HC/polygon.lua

--[[
Copyright (c) 2011 Matthias Richter

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
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]]--

module(..., package.seeall)
local Class = require(_PACKAGE .. 'class')
local vector = require(_PACKAGE .. 'vector')

----------------------------
-- Private helper functions
--
-- create vertex list of coordinate pairs
local function toVertexList(vertices, x,y, ...)
	if not x or not y then return vertices end -- no more arguments

	vertices[#vertices + 1] = vector(x, y)     -- set vertex
	return toVertexList(vertices, ...)         -- recurse
end

-- returns true if three points lie on a line
local function areCollinear(p,q,r)
	return (q - p):cross(r - p) == 0
end
-- remove vertices that lie on a line
local function removeCollinear(vertices)
	local ret = {}
	for k=1,#vertices do
		local i = k > 1 and k - 1 or #vertices
		local l = k < #vertices and k + 1 or 1
		if not areCollinear(vertices[i], vertices[k], vertices[l]) then
			ret[#ret+1] = vertices[k]
		end
	end
	return ret
end

-- get index of rightmost vertex (for testing orientation)
local function getIndexOfleftmost(vertices)
	local idx = 1
	for i = 2,#vertices do
		if vertices[i].x < vertices[idx].x then
			idx = i
		end
	end
	return idx
end

-- returns true if three points make a counter clockwise turn
local function ccw(p, q, r)
	return (q - p):cross(r - p) >= 0
end

-- unpack vertex coordinates, i.e. {x=p, y=q}, ... -> p,q, ...
local function unpackHelper(v, ...)
	if not v then return end
	return v.x,v.y,unpackHelper(...)
end

-- test if a point lies inside of a triangle using cramers rule
local function pointInTriangle(q, p1,p2,p3)
	local v1,v2 = p2 - p1, p3 - p1
	local qp = q - p1
	local dv = v1:cross(v2)
	local l = qp:cross(v2) / dv
	if l <= 0 then return false end
	local m = v1:cross(qp) / dv
	if m <= 0 then return false end
	return l+m < 1
end

-- returns starting indices of shared edge, i.e. if p and q share the
-- edge with indices p1,p2 of p and q1,q2 of q, the return value is p1,q1
local function getSharedEdge(p,q)
	local vertices = {}
	for i,v in ipairs(q) do vertices[ tostring(v) ] = i end
	for i,v in ipairs(p) do
		local w = (i == #p) and p[1] or p[i+1]
		if vertices[ tostring(v) ] and vertices[ tostring(w) ] then
			return i, vertices[ tostring(v) ]
		end
	end
end

-----------------
-- Polygon class
--
Polygon = Class{name = "Polygon", function(self, ...)
	local vertices = removeCollinear( toVertexList({}, ...) )
	assert(#vertices >= 3, "Need at least 3 non collinear points to build polygon (got "..#vertices..")")

	-- assert polygon is oriented counter clockwise
	local r = getIndexOfleftmost(vertices)
	local q = r > 1 and r - 1 or #vertices
	local s = r < #vertices and r + 1 or 1
	if not ccw(vertices[q], vertices[r], vertices[s]) then -- reverse order if polygon is not ccw
		local tmp = {}
		for i=#vertices,1,-1 do
			tmp[#tmp + 1] = vertices[i]
		end
		vertices = tmp
	end
	self.vertices = vertices
	-- make vertices immutable
	setmetatable(self.vertices, {__newindex = function() error("Thou shall not change a polygons vertices!") end})

	-- compute polygon area and centroid
	self.area = vertices[#vertices]:cross(vertices[1])
	for i = 1,#vertices-1 do
		self.area = self.area + vertices[i]:cross(vertices[i+1])
	end
	self.area = self.area / 2

	local p,q = vertices[#vertices], vertices[1]
	local det = p:cross(q)
	self.centroid = vector((p.x+q.x) * det, (p.y+q.y) * det)
	for i = 1,#vertices-1 do
		p,q = vertices[i], vertices[i+1]
		det = p:cross(q)
		self.centroid.x = self.centroid.x + (p.x+q.x) * det
		self.centroid.y = self.centroid.y + (p.y+q.y) * det
	end
	self.centroid = self.centroid / (6 * self.area)
end}

-- return vertices as x1,y1,x2,y2, ..., xn,yn
function Polygon:unpack()
	return unpackHelper( unpack(self.vertices) )
end

-- deep copy of the polygon
function Polygon:clone()
	return Polygon( self:unpack() )
end

