凸包アルゴリズムの作業を開始しましたが、ポリゴンのエッジを滑らかにするためにどのような方法を採用できるか疑問に思っていました。船体の輪郭は滑らかではありません。私がやりたいのは、頂点を通る線をより滑らかにして、角度がつかないようにすることです。
ベジエ (形状が船体の形状とはまったく異なることに気付くためだけに) と b-スプライン (やはり形状は似ていませんでした。実際、b-スプラインを閉じた形状にすることはできませんでした) を実装しようとしました。
私は失敗していて、誰かがガイダンスを提供できることを願っています。
（注！それは解決策ではありません）
極座標のラグランジュ多項式として正確な解を見つけようとしましたが、「平滑化曲線」が凸多角形の内側にあることがあることがわかりました。theta in [0:2 * pi]一次導関数の一致条件 (開始点) は、間隔の外側に移動可能な非表示の点を追加することで基本的に解決できます。しかし、上記の問題はとにかく私の頭では解決できません。
これが私の試みによるLuaスクリプトです（ qhull、rbox（qhullツールチェーンから）およびgnuplotユーティリティを使用）：
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()
2 番目の端子には、ラグランジュ多項式近似が含まれます。