summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/tex/lualatex
diff options
context:
space:
mode:
authorKarl Berry <karl@freefriends.org>2018-05-02 17:55:49 +0000
committerKarl Berry <karl@freefriends.org>2018-05-02 17:55:49 +0000
commit986752733eb91fe2ebdb22f9e2e058794042f51e (patch)
tree73b8453bccc35861fc31e00cf37f75740b42652a /Master/texmf-dist/tex/lualatex
parente8aaa9eaad9ba05d4467157da6d1162ed0c61d20 (diff)
bezierplot (13apr18)
git-svn-id: svn://tug.org/texlive/trunk@47566 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/tex/lualatex')
-rwxr-xr-xMaster/texmf-dist/tex/lualatex/bezierplot/bezierplot.lua698
-rw-r--r--Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.sty17
2 files changed, 715 insertions, 0 deletions
diff --git a/Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.lua b/Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.lua
new file mode 100755
index 00000000000..da702f987fc
--- /dev/null
+++ b/Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.lua
@@ -0,0 +1,698 @@
+#!/usr/bin/env lua
+-- Linus Romer, published 2018 under LPPL Version 1.3c
+-- version 1.0 2018-04-12
+abs = math.abs
+acos = math.acos
+asin = math.asin
+atan = math.atan
+cos = math.cos
+exp = math.exp
+e = math.exp(1)
+log = math.log
+pi = math.pi
+sin = math.sin
+sqrt = math.sqrt
+tan = math.tan
+
+-- cube root defined for all real numbers x
+function cbrt(x)
+ if x < 0 then
+ return -(-x)^(1/3)
+ else
+ return x^(1/3)
+ end
+end
+
+function sgn(x)
+ if x<0 then
+ return -1
+ elseif x>0 then
+ return 1
+ else
+ return 0
+ end
+end
+
+function round(num, decimals)
+ local result = tonumber(string.format("%." .. (decimals or 0) .. "f", num))
+ if abs(result) == 0 then
+ return 0
+ else
+ return result
+ end
+end
+
+-- 5-stencil method
+-- return from a graph from f in the form {{x,y},...}
+-- the derivatives in form {{x,y,dy/dx,ddy/ddx},...}
+function diffgraph(func,graph,h)
+ local dgraph = {}
+ local yh = func(graph[1][1]-h)
+ local yhh = func(graph[1][1]-2*h)
+ if yhh > -math.huge and yhh < math.huge -- if defined at all
+ and yh > -math.huge and yh < math.huge then
+ dgraph[1] = {graph[1][1],graph[1][2],
+ (yhh-8*yh+8*graph[2][2]-graph[3][2])/(12*h),
+ (-yhh+16*yh-30*graph[1][2]+16*graph[2][2]-graph[3][2])
+ /(12*h^2)}
+ dgraph[2] = {graph[2][1],graph[2][2],
+ (yh-8*graph[1][2]+8*graph[3][2]-graph[4][2])/(12*h),
+ (-yh+16*graph[1][2]-30*graph[2][2]+16*graph[3][2]-graph[4][2])
+ /(12*h^2)}
+ else -- take neighbour values
+ dgraph[1] = {graph[1][1],graph[1][2],
+ (graph[1][2]-8*graph[2][2]+8*graph[4][2]-graph[5][2])/(12*h),
+ (-graph[1][2]+16*graph[2][2]-30*graph[3][2]
+ +16*graph[4][2]-graph[5][2])/(12*h^2)}
+ dgraph[2] = {graph[2][1],graph[2][2],
+ (graph[1][2]-8*graph[2][2]+8*graph[4][2]-graph[5][2])/(12*h),
+ (-graph[1][2]+16*graph[2][2]-30*graph[3][2]
+ +16*graph[4][2]-graph[5][2])/(12*h^2)}
+ end
+ local l = #graph
+ for i = 3, l-2 do
