Calculates the arc length of a quadratic Bezier curve from 0 to t

quadraticBezierLength.lua

local abs = math.abs;
local log = math.log;
local sqrt = math.sqrt;

-- calculates length from 0 to t (where t is a percentage eg 1 = 100%).
local function quadraticBezierLength(t, p0, p1, p2)
	local a = p0 - 2*p1 + p2;
	local b = 2*(p1 - p0);
	local A = 4*(a:Dot(a));
	local B = 4*(a:Dot(b));
	local C = b:Dot(b);
	
	local q = B/(2*A);
	local r = sqrt(C/A - q*q);
	
	local tan0 = (0+q)/r;
	local cos0 = r/sqrt((0+q)^2 + r*r);
	local inner0 = tan0/cos0 + log(abs(tan0 + 1/cos0));
	
	local tant = (t+q)/r;
	local cost = r/sqrt((t+q)^2 + r*r);
	local innert = tant/cost + log(abs(tant + 1/cost));
	
	return r*r*sqrt(A)*0.5*(innert - inner0);
end;