The basic problem? Given a collection of points, curves, or surfaces in space, find a well-distributed arrangement that avoids (self-)intersection.
To make things interesting, you also usually have some constraints—like: the geometry is contained inside a bunny!
(3/14)
Why would you want to do this?
Well, *point* repulsion is already used in every major area of computer graphics: from image stippling to mesh generation to density estimation to fluid simulation!
But surprisingly little work has been done on repulsive curves & surfaces… (4/14)
And yet there are *all sorts* of things you can do with higher-dimensional repulsion—from better graph layouts, to generative modeling, to robotic path planning, to intersection-free modeling & illustration, to artificial tissue design, to unraveling mathematical mysteries!(5/14)
Repulsion goes far beyond (literally) traditional collision-aware design & modeling: rather than slamming on the brakes right before the moment of impact, repulsive optimization is all about finding a harmonious *global* balance of forces, long before collisions occur. (6/14)
Imagine, for instance, playing a game of pool with charged particles rather than rigid bodies.
In contrast to collision, where you can aggressively prune away distant barrier functions, the whole point of repulsive optimization is to get long-range forces into equilibrium (7/14)
Beyond just dealing with O(n²) interactions, repulsive energies lead to some rich & interesting challenges. For one thing, repulsive energies for points don't nicely generalize to curves & surfaces. For another, you have to deal w/ optimization of "fractional derivatives." (8/14)
…Rather than regurgitate everything here on Twitter, I'll point again to this talk:
Since I know it's easy to get lost in the math & switch off the video, I gave each section a level of difficulty. Don't feel bad about skipping to the fun stuff! (9/14)
You might also be interested in checking out this longer Twitter thread, which also goes deeper into the details for curves:
But the generalization to repulsive surfaces lets us do some really beautiful things like never before. (10/14)
To end with just one cool example, consider this pair of handcuffs in a "linked" and "unlinked" configuration.
Do you think it's possible to unlink the handcuffs, without breaking them apart or letting them pass through each other? (11/14)
If you said "yes," you were right (hey, you had a 50% chance!)
But can you show me how?
There are lots of beautiful drawings of this counter-intuitive transition—but unless you have a good geometric imagination, they can be pretty hard to follow. (12/14).
However, we can use repulsive optimization to make this remarkable topological phenomenon come to life.
Just minimize tangent-point energy starting in both linked and unlinked configurations, then join the two movies together (the 2nd one playing backwards). Voilà! (13/14)
The paper/videos have lots of other examples of new & different things you can do with repulsive shape optimization.
But the real limitation is *our* own creativity!
I'd love to hear what ideas this sparks for you, and what problems, challenges, & creations it leads to…(14/14)
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I realized this can be coded up in < 5 minutes in the browser via @UsePenrose, and gave it a try. Pretty darn close! 🧵
[2/n] To be clear, this 🧵 isn't about finding a better packing—or even finding it faster. Wizards like @KangarooPhysics surely have better tricks up their sleeves 🪄
Instead, it's about an awesome *tool* for quickly whipping up constraint-based graphics: penrose.cs.cmu.edu
[3/n] The "17 squares" problem provides a great demonstration of how Penrose works.
In fact, if you want to use this thread as a mini-tutorial, you can try it out at penrose.cs.cmu.edu/try/
Let's find out by trying to re-create a bunch of classic graphics images using #dalle2! A thread. 🧵 [1/n]
Left: original image
Right: DALL-E 2 image
In each case I tried many times & show the best result. Full query string given.
Let's start with a real classic: a chrome and glass ball over a checkerboard, from Turner Whitted's 1980 paper, "An Improved Illumination Model for Shaded Display": cs.drexel.edu/~david/Classes…
Pretty good job on reflection/refraction! Was hard to get the colors I wanted. [2/n]
Two more classics for the price of one: the Stanford Bunny in a Cornell Box.
Models in engineering & science have *way* more complexity in geometry/materials than what conventional solvers can handle.
But imagine if simulation was like Monte Carlo rendering: just load up a complex model and hit "go"; don't worry about meshing, basis functions, etc. [1/n]
Here's one fun example: heat radiating off of infinitely many aperiodically-arranged black body emitters, each with super-detailed geometry, and super-detailed material coefficients.
From this view alone, the *boundary* meshes have ~600M vertices. Try doing that with FEM! [3/n]
Here's another fun question: given two loops around an (infinite) pole, can you remove one loop without breaking it?
Amazingly enough... yes!
This is a surprising example of what's called an "ambient isotopy": a continuous deformation of space taking one shape to another. 1/n
People have made some great drawings of this transformation over the years (sometimes using a loop rather than an infinite pole—which is equivalent), but it can still be hard to interpolate between individual drawings in your head.
(...does a movie make it any clearer?!) 2/n
What's also fun about the motion in the movie above is that it was created without* human input: instead, the computer tries to nudge the shape around so that every point is as far as possible from itself. You can read all about it in this thread:
Disclaimer: this is a simplified exercise for a final exam and should not be used for serious scientific work! It omits important forces and uses nonphysical constants.
What's the nicest way to draw a shape with many "holes"?
We can use the principle of repulsion to explore this question: each point of the shape behaves like a charged particle, trying to repel all others. Surface tension prevents everything from shooting off to infinity. 1/n
For millennia people have been drawn to the question: what are the "nicest" possible shapes that exist?
This is really a basic question about nature: these shapes exist outside space and time; the same shapes can be discovered by civilizations anywhere in the universe. 2/n
"Nicest" could mean the most symmetric—for instance, the ancient Greeks discovered there were five so-called Platonic solids where every face and every vertex looks the same: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. 3/n