+ table.insert(dgraph,{graph[i][1],graph[i][2],
+ (graph[i-2][2]-8*graph[i-1][2]+8*graph[i+1][2]-graph[i+2][2])
+ /(12*h),
+ (-graph[i-2][2]+16*graph[i-1][2]-30*graph[i][2]
+ +16*graph[i+1][2]-graph[i+2][2])
+ /(12*h^2)})
+ end
+ yh = func(graph[l][1]+h)
+ yhh = func(graph[l][1]+2*h)
+ if yhh > -math.huge and yhh < math.huge -- if defined at all
+ and yh > -math.huge and yh < math.huge then
+ dgraph[l-1] = {graph[l-1][1],graph[l-1][2],
+ (graph[l-3][2]-8*graph[l-2][2]+8*graph[l][2]-yh)/(12*h),
+ (-graph[l-3][2]+16*graph[l-2][2]-30*graph[l-1][2]
+ +16*graph[l][2]-yh)/(12*h^2)}
+ dgraph[l] = {graph[l][1],graph[l][2],
+ (graph[l-2][2]-8*graph[l-1][2]+8*yh-yhh)/(12*h),
+ (-graph[l-2][2]+16*graph[l-1][2]-30*graph[l][2]
+ +16*yh-yhh)/(12*h^2)}
+ else
+ -- take neighbour values
+ dgraph[l] = {graph[l][1],graph[l][2],
+ (graph[l-4][2]-8*graph[l-3][2]+8*graph[l-1][2]-graph[l][2])
+ /(12*h),
+ (-graph[l-4][2]+16*graph[l-3][2]-30*graph[l-2][2]
+ +16*graph[l-1][2]-graph[l][2])/(12*h^2)}
+ dgraph[l-1] = {graph[l-1][1],graph[l-2][2],
+ (graph[l-4][2]-8*graph[l-3][2]+8*graph[l-1][2]-graph[l][2])
+ /(12*h),
+ (-graph[l-4][2]+16*graph[l-3][2]-30*graph[l-2][2]
+ +16*graph[l-1][2]-graph[l][2])/(12*h^2)}
+ end
+ -- add information about being extremum / inflection point (true/false)
+ for i = 1, l do
+ dgraph[i][5] = false -- dy/dx == 0 ? default, may change later
+ dgraph[i][6] = false -- ddy/ddx == 0 ? default, may change later
+ end
+ for i = 1, l-1 do
+ -- if no gap is inbetween
+ if not (dgraph[i+1][1] - dgraph[i][1] > 1.5*h) then
+ -- check for dy/dx == 0
+ -- if not already determined as near dy/dx=0
+ if not dgraph[i][5] then
+ if dgraph[i][3] == 0 then
+ dgraph[i][5] = true
+ elseif abs(dgraph[i][3]*dgraph[i+1][3])
+ ~= dgraph[i][3]*dgraph[i+1][3] then -- this must be near
+ if abs(dgraph[i][4]) <= abs(dgraph[i+1][4]) then
+ dgraph[i][5] = true
+ else
+ dgraph[i+1][5] = true
+ end
+ end
+ end
+ -- check for ddy/ddx == 0
+ -- if not already determined as near ddy/ddx=0
+ if not dgraph[i][6] then
+ if abs(dgraph[i][4]*dgraph[i+1][4])
+ ~= dgraph[i][4]*dgraph[i+1][4] then -- this must be near
+ if abs(dgraph[i][4]) <= abs(dgraph[i+1][4]) then
+ dgraph[i][6] = true
+ else
+ dgraph[i+1][6] = true
+ end
+ end
+ end
+ end
+ end
+ return dgraph
+end
+
+-- checks for 100 x, if the function given by funcstring
+-- fits the graph g (up to maxerror) after filling in
+-- the parameters a, b, c, d
+-- if the graph is inverted, then isinverse has to be set true
+function do_parameters_fit(a,b,c,d,funcstring,funcgraph,maxerror,isinverse)
+ local funcx = string.gsub(funcstring, "a", a)
+ local funcx = string.gsub(funcx, "b", b)
+ local funcx = string.gsub(funcx, "c", c)
+ local funcx = string.gsub(funcx, "d", d)
+ local func = assert(load("local x = ...; return "..funcx))
+ for i = 1, #funcgraph, math.max(1,math.floor(0.01*#funcgraph)) do
+ if isinverse then
+ if abs(func(funcgraph[i][2])-funcgraph[i][1])
