Bezier.lua

-- Smooth Interpolation Curve Generator
-- @author Validark
-- @original https://github.com/gre/bezier-easing
-- 		Copyright (c) 2014 Gaëtan Renaudeau, MIT License (see bottom for full license)
-- @testsite http://cubic-bezier.com/
-- @testsite http://greweb.me/bezier-easing-editor/example/

-- Bezier.new(x1, y1, x2, y2)
-- @param numbers (x1, y1, x2, y2) The control points of your curve
-- @returns function(t [b, c, d])
--	@param number t the time elapsed [0, d]
--	@param number b beginning value being interpolated (default = 0)
--	@param number c change in value being interpolated (equivalent to: ending - beginning) (default = 1)
--	@param number d duration interpolation is occurring over (default = 1)

-- These values are established by empiricism with tests (tradeoff: performance VS precision)
local NEWTON_ITERATIONS = 4
local NEWTON_MIN_SLOPE = 0.001
local SUBDIVISION_PRECISION = 0.0000001
local SUBDIVISION_MAX_ITERATIONS = 10
local K_SPLINE_TABLE_SIZE = 11

local K_SAMPLE_STEP_SIZE = 1 / (K_SPLINE_TABLE_SIZE - 1)

local Resources = require(game:GetService("ReplicatedStorage"):WaitForChild("Resources"))
local Table = Resources:LoadLibrary("Table")
local Debug = Resources:LoadLibrary("Debug")

local function Linear(t, b, c, d)
	return (c or 1)*t / (d or 1) + (b or 0)
end

local Bezier = {}

function Bezier.new(x1, y1, x2, y2)
	if not (x1 and y1 and x2 and y2) then Debug.Error("Need 4 numbers to construct a Bezier curve") end
	if not (0 <= x1 and x1 <= 1 and 0 <= x2 and x2 <= 1) then Debug.Error("The x values must be within range [0, 1]") end

	if x1 == y1 and x2 == y2 then
		return Linear
	end

	-- Precompute redundant values
	local e, f = 3*x1, 3*x2
	local g, h, i = 1 - f + e, f - 2*e, 3*(1 - f + e)
	local j, k = 2*h, 3*y1
	local l, m = 1 - 3*y2 + k, 3*y2 - 2*k

	-- Precompute samples table
	local SampleValues = {}
	for a = 0, K_SPLINE_TABLE_SIZE - 1 do
		local z = a*K_SAMPLE_STEP_SIZE
		SampleValues[a] = ((g*z + h)*z + e)*z -- CalcBezier
	end

	return function(t, b, c, d)
		if d ~= nil then t = t / d end
		t = math.clamp(t, 0, 1) -- Make sure the endpoints are correct

		local CurrentSample = K_SPLINE_TABLE_SIZE - 2

		for a = 1, CurrentSample do
			if SampleValues[a] > t then
				CurrentSample = a - 1
				break
			end
		end

		-- Interpolate to provide an initial guess for t
		local IntervalStart = CurrentSample*K_SAMPLE_STEP_SIZE
		local GuessForT = IntervalStart + K_SAMPLE_STEP_SIZE*(t - SampleValues[CurrentSample]) / (SampleValues[CurrentSample + 1] - SampleValues[CurrentSample])
		local InitialSlope = (i*GuessForT + j)*GuessForT + e

		if InitialSlope >= NEWTON_MIN_SLOPE then
			for NewtonRaphsonIterate = 1, NEWTON_ITERATIONS do
				local CurrentSlope = (i*GuessForT + j)*GuessForT + e
				if CurrentSlope == 0 then break end
				GuessForT = GuessForT - (((g*GuessForT + h)*GuessForT + e)*GuessForT - t) / CurrentSlope
			end
		elseif InitialSlope ~= 0 then
			local IntervalStep = IntervalStart + K_SAMPLE_STEP_SIZE

			for BinarySubdivide = 1, SUBDIVISION_MAX_ITERATIONS do
				GuessForT = IntervalStart + 0.5*(IntervalStep - IntervalStart)
				local BezierCalculation = ((g*GuessForT + h)*GuessForT + e)*GuessForT - t

				if BezierCalculation > 0 then
					IntervalStep = GuessForT
				else
					IntervalStart = GuessForT
					BezierCalculation = -BezierCalculation
				end

				if BezierCalculation <= SUBDIVISION_PRECISION then break end
			end
		end

		t = ((l*GuessForT + m)*GuessForT + k)*GuessForT
		return (c or 1)*t + (b or 0)
	end
end

return Table.Lock(Bezier)

--[[
Copyright (c) 2014 Gaëtan Renaudeau

Permission is hereby granted, free of charge, to any person
obtaining a copy of this software and associated documentation
files (the "Software"), to deal in the Software without
restriction, including without limitation the rights to use,
copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following
conditions:

The above copyright notice and this permission notice shall be
included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
OTHER DEALINGS IN THE SOFTWARE.
--]]