## Unit Gradient Fields: The Two-Body Field, Ξ

So far in the series, we’ve defined the basic idea that UGFs generalize SDFs and examined that when representing shapes, UGFs offer design freedom in the shapes’ normal cones. In most of the examples, we’ve shown that this freedom helps recapitulate the kinds of edge treatments we see in engineering software like rolling-ball blends and chamfers. In this post, we’ll take a look at the clearance and midsurface fields that apply to SDFs and the two-body field that applies to all UGFs.

*Use the slider to change viewing modes:*

*The clearance field, \(\ugf{A} + \ugf{B}\,\), the midsurface field, \(\ugf{A} - \ugf{B}\,\), and the two-body field: \(\twobody{\ugf{A}}{\ugf{B}} \equiv \frac{\ugf{A} - \ugf{B}}{\ugf{A} + \ugf{B}}\). The clearance and midsurface fields are overlaid to demonstrate their orthogonality.*

## Gradient Control Laboratories and LatticeRobot

With the UGF research entering a public phase, some other collaborations from the background are entering the foreground. In particular, some investigations with some friends have evolved into two new entities: an incubator, Gradient Control Laboratories and its first spinoff, LatticeRobot!

### Media coverage from CDFAM ‘23

3DPrint.com: LatticeRobot Launches a Home for Lattices, Metamaterials, and Textures

Develop3D: LatticeRobot announces community for advancing lattices in products

TCT: LatticeRobot launches engineering community for lattice research and knowledge share

## Unit Gradient Fields: The Promise of Unbreakable Geometry

Something seemed different about Sarah Frisken and Ron Perry, researchers from MERL who had ventured out to Boston’s Route 128 tech corridor to present their new modeling technology to the top geometry engineers at a leading CAD company. Frisken and Perry demonstrated that their *adaptively sampled distance fields* (“ADFs”) encoded geometry in a way where the offsets, Booleans, and rounded blends that confounded contemporaneous CAD systems like ours, would always succeed. They demonstrated organic texturing and lattices that would be unthinkable on state-of-the-art boundary representation (B-rep) solids. On the other hand, the modeling operations seemed limited to CSG operations, which had been superseded in mechanical CAD by the more expressive B-reps, and their only practical output was meshed geometry, considered inferior to B-reps. Would it be possible to combine the benefits of B-rep modeling with robust offsets, Booleans, and blends? It was 2001, I’d never seen anything like it, and I was hooked.

## Unit Gradient Fields: SDFs, UGFs, and their friends

Many readers of the last post requested a more formal definition of a UGF. Let’s look a bit more closely at the definition of an SDF and how it compares to UGFs and other useful fields in engineering applications. Some readers may find the visual concepts more intuitive than the nuances, so let’s get a feel for the territory first by examining the field at the intersection of two planes:

## Unit Gradient Fields: What do we mean by "offset"?

For those of us who work in engineering and geometric modeling, “offset” is an everyday operation. We use it in 2D and 3D to produce curves and surfaces at constant distance from other curves and surfaces. With experience, we learn that offset can be failure-prone, especially with precise B-rep solids and meshes. Implicit modeling, in particular, the signed distance field (SDF) representation of shapes, offers robust offsetting, but again, with experience, we learn that the results aren’t always what we expect.

Take these three examples of an offset rectangle, created using three different “line joining” approaches that date back to the early days of 2D graphics and are built into your browser:

## Case Study: Hyperbolic multiscale lattices for the entangled lifestyle

## 1 April 2023 (Fiction)

As mentioned yesterday, hyperbolic space is more spatially dense than Euclidean space, and therefore offers opportunities for higher performance and fidelity in engineering applications. In this case study, we’ll examine how to prepare ordinary Euclidean CAD and mesh geometry for embedding in hyperbolic space and manufacture in QE3D’s quantum entanglement production system.

### Triangles in hyperbolic space

The key unit in any structural design, including beam lattices, is a triangle. In Euclidean space, the sum of the angles of set of triangles around a vertex must total 360°. In hyperbolic space, we can increase that total angle to any number we want, even ∞!

## Hyperbolic CAD for High-Performance Engineering

## 1 April 2023 (Fiction)

As the cost of fabricating high-fidelity, quantum-collapsed geometry becomes increasingly prevalent, as an industry, we’re forced to confront the following challenge: where are we going to put all of our crap? At QE3D, we’re hard at work to collapse wave functions in our commitment to enable more engineering design space.

You may wonder: how does the superposition of quantum entanglement and machine learning scintillate more room for our everyday carry? Indeed, our technology achieves for consumer products, implantable electronics, wearable devices, and sub-dermal surveillance exactly the same advantages parachute pants achieved for break dancers. With the supremacy of the mesoscale fully realized via TPMS, spinodal decomposition, and mixed topology lattices, from what extra space might we draw additional engineering acumen?

At QE3D, we manifest our quantum technology through three regimes for AI-driven, dimensionality enhanced, spatial domains.

## Knowing versus experiencing math

On the Geometric Processing Worldwide Discord server, a frequent participant was lamenting the challenge of absorbing all the great work going on in computational geometry. It caused me to consider my own challenges with learning math.

For a long time I wanted to know math. I thought that I could learn what was out there but by glossing over the text and seeing the ideas, maintaining some sort of mental index to the math I might someday need to use. I would assume that the author’s introductory instructions to do the exercises didn’t apply to me. I usually only made it a few chapters into such texts.

41 post articles, 6 pages.