+ > maxerror then
+ return false
+ end
+ else
+ if abs(func(funcgraph[i][1])-funcgraph[i][2])
+ > maxerror then
+ return false
+ end
+ end
+ end
+ return true
+end
+
+-- f(x)=a*x^3+b*x+c
+function parameters_cubic(xp,yp,xq,yq,xr,yr,xs,ys)
+ local a = (((xp^2 * xq) * yr) - ((xp^2 * xq) * ys)
+ - ((xp^2 * xr) * yq) + ((xp^2 * xr) * ys) + ((xp^2 * xs) * yq)
+ - ((xp^2 * xs) * yr) - ((xp * xq^2) * yr) + ((xp * xq^2) * ys)
+ + ((xp * xr^2) * yq) - ((xp * xr^2) * ys) - ((xp * xs^2) * yq)
+ + ((xp * xs^2) * yr) + ((xq^2 * xr) * yp) - ((xq^2 * xr) * ys)
+ - ((xq^2 * xs) * yp) + ((xq^2 * xs) * yr) - ((xq * xr^2) * yp)
+ + ((xq * xr^2) * ys) + ((xq * xs^2) * yp) - ((xq * xs^2) * yr)
+ + ((xr^2 * xs) * yp) - ((xr^2 * xs) * yq) - ((xr * xs^2) * yp)
+ + ((xr * xs^2) * yq)) /
+ (((xp^3 * xq^2) * xr) - ((xp^3 * xq^2) * xs)
+ - ((xp^3 * xq) * xr^2) + ((xp^3 * xq) * xs^2)
+ + ((xp^3 * xr^2) * xs) - ((xp^3 * xr) * xs^2)
+ - ((xp^2 * xq^3) * xr) + ((xp^2 * xq^3) * xs)
+ + ((xp^2 * xq) * xr^3) - ((xp^2 * xq) * xs^3)
+ - ((xp^2 * xr^3) * xs) + ((xp^2 * xr) * xs^3)
+ + ((xp * xq^3) * xr^2) - ((xp * xq^3) * xs^2)
+ - ((xp * xq^2) * xr^3) + ((xp * xq^2) * xs^3)
+ + ((xp * xr^3) * xs^2) - ((xp * xr^2) * xs^3)
+ - ((xq^3 * xr^2) * xs) + ((xq^3 * xr) * xs^2)
+ + ((xq^2 * xr^3) * xs) - ((xq^2 * xr) * xs^3)
+ - ((xq * xr^3) * xs^2) + ((xq * xr^2) * xs^3))
+ local b = ((((-xp^3) * xq) * yr) + ((xp^3 * xq) * ys)
+ + ((xp^3 * xr) * yq) - ((xp^3 * xr) * ys) - ((xp^3 * xs) * yq)
+ + ((xp^3 * xs) * yr) + ((xp * xq^3) * yr) - ((xp * xq^3) * ys)
+ - ((xp * xr^3) * yq) + ((xp * xr^3) * ys) + ((xp * xs^3) * yq)
+ - ((xp * xs^3) * yr) - ((xq^3 * xr) * yp) + ((xq^3 * xr) * ys)
+ + ((xq^3 * xs) * yp) - ((xq^3 * xs) * yr) + ((xq * xr^3) * yp)
+ - ((xq * xr^3) * ys) - ((xq * xs^3) * yp) + ((xq * xs^3) * yr)
+ - ((xr^3 * xs) * yp) + ((xr^3 * xs) * yq) + ((xr * xs^3) * yp)
+ - ((xr * xs^3) * yq)) /
+ (((xp^3 * xq^2) * xr) - ((xp^3 * xq^2) * xs)
+ - ((xp^3 * xq) * xr^2) + ((xp^3 * xq) * xs^2)
+ + ((xp^3 * xr^2) * xs) - ((xp^3 * xr) * xs^2)
+ - ((xp^2 * xq^3) * xr) + ((xp^2 * xq^3) * xs)
+ + ((xp^2 * xq) * xr^3) - ((xp^2 * xq) * xs^3)
+ - ((xp^2 * xr^3) * xs) + ((xp^2 * xr) * xs^3)
+ + ((xp * xq^3) * xr^2) - ((xp * xq^3) * xs^2)
+ - ((xp * xq^2) * xr^3) + ((xp * xq^2) * xs^3)
+ + ((xp * xr^3) * xs^2) - ((xp * xr^2) * xs^3)
+ - ((xq^3 * xr^2) * xs) + ((xq^3 * xr) * xs^2)
+ + ((xq^2 * xr^3) * xs) - ((xq^2 * xr) * xs^3)
+ - ((xq * xr^3) * xs^2) + ((xq * xr^2) * xs^3))
+ local c = (((xp^3 * xq^2) * yr) - ((xp^3 * xq^2) * ys)
+ - ((xp^3 * xr^2) * yq) + ((xp^3 * xr^2) * ys)
+ + ((xp^3 * xs^2) * yq) - ((xp^3 * xs^2) * yr)
+ - ((xp^2 * xq^3) * yr) + ((xp^2 * xq^3) * ys)
+ + ((xp^2 * xr^3) * yq) - ((xp^2 * xr^3) * ys)
+ - ((xp^2 * xs^3) * yq) + ((xp^2 * xs^3) * yr)
+ + ((xq^3 * xr^2) * yp) - ((xq^3 * xr^2) * ys)
+ - ((xq^3 * xs^2) * yp) + ((xq^3 * xs^2) * yr)
+ - ((xq^2 * xr^3) * yp) + ((xq^2 * xr^3) * ys)
