Polar equations

Roblox operates using the Cartesian coordinate system. This is a gridlike system much like the tiles on the floor of a bathroom. This system is ideal for certain things, e.g., straight lines, roads, buildings, tiles. It is not as good for other things, such as spirals. Another system, called the Polar coordinate system based on an angle and the distance from the center of a map, is much better suited for this.

Introduction

The problem with this working in the Polar system is that you need to convert your results to the Cartesian system, because Roblox operates in it. This is actually easy enough to do, if you have a formula in the Polar system. Let's take the Quadrifolium from Wikipedia. Its polar equation is:

n, d = 2, 1(Quadrifolium)

n, d = 3, 2

Rhodonea curves, with n and d listed.

r = math.cos(2*theta)

So, to obtain the values for x and z, we have to multiply r as follows:

x = r * math.cos(theta)
z = r * math.sin(theta)

We now get:

x=math.cos(2*i) * math.cos(theta)
z=math.cos(2*i) * math.sin(theta)

You will see these formulas again in the script below.

Example

local n=2
local d=1
local k = n/d

for i = 1, 500, 1 do
	local x=math.cos(k*i) * math.cos(i)
	local z=math.cos(k*i) * math.sin(i)

	local p = Instance.new("Part")
	p.CFrame = CFrame.new(Vector3.new(100*x, 100, 100*z))
	p.Size = Vector3.new(8,8,8)
	p.Anchored = true
	p.BottomSurface = "Smooth"
	p.TopSurface = "Smooth"
	p.Parent = game.Workspace
	p.BrickColor = BrickColor.new(217)
end

Other curves can be set by changing the value of n and d, which in turn result in a change in k.

Archimedean spiral

This classic mathematical example has a polar formula of:

r = a + b*theta

Similarly, using these conversion formulas, one arrives at the formulas used in the script further below:

x = r*cos(theta) -- #1
y = r*sin(theta) -- #2

Archimedean spiral

local a = 1
local b = 1

for i = 1, 50, 0.1 do
	local x = (a+b*(i)) * math.cos(i)
	local z = (a+b*(i)) * math.sin(i)

	local p = Instance.new("Part")
	p.CFrame = CFrame.new(Vector3.new(100*x, 100, 100*z))
	p.Size = Vector3.new(8,8,8)
	p.Anchored = true
	p.BottomSurface = "Smooth"
	p.TopSurface = "Smooth"
	p.Parent = game.Workspace
	p.BrickColor = BrickColor.new(217)
end

3-D images

Similarly, 3-D images can be created with a system similar to polar equations, called Spherical coordinates. The formula for a sphere in polar is rho = R. Using the formula to convert Spherical coordinates to Cartesian:

x = r * math.sin(theta) * math.cos(phi)
y = r * math.sin(theta) * math.sin(phi)
z = r * math.cos(theta)

The formula rho = R is obvious below.

Hollow sphere

local R=4

for i = 1,10, 1 do
	for j = 1,10, .1 do
		local x = R*math.sin(i)*math.cos(j)
		local y = R*math.sin(i)*math.sin(j)
		local z = R*math.cos(i)

		local p = Instance.new("Part")
		p.CFrame = CFrame.new(Vector3.new(10*x, 10*y, 10*z))
		p.Size = Vector3.new(8,8,8)
		p.Anchored = true
		p.BottomSurface = "Smooth"
		p.TopSurface = "Smooth"
		p.Parent = game.Workspace
		p.BrickColor = BrickColor.new(217)
	end
end

Ellipsoids

Another example is the ellipsoid, very similar to the sphere above. Really all that's being changed are the radii (a, b, and c).

Spherical formula:

x = a*math.cos(theta)*math.sin(phi)
y = b*math.sin(theta)*math.sin(phi)
z = math.cos(phi)

Conversion:

x = r*math.sin(theta)*math.cos(phi)
y = r*math.sin(theta)*math.sin(phi)
z = r*math.cos(theta)

Cartesian formula:

x = a*math.sin(i)*math.cos(j)
y = b*math.sin(i)*math.sin(j)
z = c*math.cos(i)

a=3, b=10, c=5

Example

local a=3	-- radius
local b=10	-- radius
local c=5	-- radius

for i = 1, 50, 1 do
	for j = 1, 50, 1 do
		local x = a*math.sin(i)*math.cos(j)
		local y = b*math.sin(i)*math.sin(j)
		local z = c*math.cos(i)

		local p = Instance.new("Part")
		p.CFrame = CFrame.new(Vector3.new(30*x, 30*y, 30*z))
		p.Size = Vector3.new(8,8,8)
		p.Anchored = true
		p.BottomSurface = "Smooth"
		p.TopSurface = "Smooth"
		p.Parent = game.Workspace
		p.BrickColor = BrickColor.new(217)
	end
end

By changing the values of a, b, and c, you can come up with different shapes:

a=b=c: Sphere
a=b>c: Oblate spheroid (disk-shaped)
a=b<c: Prolate spheroid (egg-shaped)
a>b>c: Scalene ellipsoid ("three unequal sides")

See also

• The Lua math library
• Wikipedia, Quadrifolium
• Rose curve
• Polar coordinate system
• Spherical coordinate system