Constrained Mesh Enclosure Lab

Watch a surface shrink-wrap itself around a shape it is never allowed to touch.

A familiar physical intuition—wrapping a taut skin around an object—as a live optimization problem you can watch unfold in your browser. A deformable triangulated mesh flows "downhill" on an energy landscape (it wants to minimize its surface area, match a target volume, keep its triangles well-shaped, and so on), while a hard collision-detection wall guarantees it never intrudes into an inner "keep-out" shape. The result is a small, tangible instrument for studying how surfaces settle, and—more subtly—how the choice of optimizer quietly decides which of many equally-good answers you end up with.

This document is a tour for the curious reader, not a developer's manual. If you want the formulas, the derivations, or the file layout, those live in the companion documents (idea.md, math_outline.md, novelty_utility.md, and use_cases.md).

1. The Idea in Plain Terms

Imagine a fixed inner object—call it the keep-out volume, K. Now imagine a stretchy, closed surface draped loosely around it. We give that surface a set of preferences: maybe it wants to be as small as possible (a tight shrink-wrap), maybe it wants to enclose a specific amount of space, maybe it wants its facets to be uniform and its triangles to be pretty. These preferences are encoded as an energy, and the surface continually nudges itself to lower that energy.

The twist is that the surface is under a hard rule, not a gentle suggestion: it may never penetrate the inner object, and (approximately) it may never pass through itself. Rather than adding a "penalty" that merely discourages intrusion, the lab uses continuous collision detection (CCD)—the same family of techniques used to keep cloth and rigid bodies from tunneling through each other in physics simulations—to catch the exact moment any part of the surface would cross the boundary, and clips the motion right there. It is a wall, not a spring.

So the surface behaves like a shrink-wrap that flows freely in open space but slides along the forbidden geometry when it bumps into it, creeping around obstacles rather than stalling against them.

2. Why This Is Interesting

Here is the part that turns a graphics demo into a small research instrument. Several of the surface's "preferences"—edge-length uniformity in particular—have a curious property: many different surfaces satisfy them equally well. When every edge contributes the same amount to the length distribution, the relevant energy term saturates; a whole family of tessellations ties for first place. There is no single winner.

So what actually breaks the tie? It turns out that the optimizer does—the numerical method used to walk downhill. Give the lab the identical starting mesh and the identical energy, then switch from Adam to L-BFGS to QQN, and you will watch it converge on visibly different final surfaces that nonetheless score identically. I find this "fingerprinting" effect genuinely fun to watch: the optimizer's personality is written into the geometry it leaves behind.

That phenomenon is the heart of the lab. It is a clean, reproducible way to see something usually invisible—that the path an optimizer takes, and the way it negotiates the sharp corners of a constrained feasible region (where several collisions happen at once), materially shapes the outcome. This "optimizer fingerprinting" effect is not unique to the mesh; it recurs throughout the sibling geometric attractor experiments — the Geometric Entropy Lab, the Dihedral Attractors lab, and the No-Three-in-Line Lab all exhibit the same phenomenon in different guises. Wherever an energy has a large family of tied optima, the numerical method quietly becomes the selector. The mesh lab is simply the place where the tie-breaking is easiest to watch in three dimensions.

A second, quieter reason it is interesting: because the non-penetration rule is exact rather than a tunable penalty, you can adjust the surface's preferences freely without also having to balance them against a barrier penalty stiffness. The soft goals and the hard constraints stay cleanly separated, which makes the lab a good bench for isolating optimizer behavior from the usual penalty-tuning artifacts. The three optimizers on offer — Adam, L-BFGS, and QQN — are the same trio that appears across these experiments. QQN in particular (the Quadratic-Quasi-Newton method) is documented at length in its own writeup; its defining trait is a curved search path that begins tangent to the safe gradient direction before bending toward the bolder quasi-Newton stride, which gives it a characteristic personality when it meets the sharp corners of a constrained region.