+ + ((xq^2 * xs^3) * yp) - ((xq^2 * xs^3) * yr)
+ + ((xr^3 * xs^2) * yp) - ((xr^3 * xs^2) * yq)
+ - ((xr^2 * xs^3) * yp) + ((xr^2 * xs^3) * yq)) /
+ (((xp^3 * xq^2) * xr) - ((xp^3 * xq^2) * xs)
+ - ((xp^3 * xq) * xr^2) + ((xp^3 * xq) * xs^2)
+ + ((xp^3 * xr^2) * xs) - ((xp^3 * xr) * xs^2)
+ - ((xp^2 * xq^3) * xr) + ((xp^2 * xq^3) * xs)
+ + ((xp^2 * xq) * xr^3) - ((xp^2 * xq) * xs^3)
+ - ((xp^2 * xr^3) * xs) + ((xp^2 * xr) * xs^3)
+ + ((xp * xq^3) * xr^2) - ((xp * xq^3) * xs^2)
+ - ((xp * xq^2) * xr^3) + ((xp * xq^2) * xs^3)
+ + ((xp * xr^3) * xs^2) - ((xp * xr^2) * xs^3)
+ - ((xq^3 * xr^2) * xs) + ((xq^3 * xr) * xs^2)
+ + ((xq^2 * xr^3) * xs) - ((xq^2 * xr) * xs^3)
+ - ((xq * xr^3) * xs^2) + ((xq * xr^2) * xs^3))
+ local d = ((((xp^(3) * xq^(2)) * xr) * ys)
+ - (((xp^(3) * xq^(2)) * xs) * yr) - (((xp^(3) * xq) * xr^(2)) * ys)
+ + (((xp^(3) * xq) * xs^(2)) * yr) + (((xp^(3) * xr^(2)) * xs) * yq)
+ - (((xp^(3) * xr) * xs^(2)) * yq) - (((xp^(2) * xq^(3)) * xr) * ys)
+ + (((xp^(2) * xq^(3)) * xs) * yr) + (((xp^(2) * xq) * xr^(3)) * ys)
+ - (((xp^(2) * xq) * xs^(3)) * yr) - (((xp^(2) * xr^(3)) * xs) * yq)
+ + (((xp^(2) * xr) * xs^(3)) * yq) + (((xp * xq^(3)) * xr^(2)) * ys)
+ - (((xp * xq^(3)) * xs^(2)) * yr) - (((xp * xq^(2)) * xr^(3)) * ys)
+ + (((xp * xq^(2)) * xs^(3)) * yr) + (((xp * xr^(3)) * xs^(2)) * yq)
+ - (((xp * xr^(2)) * xs^(3)) * yq) - (((xq^(3) * xr^(2)) * xs) * yp)
+ + (((xq^(3) * xr) * xs^(2)) * yp) + (((xq^(2) * xr^(3)) * xs) * yp)
+ - (((xq^(2) * xr) * xs^(3)) * yp) - (((xq * xr^(3)) * xs^(2)) * yp)
+ + (((xq * xr^(2)) * xs^(3)) * yp)) /
+ (((xp^(3) * xq^(2)) * xr) -
+ ((xp^(3) * xq^(2)) * xs) - ((xp^(3) * xq) * xr^(2))
+ + ((xp^(3) * xq) * xs^(2)) + ((xp^(3) * xr^(2)) * xs)
+ - ((xp^(3) * xr) * xs^(2)) - ((xp^(2) * xq^(3)) * xr)
+ + ((xp^(2) * xq^(3)) * xs) + ((xp^(2) * xq) * xr^(3))
+ - ((xp^(2) * xq) * xs^(3)) - ((xp^(2) * xr^(3)) * xs)
+ + ((xp^(2) * xr) * xs^(3)) + ((xp * xq^(3)) * xr^(2))
+ - ((xp * xq^(3)) * xs^(2)) - ((xp * xq^(2)) * xr^(3))
+ + ((xp * xq^(2)) * xs^(3)) + ((xp * xr^(3)) * xs^(2))
+ - ((xp * xr^(2)) * xs^(3)) - ((xq^(3) * xr^(2)) * xs)
+ + ((xq^(3) * xr) * xs^(2)) + ((xq^(2) * xr^(3)) * xs)
+ - ((xq^(2) * xr) * xs^(3)) - ((xq * xr^(3)) * xs^(2))
+ + ((xq * xr^(2)) * xs^(3)))
+ return a, b, c, d
+end
+
+-- f(x)=a*x+b
+function parameters_affine(xp,yp,xq,yq)
+ local a = (yp - yq) / (xp - xq)
+ local b = ((xp * yq) - (xq * yp)) / (xp - xq)
+ return a, b
+end
+
+-- returns true iff the function is of type f(x)=a*x^3+b*x^2+c*x+d
+-- a, b, c, d being real numbers
+function is_cubic(graph,maxerror)
+ local l = #graph
+ local a, b, c, d = parameters_cubic(graph[1][1],graph[1][2],
+ graph[math.floor(l/3)][1],graph[math.floor(l/3)][2],
+ graph[math.floor(2*l/3)][1],graph[math.floor(2*l/3)][2],
+ graph[l][1],graph[l][2])
+ return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",graph,
+ maxerror,false)
+end
+
