General math

This page lists mathematical functions which are useful in game programming. They are generally short, simple, and optimal.

-- Averages an arbitrary number of angles (in radians).
function math.averageAngles(...)
	local x,y = 0,0
	for i=1,select('#',...) do local a= select(i,...) x, y = x+math.cos(a), y+math.sin(a) end
	return math.atan2(y, x)
end


-- Returns the distance between two points.
function math.dist(x1,y1, x2,y2) return ((x2-x1)^2+(y2-y1)^2)^0.5 end
-- Distance between two 3D points:
function math.dist(x1,y1,z1, x2,y2,z2) return ((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)^0.5 end


-- Returns the angle between two points.
function math.angle(x1,y1, x2,y2) return math.atan2(y2-y1, x2-x1) end


-- Returns the closest multiple of 'size' (defaulting to 10).
function math.multiple(n, size) size = size or 10 return math.round(n/size)*size end


-- Clamps a number to within a certain range.
function math.clamp(low, n, high) return math.min(math.max(low, n), high) end


-- Linear interpolation between two numbers.
function lerp(a,b,t) return (1-t)*a + t*b end
function lerp2(a,b,t) return a+(b-a)*t end

-- Cosine interpolation between two numbers.
function cerp(a,b,t) local f=(1-math.cos(t*math.pi))*.5 return a*(1-f)+b*f end


-- Normalize two numbers.
function math.normalize(x,y) local l=(x*x+y*y)^.5 if l==0 then return 0,0,0 else return x/l,y/l,l end end


-- Returns 'n' rounded to the nearest 'deci'th (defaulting whole numbers).
function math.round(n, deci) deci = 10^(deci or 0) return math.floor(n*deci+.5)/deci end


-- Randomly returns either -1 or 1.
function math.rsign() return math.random(2) == 2 and 1 or -1 end


-- Returns 1 if number is positive, -1 if it's negative, or 0 if it's 0.
function math.sign(n) return n>0 and 1 or n<0 and -1 or 0 end


-- Gives a precise random decimal number given a minimum and maximum
function math.prandom(min, max) return math.random() * (max - min) + min end


-- Checks if two line segments intersect. Line segments are given in form of ({x,y},{x,y}, {x,y},{x,y}).
function checkIntersect(l1p1, l1p2, l2p1, l2p2)
	local function checkDir(pt1, pt2, pt3) return math.sign(((pt2.x-pt1.x)*(pt3.y-pt1.y)) - ((pt3.x-pt1.x)*(pt2.y-pt1.y))) end
	return (checkDir(l1p1,l1p2,l2p1) ~= checkDir(l1p1,l1p2,l2p2)) and (checkDir(l2p1,l2p2,l1p1) ~= checkDir(l2p1,l2p2,l1p2))
end

-- Checks if two lines intersect (or line segments if seg is true)
-- Lines are given as four numbers (two coordinates)
function findIntersect(l1p1x,l1p1y, l1p2x,l1p2y, l2p1x,l2p1y, l2p2x,l2p2y, seg1, seg2)
	local a1,b1,a2,b2 = l1p2y-l1p1y, l1p1x-l1p2x, l2p2y-l2p1y, l2p1x-l2p2x
	local c1,c2 = a1*l1p1x+b1*l1p1y, a2*l2p1x+b2*l2p1y
	local det,x,y = a1*b2 - a2*b1
	if det==0 then return false, "The lines are parallel." end
	x,y = (b2*c1-b1*c2)/det, (a1*c2-a2*c1)/det
	if seg1 or seg2 then
		local min,max = math.min, math.max
		if seg1 and not (min(l1p1x,l1p2x) <= x and x <= max(l1p1x,l1p2x) and min(l1p1y,l1p2y) <= y and y <= max(l1p1y,l1p2y)) or
		   seg2 and not (min(l2p1x,l2p2x) <= x and x <= max(l2p1x,l2p2x) and min(l2p1y,l2p2y) <= y and y <= max(l2p1y,l2p2y)) then
			return false, "The lines don't intersect."
		end
	end
	return x,y
end

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