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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IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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]]--

local _PACKAGE = (...):match("^(.+)%.[^%.]+")
if not (common and common.class and common.instance) then
	class_commons = true
	require(_PACKAGE .. '.class')
end
local vector = require(_PACKAGE .. '.vector-light')

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

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

-- returns true if three vertices lie on a line
local function areCollinear(p, q, r, eps)
	return math.abs(vector.det(q.x-p.x, q.y-p.y,  r.x-p.x,r.y-p.y)) <= (eps or 1e-32)
end
-- remove vertices that lie on a line
local function removeCollinear(vertices)
	local ret = {}
	local i,k = #vertices - 1, #vertices
	for l=1,#vertices do
		if not areCollinear(vertices[i], vertices[k], vertices[l]) then
			ret[#ret+1] = vertices[k]
		end
		i,k = k,l
	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 vector.det(q.x-p.x, q.y-p.y,  r.x-p.x, r.y-p.y) >= 0
end

-- test wether a and b lie on the same side of the line c->d
local function onSameSide(a,b, c,d)
	local px, py = d.x-c.x, d.y-c.y
	local l = vector.det(px,py,  a.x-c.x, a.y-c.y)
	local m = vector.det(px,py,  b.x-c.x, b.y-c.y)
	return l*m >= 0
end

local function pointInTriangle(p, a,b,c)
	return onSameSide(p,a, b,c) and onSameSide(p,b, a,c) and onSameSide(p,c, a,b)
end

-- returns starting/ending 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,q2
local function getSharedEdge(p,q)
	local pindex = setmetatable({}, {__index = function(t,k)
		local s = {}
		t[k] = s
		return s
	end})

	-- record indices of vertices in p by their coordinates
	for i = 1,#p do
		pindex[p[i].x][p[i].y] = i
	end

	-- iterate over all edges in q. if both endpoints of that
	-- edge are in p as well, return the indices of the starting
	-- vertex
	local i,k = #q,1
	for k = 1,#q do
		local v,w = q[i], q[k]
		if pindex[v.x][v.y] and pindex[w.x][w.y] then
			return pindex[w.x][w.y], k
		end
		i = k
	end
end

-----------------
-- Polygon class
--
local Polygon = {}
function Polygon:init(...)
	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 polygon's vertices!") end})

	-- compute polygon area and centroid
	local p,q = vertices[#vertices], vertices[1]
	local det = vector.det(p.x,p.y, q.x,q.y) -- also used below
	self.area = det
	for i = 2,#vertices do
		p,q = q,vertices[i]
		self.area = self.area + vector.det(p.x,p.y, q.x,q.y)
	end
	self.area = self.area / 2

	p,q = vertices[#vertices], vertices[1]
	self.centroid = {x = (p.x+q.x)*det, y = (p.y+q.y)*det}
	for i = 2,#vertices do
		p,q = q,vertices[i]
		det = vector.det(p.x,p.y, q.x,q.y)
		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.x = self.centroid.x / (6 * self.area)
	self.centroid.y = self.centroid.y / (6 * self.area)

	-- get outcircle
	self._radius = 0
	for i = 1,#vertices do
		self._radius = math.max(self._radius,
			vector.dist(vertices[i].x,vertices[i].y, self.centroid.x,self.centroid.y))
	end
end
local newPolygon


-- return vertices as x1,y1,x2,y2, ..., xn,yn
function Polygon:unpack()
	local v = {}
	for i = 1,#self.vertices do
		v[2*i-1] = self.vertices[i].x
		v[2*i]   = self.vertices[i].y
	end
	return unpack(v)
end

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

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

		if lrx < p.x then lrx = p.x end
		if lry < p.y then lry = p.y end
	end

	return ulx,uly, lrx,lry
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(dx, dy)
	if not dy then
		dx, dy = dx:unpack()
	end
	for i,v in ipairs(self.vertices) do
		v.x = v.x + dx
		v.y = v.y + dy
	end
	self.centroid.x = self.centroid.x + dx
	self.centroid.y = self.centroid.y + dy
end

function Polygon:rotate(angle, cx, cy)
	if not (cx and cy) then
		cx,cy = self.centroid.x, self.centroid.y
	end
	for i,v in ipairs(self.vertices) do
		-- v = (v - center):rotate(angle) + center
		v.x,v.y = vector.add(cx,cy, vector.rotate(angle, v.x-cx, v.y-cy))
	end
	local v = self.centroid
	v.x,v.y = vector.add(cx,cy, vector.rotate(angle, v.x-cx, v.y-cy))
end

