Derivatives

Introduction

This tutorial is intended to introduce you to the mathematical notion of derivatives.

How will this page help me

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

What is a derivative?

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, which you see in the script below.

x=0
z=0
for i = -2.5,2.5, .01 do
x=i
z=-x^2

p = Instance.new("Part")
p.CFrame=CFrame.new(Vector3.new(100*x+100, 100*z+100, 100))*CFrame.fromAxisAngle(Vector3.new(0,0,1), -2*x)
 
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

Now, because of the holes in the graph, towards the bottom, the graph isn't going to be as smooth. The more bricks you have, the smoother your graph, but the more lag you have.

File:Infinitelysmallbricks.PNG

this is the same graph as above, but double the bricks

x=0
z=0
for i = -2.0,2.0, .005 do
x=i
z=-x^2

p = Instance.new("Part")
p.CFrame=CFrame.new(Vector3.new(100*x+100, 100*z+100, 100))*CFrame.fromAxisAngle(Vector3.new(0,0,1), -2*x)
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

What about trigonometric functions?

Yes, you can smooth out trigonometric functions, too.

Sin wave (yes, it's flipped around)

x=0
z=0
for i = -2.5,2.5, .01 do
x=i
y=math.sin(i) -- here is the formula

p = Instance.new("Part")
p.CFrame=CFrame.new(Vector3.new(10*x, 10*y, 10))*CFrame.fromAxisAngle(Vector3.new(0,0,1), math.cos(i)) 
--[[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.)

x=0
z=0
for i = -2,2, .1 do
for j = -2,2, .1 do

y=math.sin(i)+math.sin(j) -- here is the formula

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