Can you find a vector not orthogonal to N others?
I was reading something about quiver representations and some proof started with take any vector not orthogonal to any of these N vectors .
lisyarus blog15 Jun 202651 human-writing confidence
lisyarus blog
lisyarus.github.ioadded August 2026
Quivers: a year of linear algebra by drawing arrows
Disclaimer: if your first instinct was to think about string diagrams, that's not was this post is about. Cool guess, though.
lisyarus blog9 Jun 202651 human-writing confidence
Making your own programming language is easier than you think (but also harder)
In mid-December last year I started making my own programming language. It's waaay far from any production quality yet (though I did manage to write a working 1k LOC Monte-Carlo path tracer in it)...
lisyarus blog6 May 202651 human-writing confidence
It's OK to compare floating-points for equality
NB: The title of this post is an intentional clickbait. Even though I do stand for its statement, a more honest one would be something like: It's NOT OK to compare floating-points using epsilons.
lisyarus blog14 Apr 202651 human-writing confidence
A silly diffuse shading model
Happy New Year for those who celebrate! Happy Arbitrarily Chosen Point in Earth's Orbit for those how don't! We all know that the proper lighting factor (i.e.
lisyarus blog31 Dec 202551 human-writing confidence
Particle Life simulation in browser using WebGPU
If you don't care about the explanation and want a direct link to the simulation, here you go. You might know that I'm a sucker for physics simulations, and particle simulations in particular.
lisyarus blog25 May 202551 human-writing confidence
Estimating variance reduction in importance sampling
In my raytracing course at SPbU there's a lecture dedicated to importance sampling (because without it any raytraced scene is either too noisy or complete garbage, depending on the materials used).
lisyarus blog16 Mar 202551 human-writing confidence
Simulating water over terrain
If you don't want a long an boring introduction, skip directly to the virtual pipes method description.
lisyarus blog4 Feb 202551 human-writing confidence
Exploring ways to mipmap alpha-tested textures
In my village building game I'm using alpha-tested transparency for foliage — trees, bushes, grass, etc. This works simply by discarding (i.e.
lisyarus blog5 Jan 202551 human-writing confidence
Implementing a tiny CPU rasterizer | Part 6: Adding some depth
The code for this part is available in this commit. We've fixed most of the notorious problems with 3D rendering and perspective interpolation, but there's still one problem left.
lisyarus blog11 Nov 202451 human-writing confidence
Implementing a tiny CPU rasterizer | Part 4: Changing perspective
The code for this part is available in this commit. So, now we can draw some triangles and even put color gradients on them, but they're all rather flat. We want to go full 3D!
lisyarus blog8 Nov 202451 human-writing confidence
Implementing a tiny CPU rasterizer | Part 5: Fixing issues with 3D
The code for this part is available in this commit. Last time, we got a nice rotating cube: However, if we move the camera inside the cube, it looks horribly wrong: What's going on?
lisyarus blog8 Nov 202451 human-writing confidence
Implementing a tiny CPU rasterizer | Part 3: Interpolating colors
The code for this part is available in this commit. Last time we finally drew a triangle on screen, but it was filled with one fixed color.
lisyarus blog1 Nov 202451 human-writing confidence
Implementing a tiny CPU rasterizer | Part 2: Drawing a triangle
The code for this part is available in this commit. So, in the previous part we set up a test program to draw raw pixels on screen and started making our own tiny graphics API which currently only...
lisyarus blog31 Oct 202451 human-writing confidence
Implementing a tiny CPU rasterizer | Part 1: Clearing the screen
The code for this part is available in this commit. In the OpenGL course I'm teaching at a university, my students learn that the GPU uses something called rasterization to draw stuff on the...
lisyarus blog30 Oct 202451 human-writing confidence
(Yet another) Introduction to quaternions
There are thousands of articles about quaternions on the internet, but my social media said it won't hurt to have another one, so pretend you know nothing about quaternions, and let's roll.
lisyarus blog26 Oct 202451 human-writing confidence
Transforming colors with matrices
There's one trick we use at work, and now I'm using in my current medieval village building game project, which apparently isn't as well-known as I thought: transforming colors using matrices...
lisyarus blog10 Oct 202451 human-writing confidence
Demystifying multiple importance sampling
First post of the year. Hooray! This year had a truly terrible start for me. If you're into rendering, you probably know that most modern photorealistic rendering is based on path tracing, which in...
lisyarus blog25 Apr 202451 human-writing confidence
3D shape matching with quaternions
Several months ago I made a game based on a soft-body physics engine for Ludum Dare 53. It looks like this: I made a detailed blog post on how to implement such a physics engine, and one crucial...
lisyarus blog26 Dec 202351 human-writing confidence
Computing forces in a system of beams, properly
The great thing about posting stuff online is that somebody will inevitably come to tell you where you're wrong :) In my previous post I explored how one can compute the forces and stresses in a...
lisyarus blog29 Oct 202351 human-writing confidence
Computing forces in a system of beams
What I'm describing in this post turned out to be a bad mathematical model, since I've missed one crucial component.
lisyarus blog15 Oct 202351 human-writing confidence
So, you want to make a game engine
Unity's recent controversy sparked a heated debate on game engines. Some said that everyone should immediately switch to a new engine, while others replied that switching engines can take months if...
lisyarus blog15 Sept 202351 human-writing confidence
Quaternion derivatives
This is a follow-up to my earlier post about complex derivatives. I was hoping to use it for something like inverse kinematics, and I got some relatively nice formulas, but it all worked in 2D.
lisyarus blog13 Sept 202351 human-writing confidence
Complex numbers, Wirtinger derivatives and 2D inverse kinematics
Recently, I've been thinking about inverse kinematics for a certain project which I might be doing somewhere in the future.
lisyarus blog3 Aug 202351 human-writing confidence
Porting my C++ game engine to Android
The annual GMTK Jam has just finished, breaking itch.io servers again, as per tradition. I was pretty happy with how I approached the jam this time (rapid fail-early prototyping, reasonable...
lisyarus blog19 Jul 202351 human-writing confidence
Skeletal animation in glTF
Recently (well, a few months ago), I was adding support for animated models in my traffic simulation game.
lisyarus blog3 Jul 202351 human-writing confidence
Uniqueness of matrix inverses
This post is much more math-y than my usual stuff. I just got this in my head for a while and decided to write it up. I even added MathJax to my blog for that!
lisyarus blog21 Jun 202351 human-writing confidence
Making a 2D soft-body physics engine
Recently, I participated in Ludum Dare 53 and made a silly spaceship building game based on a soft-body physics engine.
lisyarus blog10 May 202351 human-writing confidence
How not to design a UI library
This post is a dive into the design of my UI library in my pet C++ game engine, as well as a lot of rants on what didn't work as well as I'd thought it would :) The library itself is located here.
lisyarus blog11 Mar 202351 human-writing confidence
Two-pass Gaussian blur coeffifients generator
This tool generates sample offsets and weights for a two-pass Gaussian blur GLSL shader that uses linear texture filtering to sample two weighted pixels using a single texture read.
lisyarus blog24 Feb 202351 human-writing confidence