+-- returns true iff the function is of type f(x)=a*x^3+b*x^2+c*x+d
+-- a, b, c, d being real numbers
+-- this takes several graph parts
+-- the idea is to have a possibility to avoid tan(x)
+function are_cubic(graphs,maxerror)
+ if is_cubic(graphs[1],maxerror) then
+ if #graphs < 2 then
+ return true
+ else -- check for the next part
+ local a, b, c, d = parameters_cubic(graphs[1][1][1],
+ graphs[1][1][2],graphs[1][math.floor(l/3)][1],
+ graphs[1][math.floor(l/3)][2],
+ graphs[1][math.floor(2*l/3)][1],
+ graphs[1][math.floor(2*l/3)][2],
+ graphs[1][l][1],graphs[1][l][2])
+ return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",
+ graphs[2],maxerror,false)
+ end
+ else
+ return false
+ end
+end
+
+-- returns true iff the inverse function is of type
+-- f(x)=a*x^3+b*x^2+c*x+d
+-- a, b, c, d being real numbers
+function is_cuberoot(graph,maxerror)
+ local l = #graph
+ local a, b, c, d = parameters_cubic(graph[1][2],graph[1][1],
+ graph[math.floor(l/3)][2],graph[math.floor(l/3)][1],
+ graph[math.floor(2*l/3)][2],graph[math.floor(2*l/3)][1],
+ graph[l][2],graph[l][1])
+ return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",graph,
+ maxerror,true)
+end
+
+-- returns true iff the function is of type f(x)=a*x^3+b*x^2+c*x+d
+-- a, b, c, d being real numbers
+-- this takes several graph parts
+-- the idea is to have a possibility to avoid tan(x)
+function are_cuberoot(graphs,maxerror)
+ if is_cuberoot(graphs[1],maxerror) then
+ if #graphs < 2 then
+ return true
+ else -- check for the next part
+ local a, b, c, d = parameters_cubic(graphs[1][1][2],
+ graphs[1][1][1],graphs[1][math.floor(l/3)][2],
+ graphs[1][math.floor(l/3)][1],
+ graphs[1][math.floor(2*l/3)][2],
+ graphs[1][math.floor(2*l/3)][1],
+ graphs[1][l][2],graphs[1][l][1])
+ return do_parameters_fit(a,b,c,d,"a*x^3+b*x^2+c*x+d",
+ graphs[2],maxerror,true)
+ end
+ else
+ return false
+ end
+end
+
+-- returns true iff function is of type f(x)=a*x+b
+-- a, b being real numbers
+function is_affine(graph,maxerror)
+ l = #graph
+ local a, b = parameters_affine(graph[1][1],graph[1][2],
+ graph[l][1],graph[l][2])
+ return do_parameters_fit(a,b,0,0,"a*x+b",graph,maxerror,false)
+end
+
+-- what is the sum of the squared error
+-- when comparing the bezier path
+-- p.. control q and r .. s
+-- with the graph g from index starti to endi
+-- (looking at the points at roughly t=.33 and t=.67)
+function squareerror(f,g,starti,endi,qx,qy,rx,ry)
+ local result = 0
+ for t = .33, .7, .34 do
+ x = (1-t)^3*g[starti][1]+3*t*(1-t)^2*qx+3*t^2*(1-t)*rx+t^3*g[endi][1]
+ y = (1-t)^3*g[starti][2]+3*t*(1-t)^2*qy+3*t^2*(1-t)*ry+t^3*g[endi][2]
+ result = result + (y-f(x))^2
+ end
+ return result
+end
+
+function pointstobezier(qx,qy,rx,ry,sx,sy,rndx,rndy)
+ return " .. controls (" .. round(qx,rndx) .. ","
+ .. round(qy,rndy) ..") and ("
+ .. round(rx,rndx) .. ","
+ .. round(ry,rndy) .. ") .. ("
+ .. round(sx,rndx) .. ","
+ .. round(sy,rndy)..")"
