Eric Lengyel

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

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

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

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.

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.

Yes, there are algebras in which a − a is essentially an infinitesimal and not exactly zero. My favorite is this:

Here, ^★ and _★ are the right and left Hodge duals.

The expression b ∨ (a^★) is the definition of the right contraction, sometimes written b ⌊ a. Likewise, (a_★) ∨ b is the left contraction a ⌋ b.

When a is a vector and b is any multivector, the geometric product a ⟑ b decomposes as

a ⟑ b = a ∧ b + b ∨ (a^★),

and b ⟑ a decomposes as

b ⟑ a = b ∧ a + (a_★) ∨ b.

If a and b are both vectors, these both reduce to

a ⟑ b = a ∧ b + a ⋅ b.

Whoa, the SIGGRAPH exhibit floor is tiny this year. This used to be a huge show! What the heck is going on?
maps.goeshow.com/acm/siggraph...

This might be the first year since 1999 that I don't attend GDC.

This is a reminder that every diagram from my book Projective Geometric Algebra Illuminated is available on Wikimedia Commons under the CC-BY-4.0 license. They can be freely used in slides, papers, books, Wikipedia articles, etc., with proper attribution.
commons.wikimedia.org/wiki/User:Er...

Thanks!!

I wrote a very thorough answer to that question here:
www.amazon.com/dp/B0CXY8C72...

Lengyel's conventions. Fuck yeah.

I feel very honored, and you can imagine the deep sense of satisfaction and justification this provides for me. The images above are from a recent NASA paper and a new textbook in aerospace engineering.

Pose Estimation in the Geometric Algebra G(3,0,1) by Russell Carpenter.

Fundamentals of Spacecraft Optical Navigation by John Christian.

I've worked very hard over many years to develop a complete and correct picture of Grassmann and geometric algebras, often in the face of extreme antagonism. The results of my efforts are now being adopted by prominent scientists and engineers in various fields.

It's got lots of great stuff with two big chapters covering mathematics (including Grassmann / geometric algebra) and another chapter covering things like radiometry, reflectance models, scattering, and ray tracing. There's substantial overlap between this field and computer graphics.

John Christian's new book Fundamentals of Spacecraft Optical Navigation arrived today.

I was recently considering this same question for the purpose of fitting horizontal strokes in a glyph to the pixel grid. I didn't find a satisfying solution.

I added LaTeX output to the latest version of Radical Pie, which was released today. It can't handle everything that Radical Pie can do, but it tries its best to faithfully reproduce equations that can be written in basic LaTeX code without requiring obscure packages.

FGED3 adds five new chapters to the series:

11 • Curves
12 • Motion Operators
13 • Models and Animation
14 • Advanced Shading
15 • Special Effects

I'm shooting for a SIGGRAPH 2026 release.

(The cover is not necessarily final.)

• There is a lot of interest in editing equations with RP and then exporting the LaTeX code.
• The Greek letter in the RP logo is not obvious, but people love it when they see it.

You can download a free trial on the website:
radicalpie.com

Some things I learned when talking to people at JMM this week:
• Most find the Radical Pie UI to be very intuitive.
• A Mac version would be useful even if it doesn't integrate with Word as it does on Windows.
• Nobody likes the Microsoft Office equation editor. And I mean nobody.

I’ve been getting a lot of great reactions from people discovering Radical Pie at the Joint Mathematics Meetings this week.

Don’t forget the green one!

I’m in Washington DC for the Joint Mathematics Meetings. The exhibits open tomorrow, and Radical Pie is right up front.

Tomorrow, I leave for Washington DC where I'll be showing the brand new Radical Pie equation editor at the Joint Mathematics Meetings conference. If you're attending, come check it out at booth #505. I'm also going to have plenty of materials about projective geometric algebra.

Version 1.2 of the Radical Pie equation editor has been released. This version contains some new features and a couple minor bug fixes. This is a free upgrade for all users.
radicalpie.com

Slug version 7.5 has been released. This update includes several small feature improvements.
sluglibrary.com

Radical Pie was designed to handle all the Unicode math symbols really well. This includes things like automatically inserting the right structures into an equation when an n-ary operator or integral symbol is typed.