[Guide] Digital Geometry

Discussion in 'General Minecraft Discussion' started by ForeverMaster, Nov 28, 2013.

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After reading this guide, do you want to see more in the future?

 Yes. No. 8 vote(s) 100.0% 0 vote(s) 0.0%
1. Over this year, I've been researching about building beyond common effortlessness in most people's Minecraft builds, like flatness, plainness, etc. I had started a thread about advanced building in Minecraft, but I have now realized that if someone new to professional building watches a video, like from some of the YouTube channels that I had listed, the result could be plagiarism. If that happens, they won't or barely gain any useful knowledge for when they want to build something unique. So, I have typed out a guide for one of the most important parts of a building: its shape(s). I hope you find this useful!

Introduction
The real world is filled with all kind of shapes, but when comes to the digital world creating them is different than simply drawing it out. All of the images and textures are made out of squares called 'pixels'. Although, the question is: 'how do expert Minecraft builders and graphics designers make shapes through an array of squares?' This question has multiple answers to it. Note that this guide doesn't include every single answer.

Creating Angles

In a digital image, an angle is represented as a slope which is a straight line. Using the formula of slope, you can find out how many identical or similar pixels/Minecraft blocks are in each column or row. The formula for slope is as follows:

m(Slope)= y2-y1(Rise)
x2-x1(Run)

Example: When y2= 2, y1= 1, x2= 2, and x1= 1.

2-1= 1 m= 1
2-1__1

Since the slope equals 1, this mean that to form the line, each square has to be right 1 and up 1. If the slope was negative, the direction of the line would be reverse for the respective axis.

Curves

In theory, circles have an indefinite number of sides, and spheres have an indefinite number of faces, edges, and vertices, but they're limited by image resolution digitaly. They're made out of curves which can be made by using different angles. A common way to make a curve is to start with a single line and making another one that isn't longer than the your starting point. But, if you were to go from a 3-pixel line to a 5-line, your curve is actually flowing in another direction which the wouldn't want to have if you want just a single curve.

For example: From a 5-pixel tall line that passes through the ordered pairs (1,1) and (1,5), you want to create a curve that flows into a 5-pixel long angle running through the ordered pairs (6, 10) and (11, 11). *Prepared lines as shown at a 1:8 pixel scale.
In this situation, the horizontal line is 6 pixels to the right and 6 pixels up from the vertical line making a 5*5 area inbetween the 2 lines. In this open space, the curve would go inside. With the amount of space, you can make a total of 2 different lines. Here's how I might form the curve.
Coming from above the vertical line, I would place a 2-pixel high line 1 pixel to the right, a single pixel diagonally from the 2-high line, and finally a 2-long line placed diagonally. *Finished Curve at a 1:8 pixel scale. *Same as before but at a 1:1 scale.
Conclusion
By following the concepts I have explained, I hope you'll be able to start planning out shapes of your builds and actually build them without having to copy from a separate image. Because of the lack of images I was able to provide, images from any of you showing that you understand what I covered would be helpful to me.

If you want to see more of these guides, cast your vote above this post. I won't be able to explain every detail of information about digital building because I'm still learning.
ToriDesu likes this.
2. LOL What you say is very true, it's how I've come to build smooth shapes.

However, it's also extremely technical. I guess directed at highly technical people? In which case they probably already know this. I learned it simply by drawing lines/curves on grid paper and seeing how the lines cut the squares, and building accordingly.

I don't think most people will understand what you wrote at all. You should include images of what you are talking about, it'll give it much more understanding. I'll post a couple shortly..

EDIT: I was going to add in some quick images using google docs to draw shapes, but couldn't figure out how to paint in parts of a grid, so I won't do it now. maybe later if i find a program to do it.
3. Want to teach my geometry class? PenguinDJ and maxthegreat2 like this.
4. I don't think I'll be able to do that. The explaination above took me the entire afternoon to type right. 5. The top post has been updated. I have some sample images there that should make it easier to understand curves.
6. Thread bump. Maybe this will give more attention.
7. Nice tutorial. Another nice formula for making arches would be the quadratic one. Useful for large arches so you don't have to guess and check.
Formula that is easiest to adjust for your needs: a(x-h)^2+k
Where (h,k) is the highest or lowest point of the arch.
cddm95ace likes this.
8. I'm having some confusion understand this. Could you tell/show me an example using my sample arch from the top post?
9. Alright, I would love to read this and I bet this will make me a better builder it's just that... I am very lazy.
10. Ignore what I said about the special form. I found a better way.

First you have the quadratic equation:
a*(x^2+bx+c)
This may look a bit different then the actual one but you will see why soon.

You need to know where you want the arch to touch the base or ground. The left side of the arch should be block 0 and the right side is however far you want it to be. You will also need to know how high you want the arch.

Next step is foiling the roots (where the arch touches the ground)
It should look like:
x^2-rx
With r being the right point where it touches the ground.
Now for the last step. After plugging that into the quadratic equation above you should have
a*(x^2-bx)
All I did was switch the r to a b and add the a* to it. Find the right-most root and divide it by 2. Make this equal m.
Make the max height you choose equal h
It should look like:
h=a*(m^2-bm)
When you are ctaully fill in numbers the only variable will be a. Solve for a by getting it on a side by itself (simplify the parenthesis and divide h by it what is in the parenthesis.

Simple multiply everything by a and put the x back in. Just get a table and place blocks to mimic the graph.

I hope this helps.