3. A Little Background

None of the individual ingredients here are new, and I want to be honest about that. Continuous collision detection comes from cloth and rigid-body simulation. The surface energies (area, volume, Laplacian smoothing, triangle-quality terms) are textbook geometry processing. Trust-region descent—taking cautious steps where the landscape is stiff and bold strides where it is calm—is classical numerical optimization. Even the entropy-based "many optima tie" flavor is borrowed from a sibling Geometric Entropy Lab.

What is new is the combination: using an exact collision wall to constrain an optimizer (not a physics simulation), and then treating the optimizer's identity as the mechanism that selects among tied solutions. One specific, defensible detail matters here—when a step gets truncated or projected by a collision, the stateful optimizers (L-BFGS, QQN) have their accumulated "memory" of the landscape's curvature reset for the affected vertices, because curvature estimates measured across a discontinuous correction are simply garbage. That reset discipline is the same lesson, transported into a collision setting, that shows up elsewhere in this collection of experiments.

4. Using the Lab

The lab is a single web page. You press Run and watch, or Step through one iteration at a time, or Reset to rebuild the initial shapes. The controls let you conduct your own little experiments:

• Optimizer — switch between Adam, L-BFGS, and QQN. This is the dial to turn if you want to see the fingerprinting effect for yourself.
• Energy weights — a set of sliders (the "λ" values) that set how much the surface cares about area, volume, fidelity to landmark points, edge-length uniformity, triangle quality, and smoothness. There is also a Target Volume to aim for when the volume preference is active.
• δ_safe — how much clearance to keep from the inner keep-out shape.
• Self-collision — toggle an approximate check that keeps the surface from passing through itself.
• Reset optimizer on truncation — the curvature-reset discipline described above (recommended on).
• Retriangulate — a conservative pass that improves the mesh's connectivity, applied only well away from the forbidden geometry.

In the viewport you can drag to orbit around the scene and use the mouse wheel to zoom in and out for a closer look at how the surface meets the constraint.

A panel of live metrics reports what the surface is doing moment to moment: its surface area, enclosed volume, how uniform its edges have become, its triangle-quality and smoothness energies, the smallest gap to the keep-out shape (which should never drop below your clearance setting), how many active contacts are in play, and how many steps have been taken or rejected.

5. Who Might Find This Useful

• The curious and the visual learners. If you have ever wanted to see the difference between a "hard wall" constraint and a "soft penalty," or to understand why an optimizer that remembers curvature can get confused when it slams into a boundary, this lab makes those abstractions concrete and three-dimensional.
• Educators. It is a compact teaching vehicle for several ideas at once: projection versus penalty constraint handling, why stateful optimizers need resets across discontinuities, how continuous collision detection prevents tunneling, and the intuition behind degenerate optima and optimizer fingerprinting—all with a tangible artifact on screen.
• Researchers and practitioners in optimization and geometry. As a controlled instrument, it isolates constrained-descent dynamics from penalty-tuning noise, and it surfaces (deliberately) the stiff multi-contact situations where naive descent plus naive collision response tend to fail.

A candid caveat, in the interest of setting expectations: this is a research and teaching instrument, not a production geometry tool. Dedicated shrink-wrap and remeshing pipelines will produce cleaner results faster for real-world models, and the hand-rolled collision code is tuned for tractability on modest meshes rather than raw scale. What it offers instead is clarity—a place to watch constrained optimization think out loud.

6. Where to Go Next

If this piqued your interest, the companion documents go deeper: idea.md lays out the full concept, math_outline.md collects the formulas and the open analysis questions, and novelty_utility.md and use_cases.md place the lab in context and sketch the scenarios—uniform tessellations, faceted panels, volume-matched enclosures, minimal-area wraps—where each combination of preferences becomes interesting.

I'm looking forward to seeing what people notice when they start switching optimizers and watching the surfaces diverge. Enjoy!

License

Part of the experiments collection. See the repository root for license details.