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[Shurikane]

tl;dr: Somehow I don't think I've thought my cunning plan all the way through.

My current fortress is called Project Superbowl II.  I shall surround a pipe volcano with an artificial, circular bowl.

And I thought "why not make the bowl into a Division By Zero shape?"

Which is all fine and good, except I lack a blueprint. 

I could just wing it, but I figured it would be prudent to crowdsource the design a little so I don't screw it up.

So.  Here's what I know:

• From the center, the distance to the closest edge of the pipe is 9 squares, diagonally.  Therefore, the very bottom of my bell-mouth will have a radius of 9.
• I have 42 Z-levels of space to the top of the map.  This is shorter than the distance to the map edges, thus will be my limiting factor in how large the bell-mouth is.  I need one Z-level of leeway, as the very top will house some structures that I want to keep covered and sealed - thus, walls and top-flooring that's accessible by the dwarves to build.

I am not entirely certain how to approach the architecture of this, mathematically speaking.  Ideally, I'll have a plan for each Z-level so as to place my walls and my ramps at the right spot in order to have a structure that looks nice and remains as circular as a 3D-pixelated thing can be.  Perhaps drunkenness is speaking in my stead but anyway.  My geometry skills are lacking.  What do?

[Pseudo]

Related:



Unfortunately, that doesn't give the actual profile. Nonetheless, still useful.

What I'd do is the following:

Construct a function that gives the radius given the input z-level. Then draw a circle at each z-level with the given radius. You may need to add extra walls to connect z-levels. Not perfect, but meh.

Some thoughts:

What radius do you want the top to be? I'd think 8.35 / 2.35 * 9 ~= 32 blocks, but you may not have that much space...

[Shurikane]

Knowing that a circle is...

x² + y² = r²

And that r will increase faster and faster as I go up in Z-levels.  After some googling, a Gaussian function will do nicely.  The textbook "bell curve" function.


Let's focus on the radius only, putting what's above aside for the moment.  Since the radius will increase as per a Gaussian function, the formula of which shall be:

r = a^{(-((Z-b)2/(2c2)))}

Solving for Z-level #1, I get:

r = a^{(-((1-b)2/(2c2)))} ≈ 9

Solving at Z-level #40, I'll get:

r = a^{(-((40-b)2/(2c2)))} ≈ 50..60

I plugged between 50 and 60 as my desired final top radius.  Following that, I went futzed around with the coefficients a, b and c to obtain a curve that would meet the two end points nicely.  Furthermore, at the end, I round the result to the nearest integer.

After some goofing around in Excel, I obtained this:

r = 0.1^{(-((Z-4)2/(2*202)))}

Graphed, this gives me a beautiful, sharp curve.  To increase its sharpness, increase b and decrease c.

This should be even better:

r = 0.1^{(-((Z-10.1)2/(2*162)))}

Fantastic!  At Z-level 40, rounded, I end up with a circle of gloooooorious radius 65.  I surveyed the terrain and this'll mean my bell-mouth will finish right above my trade depot, giving the impression to incoming traders that the fortress IS the bell-mouth.


Let's plug this all back into my original circle formula:

x² + y² = (0.1^{(-((Z-4)2/(2*202)))})²

With each Z-level, starting at 1, my rounded radius will be:

10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
11
11
11
11
11
12
12
13
13
14
15
16
18
20
22
25
29
35
42
52
65

Easy peasy.

And now, I think I'm ready to build that thing!

[contheman]

Do post some pictures when you finish this Dwarven-American master piece. I always knew that the Super Bowl was lacking flowing lava.

[Torrenal]

Do post some pictures when you finish this Dwarven-American master piece. I always knew that the Super Bowl was lacking flowing lava.

So uh,  do shoulder pads and helmets help you swim in lava any better?

[contheman]

Do post some pictures when you finish this Dwarven-American master piece. I always knew that the Super Bowl was lacking flowing lava.

So uh,  do shoulder pads and helmets help you swim in lava any better?

If I said yes, would you Photoshop it and post it on this page?

[Pseudo]

Shurikane, that looks a bit... Off... to me.



For comparison:



An "actual" spillway doesn't flare nearly as much.

[Shurikane]

Do you know what the position of the top of the black curve is?  I'm eyeballing it at about 30 but I was curious to know which equation you used to get the graph.

Looking at it with a fresh head, I realize that having over 10 tiles of radius difference at the top will look great but be functionally difficult to work with.

[Pseudo]

That was just ripped straight from the image I posted (or rather, cleaning up the profile a bit.)

Give me a minute and I'll see if I can get some numbers.