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Yeah, I suppose I'm thinking from a projective / homogeneous coordinates perspective, where W is special. Similar things happen in spacetime as well, where nothing is projective but the time coordinate T is special. It seems like the universe really wants things to come in threes.

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So if you have {1, X, Y, Z, W, YZ, ZX, XY, WX, WY, WZ, WYZ, WZX, WXY, XYZ, XYZW} in 4D, then you'd also have {U, XU, YU, ZU, WU, YZU, ZXU, XYU, WXU, WYU, WZU, WYZU, WZXU, WXYU, XYZU, XYZWU} in 5D, where U is your fifth axis direction. The ten bivector elements give you all your 2D planes.

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The way I handle this is to take everything you have in 4D and add a copy in which each element is multiplied by the fifth dimension on the right. (This is discussed on page 183 in PGA Illuminated.)

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This applies to dual quaternion motion operators as well. A positive displacement in a screw transformation goes in the direction of the {WX, WY, WZ} components of the screw axis (which is a bivector line).

I prefer the first one because, when a line is constructed as a bivector by multiplying two points P and Q together, the direction from P to Q will show up on the {WX, WY, WZ} basis elements. If you were to use {XW, YW, ZW}, then the direction would be negated.

As for the planes involving W, you could go either way as long as you're consistent. So it's either {WX, WY, WZ} or {XW, YW, ZW}.

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Hi -- To be consistent with XY and YZ, you'd definitely want to continue using ZX, just as you said for 3D compatibility. That keeps for orientation the same for each of the X, Y, and Z axes, and it gives consistent signs to quaternion components.

Eric Lengyel @ericlengyel.bsky.social · 25/07/2026

The slides from my talks at the Better Software Conference are now available here: terathon.com/lengyel/#spe...

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Eric Lengyel @ericlengyel.bsky.social · 24/07/2026

I just returned home from an amazing week at the Better Software Conference in Sweden. So many great people there, and very impressive content.

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Eric Lengyel @ericlengyel.bsky.social · 12/07/2026

Apparently, there will be a link for the live Twitch stream on the website: bettersoftwareconference.com

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Eric Lengyel @ericlengyel.bsky.social · 11/07/2026

Their channel. I will post links.

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Eric Lengyel @ericlengyel.bsky.social · 11/07/2026

I'm giving two talks at the Better Software Conference in Sweden next week: 1) Software Design in the Radical Pie Equation Editor 2) Math and Container Libraries in the C4 Game Engine Architecture I believe videos will be available, and I will post slides.

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Eric Lengyel @ericlengyel.bsky.social · 10/07/2026

Do you use the equation editor built into Microsoft Word? If so, you can put an end to your suffering by upgrading to Radical Pie. radicalpie.com

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Eric Lengyel @ericlengyel.bsky.social · 07/07/2026

Version 1.12 of the Radical Pie equation editor has been released. This version contains some major new chemistry features that were used to construct the diagrams below. This is once again a free upgrade.

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Eric Lengyel @ericlengyel.bsky.social · 18/06/2026

On the topic of Adobe, I recently reported a bug in Illustrator that appeared in the latest update (version 30.5.1). I got a surprisingly fast response and acknowledgement that they were able to reproduce it. Microsoft should take notes.

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Eric Lengyel @ericlengyel.bsky.social · 18/06/2026

Adobe is now a double-licensee of the Slug Library, having just licensed it for a second software product. Nothing says a customer is happy more than repeat business!

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Eric Lengyel @ericlengyel.bsky.social · 03/06/2026

I'm going to be speaking at the Better Software Conference in Sweden next month. Details to come. bettersoftwareconference.com

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Eric Lengyel @ericlengyel.bsky.social · 28/04/2026

Btw, more about relativistic quaternions here: terathon.com/blog/relativ...

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Eric Lengyel @ericlengyel.bsky.social · 27/04/2026

I have posted the slides from my talk at the Virginia Tech geometry seminar last week: terathon.com/pga_relativi...

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Eric Lengyel @ericlengyel.bsky.social · 21/04/2026

I’m at Virginia Tech again this week to speak in the math career connections series. I’ll be telling students all about what I’ve been doing with my VT math degrees over the past 30 years.

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Eric Lengyel @ericlengyel.bsky.social · 12/04/2026

I posted an unusual entry in my blog that reminisces about my first professional software engineering job before I entered the games industry in the mid-90s. Bespoke Software for Oscillating Bob Viscometry terathon.com/blog/viscome...

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Eric Lengyel @ericlengyel.bsky.social · 12/04/2026

Porting to the Mac (without MS Word integration) would actually be rather straightforward. It's deploying on the Mac that's difficult.

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Eric Lengyel @ericlengyel.bsky.social · 11/04/2026

Several non-Windows users have now told me that they bought a Windows machine just so they could run Radical Pie. Pretty cool! radicalpie.com

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Eric Lengyel @ericlengyel.bsky.social · 02/04/2026

NASA has a live stream of the Artemis II mission that will run for the entire duration: www.youtube.com/watch?v=m3kR...

youtube.com

NASA's Artemis II Live Mission Coverage (Official Broadcast)

YouTube video by NASA

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Eric Lengyel @ericlengyel.bsky.social · 25/03/2026

Yes, it is. Just log into your account, and you can download the new version.

Eric Lengyel @ericlengyel.bsky.social · 25/03/2026

Radical Pie 1.8 was released today, and it includes the ability to parse LaTeX. If you have a big equation already written in LaTeX, then you can just copy/paste the plain text into the editor, and Radical Pie will lay it out visually. radicalpie.com

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Eric Lengyel @ericlengyel.bsky.social · 25/03/2026

Currently working on fur rendering for Chapter 15 in FGED3. Thor, the dog of thunder, is helping with some real-world BRDF references.

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Eric Lengyel @ericlengyel.bsky.social · 17/03/2026

New blog post: A Decade of Slug This talks about the evolution of the Slug font rendering algorithm, and it includes an exciting announcement: The patent has been dedicated to the public domain. terathon.com/blog/decade-...

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Eric Lengyel @ericlengyel.bsky.social · 09/03/2026

The equation editor called Radical Pie that I developed throughout 2025 has been getting a lot of nice feature updates over the four months since its initial release. We're already up to version 1.7. Free trial available on radicalpie.com.

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Eric Lengyel @ericlengyel.bsky.social · 02/03/2026

I've written a new blog post called "Dual Approaches to Projective Geometric Algebra". terathon.com/blog/dual-pg...

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Eric Lengyel @ericlengyel.bsky.social · 27/02/2026

I wrote a short blog post about the special mathematical / scientific font included with the Radical Pie equation editor. terathon.com/blog/radical...

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At some point, you neglect infinitesimals. The notion of equality changes from something exact to something that's infinitely close.

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Then 1 − 1 = 1 + (−1) = 000...001 + 111...111 = 000...000.000...001, which is the smallest representable number. Now let n be arbitrarily large, and this smallest number looks like an infinitesimal. If you want to see something neat, sum 2^n terms of Grandi's series like this.

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Imagine an n.n-bit fixed-point number representation where addition is defined as usual for two's complement values, but any carryout on the left of the whole part wraps around to the far right of the fractional part.

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