Derivatives

Goal

This page will help smooth out your Terrains, if you use mathematical formulae to create them. In ROBLOX, however, we do not work with mathematical "points", but "bricks", which occupy space. Therefore, in order to smooth out terrains and functions, you will want to know the slopes of your functions... to know the slopes of your terrains.

An example of how you can use Derivatives

Introduction

A derivative is the slope of a function, or, phrased differently, the difference in the height divided by the difference in width at that point.

The Ramps article provides a method on how to rotate a brick:

game.Workspace.slope.CFrame = CFrame.new(Vector3.new(0, 100, 0))
                            * CFrame.fromAxisAngle(Vector3.new(0, 0, 1), math.pi / 2)

Here we have the brick being created in its position in the first CFrame, and the rotation in the second CFrame. This is where the derivative comes in. In the script below, z=-x^2. The derivative for this is -2*x. (Why does -x work better than -2*x???)

function f(x)            return -x^2 end
function f_derivative(x) return 2*x end

for x = -1.8, 1.8, .01 do
    local y = f(x)
    local dydx = f_derivative(x)
    local slopeAngle = math.atan(dydx)

    local p = Instance.new('Part')
    p.CFrame = CFrame.new(100 * Vector.new(x, y, 0))
             * CFrame.fromAxisAngle(Vector3.new(0, 0, 1), slopeAngle)
    p.Size = Vector3.new(1, 10, 1)
    p.Anchored = true
    p.BottomSurface = 'Smooth'
    p.TopSurface = 'Smooth'
    p.Parent = game.Workspace
    p.BrickColor = BrickColor.new(217)
end
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You can smooth out trigonometric functions, too.

Sin wave (yes, it's flipped around)

Example

function f(x)            return math.sin(x) end
function f_derivative(x) return math.cos(x) end

for x = -2.5, 2.5, .01 do
    local y = f(x)
    local dydx = f_derivative(x)
    local slopeAngle = math.atan(dydx)

    local p = Instance.new('Part')
    p.CFrame = CFrame.new(Vector3.new(10 * x, 10 * y, 10)) * CFrame.fromAxisAngle(Vector3.new(0, 0, 1), slopeAngle) 

    --cos is the derivative of sin

    p.Size = Vector3.new(1,1,1)
    p.Anchored = true
    p.BottomSurface = 'Smooth'
    p.TopSurface = 'Smooth'
    p.Parent = game.Workspace
    p.BrickColor = BrickColor.new(217)
end

3D Terrain

Smoothed out 3D terrain

This is the sinx+siny formula from Terrain Generation, only smoothed out now. (This needs to be double checked.)

Example

local x, y = 0, 0
for i = -2,2, .1 do
    for j = -2,2, .1 do
        y = math.sin(i) + math.sin(j) -- here is the formula

        local p = Instance.new('Part')
        p.CFrame=CFrame.new(Vector3.new(10 * i, 10 * y, 10 * j)) * CFrame.fromAxisAngle(Vector3.new(0, 0, 1), math.cos(i) + math.cos(j)) 
        p.Size = Vector3.new(1,1,1)
        p.Anchored = true
        p.BottomSurface = "Smooth"
        p.TopSurface = "Smooth"
        p.Parent = game.Workspace
        p.BrickColor = BrickColor.new(217)
    end
end

See Also

• Ramps
• Terrain Generation
• Wikipedia article on Derivatives