How Computers Draw Weird Shapes (Marching Squares)

In this video, we start with an interesting animation of blobby objects which we introduce as metaballs. There's a lot of surprisingly intricate ideas behind making these objects render on a screen. We'll see how folks in computer graphics attempted to solve this problem through a really elegant algorithm called marching squares. Marching squares is a really powerful algorithm that allows you to render any implicit function. But what's even more impressive in my opinion is the many clever shifts in perspective that allowed a vague problems such as this one to be transformed into a clear, well-defined, and solvable problem.

0:00 Introduction
3:29 Circles and Ellipses
4:57 Defining the Problem
6:00 A Guessing Game
8:29 Contours around Two Points
10:35 Sampling The Space
12:32 Breaking Down Cases
15:00 A Clever Optimization
17:20 How Marching Squares Works
18:59 Parallel Marching Squares
20:21 How Do Metaballs Work?
24:59 Marching Cubes
25:58 Some Parting Thoughts

References/Additional Resources:
jamie-wong.com/2014/08/19/metaballs-and-marching-s… - the initial inspiration for the framework of this video, great introduction to metaballs and how they can be rendered using marching squares.

www.geisswerks.com/ryan/BLOBS/blobs.html - great resource on implementing metaballs and some of the physics inspirations behind implicit functions

Original Marching cubes paper: www.researchgate.net/publication/202232897_Marchin…

Sebastian Lague Video on Marching Cubes:
   • Coding Adventure: Marching Cubes  

Further reading on polynomial approximations of metaball implicit functions:
www.researchgate.net/publication/242914163_Data_St…

Implementation Help for Marching Cubes
paulbourke.net/geometry/implicitsurf/

Excellent lecture by Casey Muratori about Marching Cubes:    • "Papers I Have Loved" by Casey Muratori  

courses.lumenlearning.com/physics/chapter/19-4-equ… - some of the physics behind equipotential lines in electric fields

Support: www.patreon.com/reducible
Twitter: twitter.com/Reducible20

This video wouldn't be possible without the open source library manim created by 3blue1brown and maintained by Manim Community.
The Manim Community Developers. (2021). Manim – Mathematical Animation Framework (Version v0.11.0) [Computer software]. www.manim.community/

Here is link to the repository that contains the code used to generate the animations in this video: github.com/nipunramk/Reducible

All music in the video is from Aakash Gandhi