
for _, point in ipairs(chcc) do 
    gnuplot:write(string.format('%f %f %f %f\n', pole[1], pole[2], point[1], point[2])) 
end 
gnuplot:write('e\n') 
gnuplot:flush(); 

gnuplot:write('reset;\n') 
gnuplot:write('set terminal wxt 2;\n') 
gnuplot:write('set border 0;\n') 
gnuplot:write('unset xtics;\n') 
gnuplot:write('unset ytics;\n') 
gnuplot:write('set polar;\n') 
gnuplot:write('set grid polar;\n') 
gnuplot:write('set trange [-pi:2 * pi];\n') 
gnuplot:write(string.format('set rrange [-0:%f];\n', rho_max)) 
gnuplot:write('set size square;\n') 
gnuplot:write('set view equal xy;\n') 
-- gnuplot:write(string.format('set xlabel "%f";\n', rho_mean - 1.0)) 
gnuplot:write(string.format('set arrow 1 from 0,0 to %f,%f;\n', rho_max * math.cos(theta_min), rho_max * math.sin(theta_min))) 
gnuplot:write(string.format('set label 1 " origin" at %f,%f left rotate by %f;\n', rho_max * math.cos(theta_min), rho_max * math.sin(theta_min), math.deg(theta_min))) 
gnuplot:write('plot "-" using 1:2:3:4 notitle with vectors, "-" using 1:2 notitle with lines, "-" using 1:2 notitle with lines;\n') 
for _, point in ipairs(chpc) do 
    gnuplot:write(string.format('0 0 %f %f\n', point[1] * math.pi, point[2])) 
end 
gnuplot:write('e\n') 
for _, point in ipairs(chpc) do 
    gnuplot:write(string.format('%f %f\n', point[1] * math.pi, point[2])) 
end 
gnuplot:write('e\n') 
do 
    local points_count = 512 
    local dx = 2.0/points_count 
    local x = 0.0 
    for i = 1, points_count do 
     gnuplot:write(string.format('%f %f\n', x * math.pi, solution:polynomial(x))) 
     x = x + dx 
    end 
    gnuplot:write('e\n') 
end 
gnuplot:flush(); 

gnuplot:write('reset;\n') 
gnuplot:write('set terminal wxt 3;\n') 
gnuplot:write(string.format('set xrange [-1:2];\n')) 
gnuplot:write(string.format('set yrange [0:2];\n')) 
gnuplot:write(string.format('set size ratio %f;\n', rho_max/3.0)) 
gnuplot:write(string.format('set xtics %f;\n', 0.5)) 
gnuplot:write(string.format('set ytics %f;\n', 0.5)) 
gnuplot:write('set grid xtics ytics;\n') 
gnuplot:write(string.format('set arrow 1 nohead from 0,%f to 2,%f linetype 3;\n', chpc[1][2], chpc[1][2])) 
gnuplot:write(string.format('set label 1 "glue points " at 0,%f right;\n', chpc[1][2])) 
gnuplot:write('plot "-" using 1:2 notitle with lines, "-" using 1:2 notitle with lines;\n') 
for _, point in ipairs(chpc) do 
    gnuplot:write(string.format('%f %f\n', point[1], point[2])) 
end 
gnuplot:write('e\n') 
do 
    local points_count = 512 
    local dx = 2.0/points_count 
    local x = 0.0 
    for i = 1, points_count do 
     gnuplot:write(string.format('%f %f\n', x, solution:polynomial(x))) 
     x = x + dx 
    end 
    gnuplot:write('e\n') 
end 
gnuplot:flush(); 

os.execute('pause'); 
gnuplot:write('exit\n'); 
gnuplot:flush(); 
gnuplot:close() 

第二端子包含拉格朗日多項式近似。