## Wedge Products, Dot Products, Scalars, and Norms

I recently posted a couple of articles talking about what we should do when we take a wedge product or dot product and one or both of the operands is a scalar. Those posts were motivated by a quest for a fundamental “geometric property” shared by the meaningful objects in projective geometric algebra (PGA), and […]

## So You Want to Write a Technical Book

If you’re knowledgeable in a technical field, writing a book to teach others a few things can be a rewarding experience on many different levels. With the many avenues available for self-publishing these days, an important question to ask yourself is “Do I need a publisher?”. The answer depends on your particular situation and your […]

## Symmetries in Projective Geometric Algebra

This is a continuation of the post Projective Geometric Algebra Done Right from earlier this year. That post introduces the geometric antiproduct and the notation ⟑ and ⟇ used in this post. As before, everything in this post assumes familiarity with Chapter 4 of FGED1. A large two-page reference poster containing a ton of information […]

## Projective Geometric Algebra Done Right

[This is part one of a two-part series. The second part can be found here: Symmetries in Projective Geometric Algebra.] In the last chapter of FGED1, I provided a practical introduction to projective Grassmann algebra. Most people working in computer graphics are familiar with four-dimensional homogeneous coordinates, and projective Grassmann algebra extends this concept to […]

## RGB/XYZ Conversion Matrix Accuracy

The second volume of the Foundations of Game Engine Development series (FGED2) contains a derivation of the 3×3 matrices that transform between the standard RGB color space and the CIE XYZ color space. The resulting matrix entries that I published are a little different from the numbers published in both Real-Time Rendering, 3rd edition (RTR3) […]

## Dynamic Glyph Dilation

Slug renders glyphs on the GPU directly from the Bézier curves that define them. No matter how complex a glyph is, only a bounding box consisting of two triangles (or a tighter bounding polygon with up to eight vertices and six triangles) is rendered by the GPU, and the pixel shader figures out how much […]

## A Mathematical Keyboard Layout

I’ve often found myself typing basic mathematical symbols into Twitter comments, such as subscripts, exponents, infinity, the dot product, the cross product, and especially the minus sign, because I can’t stand using a hyphen to mean negative or subtract. I had enabled hex Unicode input on the numeric keypad, which requires a registry change, and […]

## Some Thoughts about Rvalue References in C++

C++ has always had these things called lvalues and rvalues that refer to expressions that can typically appear on the left or right side of an assignment. In C++11, rvalues were divided into two groups called xvalues (“expiring” rvalues) and prvalues (“pure” rvalues) so that the mess now known as rvalue references could be implemented […]

## Some Thoughts about Aliasing in C++

The C++ standard explicitly states when pointers can alias at the end of Section 3.10. It does not allow simple things like this: float f;…int i = *reinterpret_cast<int *>(&f); Pointers to int and float are not related in a way that supports aliasing, and this is technically undefined behavior. At the end of Section 5.2.10, […]

## Thoughts About the ZP+ Alorithm

A couple of months ago, a paper entitled ZP+: Correct Z-pass Stencil Shadows came across my inbox. I gave it a cursory scan, but nothing earth-shattering jumped out at me, so I didn’t give it much more thought. Recently, the paper has been brought to my attention several more times, so I decided to take […]