+end
+
+-- take end points of a graph g of the function f
+-- (from indices starti to endi)
+-- without extrema or inflection points inbetween
+-- and try to approximate it with a cubic bezier curve
+-- (round to rndx and rndy when printing)
+function graphtobezierapprox(f,g,starti,endi,rndx,rndy,maxerror)
+ local px = g[starti][1]
+ local py = g[starti][2]
+ local dp = g[starti][3]
+ local sx = g[endi][1]
+ local sy = g[endi][2]
+ local ds = g[endi][3]
+ -- we compute the corner point c, where the controls would meet
+ local cx = ((dp * px) - (ds * sx) - py + sy) / (dp - ds)
+ local cy = (dp * ((ds * px) - (ds * sx) - py + sy) / (dp - ds)) + py
+ -- now we slide q between p and c & r between s and c
+ -- and search for the best qx and best rx
+ local qx = px+.05*(cx-px)
+ local qy = py+.05*(cy-py)
+ local rx = sx+.05*(cx-sx)
+ local ry = sy+.05*(cy-sy)
+ local err = squareerror(f,g,starti,endi,qx,qy,rx,ry)
+ for i = 2, 19 do
+ for j = 2, 19 do
+ xa = px+i*.05*(cx-px)
+ ya = py+i*.05*(cy-py)
+ xb = sx+j*.05*(cx-sx)
+ yb = sy+j*.05*(cy-sy)
+ -- now check, if xa and xb fit better
+ -- at roughly t=0.33 and t=0.66 for f(x)
+ -- than the last qx and rx did
+ -- (sum of squares must be smaller)
+ if squareerror(f,g,starti,endi,xa,ya,xb,yb) < err then
+ qx = xa
+ qy = ya
+ rx = xb
+ ry = yb
+ err = squareerror(f,g,starti,endi,qx,qy,rx,ry)
+ end
+ end
+ end
+ -- check if it is close enough: (recycling err, xa, ya)
+ err = 0
+ for t = .1, .9, .1 do
+ xa = (1-t)^3*g[starti][1]+3*t*(1-t)^2*qx+3*t^2*(1-t)*rx+t^3*g[endi][1]
+ ya = (1-t)^3*g[starti][2]+3*t*(1-t)^2*qy+3*t^2*(1-t)*ry+t^3*g[endi][2]
+ if abs(ya-f(xa)) > err then
+ err = abs(ya-f(xa))
+ end
+ end
+ if err <= maxerror then
+ return pointstobezier(qx,qy,rx,ry,sx,sy,rndx,rndy)
+ else
+ -- search for an intermediate point where the graph has the same
+ -- slope as the line from the start point to the end point:
+ local interindex = math.floor(.5*starti+.5*endi) -- will change
+ for i = starti + 1, endi - 1 do
+ if abs(g[i][3]-(g[endi][2]-g[starti][2])
+ /(g[endi][1]-g[starti][1]))
+ < abs(g[interindex][3]-(g[endi][2]-g[starti][2])
+ /(g[endi][1]-g[starti][1])) then
+ interindex = i
+ end
+ end
+ return graphtobezierapprox(f,g,starti,interindex,rndx,rndy,maxerror)
+ .. graphtobezierapprox(f,g,interindex,endi,rndx,rndy,maxerror)
+ end
+end
+
+-- like above but exact for quadratic and cubic (if not inverse)
+-- resp. exact for squareroot and cuberoot (if inverse)
+function graphtobezier(g,starti,endi,rndx,rndy,isinverse)
+ local px = g[starti][1]
+ local py = g[starti][2]
+ local dp = g[starti][3]
+ local sx = g[endi][1]
+ local sy = g[endi][2]
+ local ds = g[endi][3]
+ local qx = px+(sx-px)/3
+ local rx = px+2*(sx-px)/3
+ local qy = py+(qx-px)*dp
+ local ry = sy+(rx-sx)*ds
+ if isinverse then
+ return pointstobezier(qy,qx,ry,rx,sy,sx,rndy,rndx)
+ else
+ return pointstobezier(qx,qy,rx,ry,sx,sy,rndx,rndy)
+ end
+end
+
+-- reverses a path p e.g. when p = "(0,1) -- (2,3)"
+-- it returns "(2,3) -- (0,1)"
+-- or when p = "(0,1) .. controls (2,3) and (4,5) .. (6,7)"
+-- it returns "(6,7) .. controls (4,5) and (2,3) .. (0,1)"
+function reversepath(p)
+ local r = "" -- will become the reverse path
+ local temppoint ="" -- will store temporal points like "(0,1)"
+ local tempbetween = "" -- will store things like " .. controls "
+ for i = 1, #p do
+ local c = string.sub(p,i,i)
+ if c == "(" then
+ if tempbetween == " .. " then
+ r = " .. controls " .. r
+ elseif tempbetween == " .. controls " then
