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Introduction

This is an advanced Bézier curves module for Luau, which supports Vector3 curves of any degree. It provides all functions that you need for game development.

Although it encompasses many features, it is very lightweight. It won't waste any calculations on things you don't use.

Installation​

Get the module here and insert into your game via the Toolbox. You can also just paste this in the command bar:

game:GetObjects("rbxassetid://11467361559")[1].Parent=game.ReplicatedStorage

How to create a Bézier curve​

There are multiple constructors:

Bezier.new is the default one, which simply requires an array of Vector3 points.

Bezier.fromParts can be used to create a curve from a folder of Parts. The parts must have a number as their name (1, 2, 3, ...) so the order of the points is clear.

Bezier.random allows you to generate a random curve based on the generation settings.

local Bezier = require(game.ReplicatedStorage.Bezier)

local curve = Bezier.fromParts(workspace.BezierCurve)

Get a point on the curve​

tip

For testing and visualization purposes, you can get my visualization module here.

The recommended way to calculate a point on the curve is BezierCurve:DeCasteljau (named after De Casteljau's algorithm).

It takes a t value of 0 to 1 as input, where 0 is the first point of the curve and 1 the last.

You can visualize the curve with this code:

local Bezier = require(game.ReplicatedStorage.Bezier)
local curve = Bezier.fromParts(workspace.BezierCurve)

local function createPoint(pos: Vector3, color: Color3?)
    local part = Instance.new("Part")
    part.Anchored = true
    part.Color = color or Color3.fromRGB(170, 0, 0)
    part.Shape = Enum.PartType.Ball
    part.Position = pos
    part.Size = Vector3.new(1, 1, 1)
    part.Name = "Point"
    part.TopSurface = Enum.SurfaceType.Smooth
    part.BottomSurface = Enum.SurfaceType.Smooth
    part.Parent = workspace
end

local iterations = 50
for i = 0, iterations do
    local t = i / iterations
    createPoint(curve:DeCasteljau(t))
end

Arc Length Parameterization​

If you visualize the path of the curve with the code above, you will notice that the points don't have an equal distance to each other. They are not evenly distributed although the points are spread at equal intervals.

This means that t = 0.5 is usually not at 50% of the Bézier curve's length.

This can be solved approximately using arc length parameterization. I created the function ConvertT, which takes a length as input and returns the time t at which this length occurs.

Now we can distribute the points evenly along the curve:

local iterations = 50
for i = 0, iterations do
    local t = i / iterations
    local convertedT = curve:ConvertT(t)
    createPoint(curve:DeCasteljau(convertedT))
end

Continue to the API section here!

Things to consider​

  • There are no checks of the arguments you pass in functions. You should look at the type annotations and understand what the function does. Furthermore, you can look at the source code.
  • The use of GetExtrema requires the installation of a complex numbers module, more on that in the API section.
  • It is always possible to modify the control points. However, if you use the functions ConvertT, GetNormal or the property Length, you have to call BezierCurve:UpdateLUT after the modifications. This is due to the fact that the internal lookup tables won't be accurate anymore.