local G = love.graphics

local function atanh(x)
	return (1/2) * math.log((1+x)/(1-x))
end

local function solve_for_A(r)
	local A
	if r < 3 then
		A = math.sqrt(6 * (r-1))
	else
		A = math.log(2*r) + math.log(math.log(2*r))
	end
	for i=1,5 do
		A = A - (math.sinh(A) - r*A)/(math.cosh(A)-r)
	end
	return A
end

local function plot(f,x0,xn)
	local n = 30
	x0 = x0 or G.inverseTransformPoint(0,0)
	xn = xn or G.inverseTransformPoint(G.getDimensions(),0)

	if xn < x0 then
		xn,x0 = x0,xn
	end

	local h = (xn-x0)/n
	local ps = {}
	for i=0,n do
		local x = x0 + i*h
		local y = f(x)
		table.insert(ps,x)
		table.insert(ps,y)
	end
	return ps
end

local function catenary0(x1,y1,x2,y2, L)
	-- https://math.stackexchange.com/a/3557768
	y1 = -y1
	y2 = -y2
	local dx = x2-x1
	local dy = y2-y1
	local mx = (x1+x2)/2
	local my = (y1+y2)/2

	local D = math.sqrt(dx^2+dy^2)
	L = L or D+(1/(D+0.3))

	local r = math.sqrt(L^2 - dy^2)/dx
	local A = solve_for_A(r)

	local a = dx/(2*A)
	local b = mx - a*atanh(dy/L)
	local c = my - L/(2*math.tanh(A))

	return function(x) return -(a * math.cosh((x-b)/a) + c) end
end


local function catenary(x1,y1, x2,y2, L)
	local D = math.sqrt((x2-x1)^2 + (y2-y1)^2)
	-- L = L or D+(1/(D+0.3))
	L = L or D+(0.1/(D+0.1))
	-- L = D + 0.1

	if x2 < x1 then
		x1,x2 = x2,x1
		y1,y2 = y2,y1
	end

	if math.abs(x1-x2) < 0.0001 then
		x1 = x1 - 0.01
		x2 = x2 + 0.01
	end

	local f = catenary0(x1,y1,x2,y2,L)
	return plot(f, x1,x2)
end

return {
	catenary = catenary
}


--[[
local Ps = {{-1,1}, {1,1}}

function love.update(dt)
	for b = 1,2 do
		if love.mouse.isDown(b) then
			Ps[b] = {G.inverseTransformPoint(love.mouse.getPosition())}
		end
	end
end


function love.draw()
	local W,H = G.getDimensions()
	G.clear(1,1,1)
	G.setColor(0,0,0)
	G.origin()
	local dx = Ps[1][1] - Ps[2][1]
	local dy = Ps[1][2] - Ps[2][2]
	local D = math.sqrt(dx^2+dy^2)
	local L = D+(1/(D+0.3))
	-- local L = D+1
	G.print(("%.2f\n%.2f\n%.2f"):format(L,D,L-D),10,10)
	G.setLineWidth(0.01)
	G.translate(W/2,H/2)
	G.scale(50)

	G.setColor(0.8,0.8,0.8)
	for i=-10,10 do
		G.line(-10,i,10,i)
		G.line(i,-10,i,10)
	end

	G.circle('line',0,0,1)

	-- G.setColor(0,0,0)
	-- -- plot(function(x) return x^2 end)
	-- plot(math.tanh)
	-- G.setColor(0,1,0)
	-- plot(atanh)

	G.setColor(0,0,0)
	G.line(catenary(Ps[1][1],Ps[1][2], Ps[2][1], Ps[2][2]))

	G.setColor(1,0,0)
	G.circle('fill',Ps[1][1],Ps[1][2],0.05)
	G.setColor(0,0,1)
	G.circle('fill',Ps[2][1],Ps[2][2],0.05)
end
--]]