graphics - 光滑凸包

Question

我已经开始使用凸包算法，并且想知道我可以采用什么方法来平滑多边形边缘。船体轮廓不光滑。我想做的是使通过顶点的线条更平滑，这样它们就不会倾斜。

我试图实现贝塞尔曲线（只是意识到形状与船体的形状完全不同）和 b 样条曲线（再次形状完全不同，实际上我无法使 b 样条曲线成为闭合形状）。

我失败了，希望有人可以提供指导。

Answer 1

（注意！这不是解决方案）

我试图在极坐标中找到拉格朗日多项式的精确解，但发现有时“平滑曲线”位于凸多边形内。theta in [0:2 * pi]通过在区间外添加额外的可移动不可见点，可以从根本上解决一阶导数匹配条件（在起点） 。但在我看来，上述问题无论如何都无法解决。

这是我尝试的Lua脚本（使用qhull、rbox（来自qhull工具链）和gnuplot实用程序）：

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()

第二个终端包含拉格朗日多项式逼近。

Answer 2

我会这样处理它，使用你的例子：

1. 从最长的外部段开始（在您的示例中，这是左下角） - 我们保持笔直；
2. 想象在长线的底端有一个圆圈，朝内；
3. 该圆的切线可以延伸到下一点；
4. 在下一种情况下（右下圆），没有切线连接到下一个点，因此使用另一个圆并在切线处连接圆；
5. 以这种方式继续。 所以，你先画一个圆弧，然后画一条直线，然后重复。

你的圆圈大小决定了整体的平滑度。但是当然，如​​果它们太大，您将需要降低一些分数。