+ r = " .. " .. r
+ else
+ r = tempbetween .. r
+ end
+ tempbetween = ""
+ temppoint = "("
+ elseif c == ")" then
+ r = temppoint .. ")" .. r
+ temppoint = ""
+ else
+ if temppoint == "" then -- not reading a point
+ tempbetween = tempbetween .. c
+ else
+ temppoint = temppoint .. c
+ end
+ end
+ end
+ return r
+end
+
+-- main function
+function bezierplot(functionstring,xmin,xmax,ymin,ymax)
+ local fstringreplaced = string.gsub(functionstring, "%*%*", "^")
+ local f = assert(load("local x = ...; return " .. fstringreplaced))
+ local isreverse = false
+ if xmin > xmax then
+ isreverse = true
+ end
+ xmin, xmax = math.min(xmin,xmax), math.max(xmin,xmax)
+ local xstep = (xmax-xmin)/20000
+ -- the output of the x coordinates will be rounded to rndx digits
+ local rndx = math.max(0,math.floor(4.5-log(xmax-xmin)/log(10)))
+ local xerror = abs(xmax-xmin)/(100*10^rndx)
+ ymin, ymax = math.min(ymin,ymax), math.max(ymin,ymax)
+ -- the output of the x coordinates will be rounded to rndy digits
+ local rndy = math.max(0,math.floor(4.5-log(ymax-ymin)/log(10)))
+ local yerror = (ymax-ymin)/(100*10^rndy)
+ -- determine parts of the graph that are inside window
+ local graphs = {}
+ local outside = true -- value is outside window
+ local i = 0
+ local j = 0
+ for n = 0, 20000 do
+ local x = xmin + n/20000*(xmax-xmin)
+ local y = f(x)
+ if y >= ymin-yerror and y <= ymax+yerror then -- inside
+ if outside then -- if it was outside before
+ outside = false
+ j = 0
+ i = i + 1
+ graphs[i] = {}
+ end
+ j = j + 1
+ graphs[i][j] = {x,y}
+ else
+ outside = true
+ end
+ end
+
+ local functiontype = "unknown"
+ local bezierstring = ""
+
+ -- go through the connected parts
+ for part = 1, #graphs do
+ local d = diffgraph(f,graphs[part],xstep)
+ -- check for type of function (only for the first part)
+ if part == 1 then
+ if is_affine(d,yerror) then
+ functiontype = "affine"
+ elseif are_cubic(graphs,yerror) then
+ functiontype = "cubic"
+ elseif are_cuberoot(graphs,xerror) then
+ functiontype = "cuberoot"
+ end
+ end
+ if functiontype ~= "cuberoot" then -- start with initial point
+ bezierstring = bezierstring .. "(" .. round(d[1][1],rndx)
+ .. "," .. round(d[1][2],rndy) .. ")"
+ end
+ if functiontype == "affine" then
+ bezierstring = bezierstring .. " -- (" .. round(d[#d][1],
+ rndx) .. "," .. round(d[#d][2],rndy) ..")"
+ elseif functiontype == "cubic" then
+ local startindex = 1
+ local extremainbetween = false
+ for k = 2, #d do
+ if d[k][5] then -- extrema
+ extremainbetween = true
+ bezierstring = bezierstring
+ .. graphtobezier(d,startindex,k,rndx,rndy,false)
+ startindex = k
+ end
+ end
+ if not extremainbetween then
+ for k = 2, #d do
+ if d[k][6] then -- inflection point
+ -- check, if the controlpoints are outside
+ -- of the bounding box defined by the vertices
+ -- (d[1][1],d[1][2]) and (d[#d][1],d[#d][2])
+ local qx = d[1][1]+(d[#d][1]-d[1][1])/3
+ local rx = d[1][1]+2*(d[#d][1]-d[1][1])/3
+ local qy = d[1][2]+(qx-d[1][1])*d[1][3]
+ local ry = d[#d][2]+(rx-d[#d][1])*d[#d][3]
+ if math.max(qy,ry) > ymax
+ or math.min(qy,ry) < ymin then
+ bezierstring = bezierstring ..graphtobezier(
+ d,startindex,k,rndx,rndy,false)
+ startindex = k
+ end
+ end
+ end
+ end
+ if startindex ~= #d then -- if no special points inbetween
+ bezierstring = bezierstring
+ .. graphtobezier(d,startindex,#d,rndx,rndy,false)
+ end
+ elseif functiontype == "cuberoot" then
+ -- we determine a, b, c, d and then
+ -- get x' = 3ay^2+2by+c
+ local a, b, c, dd = parameters_cubic(