-- get bounding box
function Polygon:getBBox()
	local ul = self.vertices[1]:clone()
	local lr = ul:clone()
	for i=2,#self.vertices do
		local p = self.vertices[i]
		if ul.x > p.x then ul.x = p.x end
		if ul.y > p.y then ul.y = p.y end

		if lr.x < p.x then lr.x = p.x end
		if lr.y < p.y then lr.y = p.y end
	end

	return ul.x,ul.y, lr.x,lr.y
end

-- a polygon is convex if all edges are oriented ccw
function Polygon:isConvex()
	local function isConvex()
		local v = self.vertices
		if #v == 3 then return true end

		if not ccw(v[#v], v[1], v[2]) then
			return false
		end
		for i = 2,#v-1 do
			if not ccw(v[i-1], v[i], v[i+1]) then
				return false
			end
		end
		if not ccw(v[#v-1], v[#v], v[1]) then
			return false
		end
		return true
	end

	-- replace function so that this will only be computed once
	local status = isConvex()
	self.isConvex = function() return status end
	return status
end

function Polygon:move(direction, dy)
	if dy then direction = vector(direction,dy) end
	for i,v in ipairs(self.vertices) do
		self.vertices[i] = self.vertices[i] + direction
	end
	self.centroid = self.centroid + direction
end

function Polygon:rotate(angle, center, cy)
	local center = center or self.centroid
	if cy then center = vector(cy) end
	for i,v in ipairs(self.vertices) do
		self.vertices[i] = (self.vertices[i] - center):rotate_inplace(angle) + center
	end
end

-- triangulation by the method of kong
function Polygon:triangulate()
	if #self.vertices == 3 then return {self:clone()} end
	local triangles = {} -- list of triangles to be returned
	local concave = {}   -- list of concave edges
	local adj = {}       -- vertex adjacencies
	local vertices = self.vertices

	-- retrieve adjacencies as the rest will be easier to implement
	for i,p in ipairs(vertices) do
		local l = (i == 1) and vertices[#vertices] or vertices[i-1]
		local r = (i == #vertices) and vertices[1] or vertices[i+1]
		adj[p] = {p = p, l = l, r = r} -- point, left and right neighbor
		-- test if vertex is a concave edge
		if not ccw(l,p,r) then concave[p] = p end
	end

	-- and ear is an edge of the polygon that contains no other
	-- vertex of the polygon
	local function isEar(p1,p2,p3)
		if not ccw(p1,p2,p3) then return false end
		for q,_ in pairs(concave) do
			if pointInTriangle(q, p1,p2,p3) then return false end
		end
		return true
	end

	-- main loop
	local nPoints, skipped = #vertices, 0
	local p = adj[ vertices[2] ]
	while nPoints > 3 do
		if not concave[p.p] and isEar(p.l, p.p, p.r) then
			triangles[#triangles+1] = Polygon( unpackHelper(p.l, p.p, p.r) )
			if concave[p.l] and ccw(adj[p.l].l, p.l, p.r) then
				concave[p.l] = nil
			end
			if concave[p.r] and ccw(p.l, p.r, adj[p.r].r) then
				concave[p.r] = nil
			end
			-- remove point from list
			adj[p.p] = nil
			adj[p.l].r = p.r
			adj[p.r].l = p.l
			nPoints = nPoints - 1
			skipped = 0
			p = adj[p.l]
		else
			p = adj[p.r]
			skipped = skipped + 1
			assert(skipped <= nPoints, "Cannot triangulate polygon (is the polygon intersecting itself?)")
		end
	end
	triangles[#triangles+1] = Polygon( unpackHelper(p.l, p.p, p.r) )

	return triangles
end

-- return merged polygon if possible or nil otherwise
function Polygon:mergedWith(other)
	local p,q = getSharedEdge(self.vertices, other.vertices)
	if not (p and q) then return nil end

	local ret = {}
	for i = 1, p do ret[#ret+1] = self.vertices[i] end
	for i = 2, #other.vertices-1 do
		local k = i + q - 1
		if k > #other.vertices then k = k - #other.vertices end
		ret[#ret+1] = other.vertices[k]
	end
	for i = p+1,#self.vertices do ret[#ret+1] = self.vertices[i] end
	return Polygon( unpackHelper( unpack(ret) ) )
end

-- split polygon into convex polygons.
-- note that this won't be the optimal split in most cases, as
-- finding the optimal split is a really hard problem.
-- the method is to first triangulate and then greedily merge
-- the triangles.
function Polygon:splitConvex()
	-- edge case: polygon is a triangle or already convex
	if #self.vertices <= 3 or self:isConvex() then return {self:clone()} end

	local convex = self:triangulate()
	local i = 1
	repeat
		local p = convex[i]
		local k = i + 1
		while k <= #convex do
			local _, merged = pcall(function() return p:mergedWith(convex[k]) end)
			if merged and merged:isConvex() then
				convex[i] = merged
				p = convex[i]
				table.remove(convex, k)
			else
				k = k + 1
			end
		end
		i = i + 1
	until i >= #convex
	
	return convex
end

-- module() as shortcut to module.Polygon()
do
	local m = getmetatable(_M)
	m.__call = function(_, ...) return Polygon(...) end
end