HC/gjk.lua

--[[
Copyright (c) 2012 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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AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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]]--

local _PACKAGE = (...):match("^(.+)%.[^%.]+")
local vector  = require(_PACKAGE .. '.vector-light')
local huge, abs = math.huge, math.abs

local function support(shape_a, shape_b, dx, dy)
	local x,y = shape_a:support(dx,dy)
	return vector.sub(x,y, shape_b:support(-dx, -dy))
end

-- returns closest edge to the origin
local function closest_edge(simplex)
	local e = {dist = huge}

	local i = #simplex-1
	for k = 1,#simplex-1,2 do
		local ax,ay = simplex[i], simplex[i+1]
		local bx,by = simplex[k], simplex[k+1]
		i = k

		local ex,ey = vector.perpendicular(bx-ax, by-ay)
		local nx,ny = vector.normalize(ex,ey)
		local d = vector.dot(ax,ay, nx,ny)

		if d < e.dist then
			e.dist = d
			e.nx, e.ny = nx, ny
			e.i = k
		end
	end

	return e
end

local function EPA(shape_a, shape_b, simplex)
	-- make sure simplex is oriented counter clockwise
	local cx,cy, bx,by, ax,ay = unpack(simplex)
	if vector.dot(ax-bx,ay-by, cx-bx,cy-by) < 0 then
		simplex[1],simplex[2] = ax,ay
		simplex[5],simplex[6] = cx,cy
	end

	-- the expanding polytype algorithm
	local last_diff_dist = huge
	while true do
		local e = closest_edge(simplex)
		local px,py = support(shape_a, shape_b, e.nx, e.ny)
		local d = vector.dot(px,py, e.nx, e.ny)

		local diff_dist = d - e.dist
		if diff_dist < 1e-6 or abs(last_diff_dist - diff_dist) < 1e-10 then
			return -d*e.nx, -d*e.ny
		end
		last_diff_dist = diff_dist

		-- simplex = {..., simplex[e.i-1], px, py, simplex[e.i]
		table.insert(simplex, e.i, py)
		table.insert(simplex, e.i, px)
	end
end

--   :      :     origin must be in plane between A and B
-- B o------o A   since A is the furthest point on the MD
--   :      :     in direction of the origin.
local function do_line(simplex)
	local bx,by, ax,ay = unpack(simplex)

	local abx,aby = bx-ax, by-ay

	local dx,dy = vector.perpendicular(abx,aby)

	if vector.dot(dx,dy, -ax,-ay) < 0 then
		dx,dy = -dx,-dy
	end
	return simplex, dx,dy
end

-- B .'
--  o-._  1
--  |   `-. .'     The origin can only be in regions 1, 3 or 4:
--  |  4   o A 2   A lies on the edge of the MD and we came
--  |  _.-' '.     from left of BC.
--  o-'  3
-- C '.
local function do_triangle(simplex)
	local cx,cy, bx,by, ax,ay = unpack(simplex)
	local aox,aoy = -ax,-ay
	local abx,aby = bx-ax, by-ay
	local acx,acy = cx-ax, cy-ay

	-- test region 1
	local dx,dy = vector.perpendicular(abx,aby)
	if vector.dot(dx,dy, acx,acy) > 0 then
		dx,dy = -dx,-dy
	end
	if vector.dot(dx,dy, aox,aoy) > 0 then
		-- simplex = {bx,by, ax,ay}
		simplex[1], simplex[2] = bx,by
		simplex[3], simplex[4] = ax,ay
		simplex[5], simplex[6] = nil, nil
		return simplex, dx,dy
	end

	-- test region 3
	dx,dy = vector.perpendicular(acx,acy)
	if vector.dot(dx,dy, abx,aby) > 0 then
		dx,dy = -dx,-dy
	end
	if vector.dot(dx,dy, aox, aoy) > 0 then
		-- simplex = {cx,cy, ax,ay}
		simplex[3], simplex[4] = ax,ay
		simplex[5], simplex[6] = nil, nil
		return simplex, dx,dy
	end

	-- must be in region 4
	return simplex
end

local function GJK(shape_a, shape_b)
	local ax,ay = support(shape_a, shape_b, 1,0)
	if ax == 0 and ay == 0 then
		-- only true if shape_a and shape_b are touching in a vertex, e.g.
		--  .---                .---.
		--  | A |           .-. | B |   support(A, 1,0)  = x
		--  '---x---.  or  : A :x---'   support(B, -1,0) = x
		--      | B |       `-'         => support(A,B,1,0) = x - x = 0
		--      '---'
		-- Since CircleShape:support(dx,dy) normalizes dx,dy we have to opt
		-- out or the algorithm blows up. In accordance to the cases below
		-- choose to judge this situation as not colliding.
		return false
	end

	local simplex = {ax,ay}
	local n = 2
	local dx,dy = -ax,-ay

	-- first iteration: line case
	ax,ay = support(shape_a, shape_b, dx,dy)
	if vector.dot(ax,ay, dx,dy) <= 0 then
		return false
	end

	simplex[n+1], simplex[n+2] = ax,ay
	simplex, dx, dy = do_line(simplex, dx, dy)
	n = 4

	-- all other iterations must be the triangle case
	while true do
		ax,ay = support(shape_a, shape_b, dx,dy)

		if vector.dot(ax,ay, dx,dy) <= 0 then
			return false
		end

		simplex[n+1], simplex[n+2] = ax,ay
		simplex, dx, dy = do_triangle(simplex, dx,dy)
		n = #simplex

		if n == 6 then
			return true, EPA(shape_a, shape_b, simplex)
		end
	end
end

return GJK