+ d[math.floor(.2*l)][2], d[math.floor(.2*l)][1],
+ d[math.floor(.4*l)][2], d[math.floor(.4*l)][1],
+ d[math.floor(.6*l)][2], d[math.floor(.6*l)][1],
+ d[math.floor(.8*l)][2], d[math.floor(.8*l)][1])
+ -- now recalculate the graph with the inverse function:
+ -- we can increase the accuracy
+ xstep = (ymax-ymin)/100000 -- inverse redefinition
+ local finverse = assert(load("local x = ...; return "
+ ..a.."*x^3+"..b.."*x^2+"..c.."*x+"..dd))
+ local graphinverse = {}
+ local i = 1
+ for y = ymin, ymax, xstep do
+ local x = finverse(y)
+ if x > xmin and x < xmax -- inside
+ and abs(y-f(x)) < (ymax-ymin)/(100*10^rndy) then
+ graphinverse[i] = {y,x}
+ i = i + 1
+ end
+ end
+ d = diffgraph(finverse,graphinverse,xstep)
+ bezierstring = bezierstring .. "(" .. round(d[1][2],rndy)
+ .. "," .. round(d[1][1],rndx) .. ")" -- initial point
+ local startindex = 1
+ for k = 2, #d do
+ if d[k][6] then -- inflection point
+ -- check, if the controlpoints are outside
+ -- of the bounding box defined by the vertices
+ -- (d[1][1],d[1][2]) and (d[#d][1],d[#d][2])
+ local qx = d[1][1]+(d[#d][1]-d[1][1])/3
+ local rx = d[1][1]+2*(d[#d][1]-d[1][1])/3
+ local qy = d[1][2]+(qx-d[1][1])*d[1][3]
+ local ry = d[#d][2]+(rx-d[#d][1])*d[#d][3]
+ if math.max(qy,ry) > xmax
+ or math.min(qy,ry) < xmin then
+ bezierstring = bezierstring..graphtobezier(
+ d,startindex,k,rndx,rndy,true)
+ startindex = k
+ end
+ end
+ end
+ if startindex ~= #d then -- if no special points inbetween
+ bezierstring = bezierstring
+ .. graphtobezier(d,startindex,#d,rndx,rndy,true)
+ end
+ else
+ -- standard case (nothing special)
+ local startindex = 1
+ for k = 2, #d do
+ if d[k][5] or d[k][6] then -- extrema and inflection points
+ bezierstring = bezierstring .. graphtobezierapprox(
+ f,d,startindex,k,rndx,rndy,(ymax-ymin)/(0.5*10^rndy))
+ startindex = k
+ end
+ end
+ if startindex ~= #d then -- if no special points inbetween
+ bezierstring = bezierstring .. graphtobezierapprox(f,d,
+ startindex,#d,rndx,rndy,(ymax-ymin)/(0.5*10^rndy))
+ end
+ end
+ end
+ if isreverse then
+ return reversepath(bezierstring)
+ else
+ return bezierstring
+ end
+end
+
+-- main program --
+
+if not pcall(debug.getlocal, 4, 1) then
+ if #arg >= 1 then
+ local xmin = -5
+ local xmax = 5
+ if #arg >= 2 then xmin = arg[2] end
+ if #arg >= 3 then
+ if arg[3] == arg[2] then
+ xmax = xmin + 10
+ else
+ xmax = arg[3]
+ end
+ end
+ local ymin = -5
+ local ymax = 5
+ if #arg >= 4 then ymin = arg[4] end
+ if #arg >= 5 then
+ if arg[5] == arg[4] then
+ ymax = ymin + 10
+ else
+ ymax = arg[5]
+ end
+ end
+ print(bezierplot(arg[1],xmin,xmax,ymin,ymax))
+ end
+end
+
+
+
diff --git a/Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.sty b/Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.sty
new file mode 100644
index 00000000000..66f404dfd34
--- /dev/null
+++ b/Master/texmf-dist/tex/lualatex/bezierplot/bezierplot.sty
@@ -0,0 +1,17 @@
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{bezierplot}[2018/04/12 bezierplot]
+\RequirePackage{xparse}
+\RequirePackage{iftex}
+\ifLuaTeX
+ \directlua{require("bezierplot")}
+ \DeclareExpandableDocumentCommand{\xbezierplot}{O{-5} O{5} O{-5} O{5} m}{%
+ \directlua{tex.sprint(bezierplot("#5",#1,#2,#3,#4))}
+ }
+\else
+ \let\xpandblinpt\@@input
+ \DeclareExpandableDocumentCommand{\xbezierplot}{O{-5} O{5} O{-5} O{5} m}{%
+ \xpandblinpt|"bezierplot '#5' #1 #2 #3 #4"
+ }
+\fi
+\providecommand\bezierplot{\romannumeral`\^^@\xbezierplot}
+\endinput \ No newline at end of file