function Polygon:scale(s, cx,cy)
	if not (cx and cy) then
		cx,cy = self.centroid.x, self.centroid.y
	end
	for i,v in ipairs(self.vertices) do
		-- v = (v - center) * s + center
		v.x,v.y = vector.add(cx,cy, vector.mul(s, v.x-cx, v.y-cy))
	end
	self._radius = self._radius * s
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 q ~= p1 and q ~= p2 and q ~= p3 and 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
			-- polygon may be a 'collinear triangle', i.e.
			-- all three points are on a line. In that case
			-- the polygon constructor throws an error.
			if not areCollinear(p.l, p.p, p.r) then
				triangles[#triangles+1] = newPolygon(p.l.x,p.l.y, p.p.x,p.p.y, p.r.x,p.r.y)
				skipped = 0
			end

			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

	if not areCollinear(p.l, p.p, p.r) then
		triangles[#triangles+1] = newPolygon(p.l.x,p.l.y, p.p.x,p.p.y, p.r.x,p.r.y)
	end

	return triangles
end

-- return merged polygon if possible or nil otherwise
function Polygon:mergedWith(other)
	local p,q = getSharedEdge(self.vertices, other.vertices)
	assert(p and q, "Polygons do not share an edge")

	local ret = {}
	for i = 1,p-1 do
		ret[#ret+1] = self.vertices[i].x
		ret[#ret+1] = self.vertices[i].y
	end

	for i = 0,#other.vertices-2 do
		i = ((i-1 + q) % #other.vertices) + 1
		ret[#ret+1] = other.vertices[i].x
		ret[#ret+1] = other.vertices[i].y
	end

	for i = p+1,#self.vertices do
		ret[#ret+1] = self.vertices[i].x
		ret[#ret+1] = self.vertices[i].y
	end

	return newPolygon(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 success, merged = pcall(function() return p:mergedWith(convex[k]) end)
			if success 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

function Polygon:contains(x,y)
	-- test if an edge cuts the ray
	local function cut_ray(p,q)
		return ((p.y > y and q.y < y) or (p.y < y and q.y > y)) -- possible cut
			and (x - p.x < (y - p.y) * (q.x - p.x) / (q.y - p.y)) -- x < cut.x
	end

	-- test if the ray crosses boundary from interior to exterior.
	-- this is needed due to edge cases, when the ray passes through
	-- polygon corners
	local function cross_boundary(p,q)
		return (p.y == y and p.x > x and q.y < y)
			or (q.y == y and q.x > x and p.y < y)
	end

	local v = self.vertices
	local in_polygon = false
	local p,q = v[#v],v[#v]
	for i = 1, #v do
		p,q = q,v[i]
		if cut_ray(p,q) or cross_boundary(p,q) then
			in_polygon = not in_polygon
		end
	end
	return in_polygon
end

function Polygon:intersectsRay(x,y, dx,dy)
	--local p = vector(x,y)
	--local v = vector(dx,dy)
	local nx,ny = vector.perpendicular(dx,dy)
	local wx,xy,det

	local tmin = math.huge
	local q1,q2 = nil, self.vertices[#self.vertices]
	for i = 1, #self.vertices do
		q1,q2 = q2,self.vertices[i]
		wx,wy = q2.x - q1.x, q2.y - q1.y
		det = vector.det(dx,dy, wx,wy)

		if det ~= 0 then
			-- there is an intersection point. check if it lies on both
			-- the ray and the segment.
			local rx,ry = q2.x - x, q2.y - y
			local l = vector.det(rx,ry, wx,wy) / det
			local m = vector.det(dx,dy, rx,ry) / det
			if l >= 0 and m >= 0 and m <= 1 then
				-- we cannot jump out early here (i.e. when l > tmin) because
				-- the polygon might be concave
				tmin = math.min(tmin, l)
			end
		else
			-- lines parralel or incident. get distance of line to
			-- anchor point. if they are incident, check if an endpoint
			-- lies on the ray
			local dist = vector.dot(q1.x-x,q1.y-y, nx,ny)
			if dist == 0 then
				local l = vector.dot(dx,dy, q1.x-x,q1.y-y)
				local m = vector.dot(dx,dy, q2.x-x,q2.y-y)
				if l >= 0 and l >= m then
					tmin = math.min(tmin, l)
				elseif m >= 0 then
					tmin = math.min(tmin, m)
				end
			end
		end
	end
	return tmin ~= math.huge, tmin
end

Polygon = common.class('Polygon', Polygon)
newPolygon = function(...) return common.instance(Polygon, ...) end
return Polygon