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|
#!/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
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