One would have thought that the FAT12 file system was safely a relic of the 1980s and 1990s, when it was used as the default file system for floppy disks and very early hard disks. FAT12 would be entirely impractical today, since it can only cover a maximum of 32 MB of disk space. However, I was surprised to find it very much alive today, in the most unlikely of places.When I go running, I use my trusty Garmin Forerunner 10 watch, which uses GPS to record my position and pace during the run. When I connect the watch to the USB port on my PC, it appears as a mass storage device, and allows me to retrieve the workout files (stored in the FIT format). It hadn’t occurred to me until now to check out the finer details of this mass storage device, but there were a few things that surprised me:
The entire size of the watch’s flash memory is actually just one megabyte! I’m guessing this is because they want to discourage users from using the watch for general storage (i.e. dumping of photos, documents, and so on), thereby unnecessarily wearing out the flash memory. It also encourages the user to offload the workout files relatively often, in case the memory ever gets corrupted or the watch is lost. It’s also possible that this watch uses a more expensive type of flash memory (one that is more resilient to wear and tear), which would make it prohibitively expensive to provide multi-gigabyte sizes that we breezily expect in today’s USB flash drives.
You guessed it: it uses FAT12 to organize the files in the flash memory. Because why not! With a total disk space of 1 MB, this is really the simplest and most compatible solution they could have chosen.
Therefore, hats off to Garmin for not overcomplicating things, and making use of a tried and tested solution that is sure to remain compatible and future-proof.
Lately I’ve been studying up on ray tracing, and one of my goals has been to build a nonlinear ray tracer — that is, a ray tracer that works in curved space, for example space that is curved by a nearby black hole. (See the finished source code!)
In order to do this, the path of each ray must be calculated in a stepwise fashion, since we can no longer rely on the geometry of straight lines in our world. With each step taken by the ray, the velocity vector of the ray is updated based on an equation of motion determined by a “force field” present in our space.
This idea has certainly been explored in the past, notably by Riccardo Antonelli, who derived a very clever and simple equation for the force field that guides the motion of the ray in the vicinity of a black hole, namely
I decided to use the above equation in my own ray tracer because it’s very efficient computationally (and because I’m not nearly familiar enough with the mathematics of GR to have derived it myself). The equation models a simple Schwarzschild black hole (non-rotating, non-charged) at the origin of our coordinate system. The simplicity of the equation has the tradeoff that the resulting images will be mostly unphysical, meaning that they’re not exactly what a real observer would “see” in the vicinity of the black hole. Instead, the images must be interpreted as instantaneous snapshots of how the light bends around the black hole, with no regard for redshifting or distortions relative to the observer’s motion.
Nevertheless, this kind of ray tracing provides some powerful visualizations that help us understand the behavior of light around black holes, and help demystify at least some of the properties of these exotic objects.
My goal is to build on this existing work, and create a ray tracer that is more fully featured, with support for other types of objects in addition to the black hole. I also want it to be more extensible, with the ability to plug in different equations of motion, as well as to build more complex scenes, or even to build scenes algorithmically. So, now that my work on this ray tracer has reached a semi-publishable state, let’s dive into all the things it lets us do.
The ray tracer supports an accretion disk that is either textured or plain-colored. It also supports multiple disks, at arbitrary radii from the event horizon, albeit restricted to the horizontal plane around the black hole. The collision point of the ray with the disk is calculated by performing a binary search for the exact intersection. If we don’t search for the precise point of intersection, we would see artifacts due to the “resolution” of the steps taken by each ray (notice the jagged edges at the bottom of the disk):
Once the intersection search is implemented, the lines and borders become nice and crisp:
We can also apply different colors to the top and bottom of the disk. Observe that the black hole distorts the disk in a way that makes the bottom (colored in green) appear around the lower semicircle of the photon sphere, even though we’re looking at the disk from above:
Note that the dark black circle is not the event horizon, but is actually the photon sphere. This is because photons that cross into the photon sphere from the outside cannot escape. (Only photons that are emitted outward from inside the photon sphere can be seen by an outside observer.)
If we zoom in on the right edge of the photon sphere, we can see higher-order images of the disk appear around the sphere (second- and even third-order images are visible). These are rays of light that have circled around the photon sphere one or more times, and eventually escaped back to the observer.
And here is the same image with a more realistic-looking accretion disk:
Great! Now that we have the basics out of the way, it’s time to get a little more crazy with ray tracing arbitrary materials around the black hole.
The ray tracer allows adding an unlimited number of spheres, positioned anywhere (outside the event horizon, that is!) and either textured or plain-colored. Here is a scene with one hundred “stars” randomly positioned in an “orbit” around the black hole (click to view larger versions of the images):
Notice once again how we can see second- and third-order images of the spheres as we get closer to the photon sphere. By the way, here is a similar image of stars around the black hole, but with the curvature effects turned off (as if the black hole did not curve the surrounding space):
And here is a video, generated using the ray tracer, that shows the observer circling around the black hole with stars in its vicinity. Once again, this is not a completely realistic physical picture, since the stars are not really “orbiting” around the black hole, but rather it’s a series of snapshots taken at different angles:
Notice how the spherical stars are distorted around the Einstein ring, as well as how the background sky is affected by the curvature.
And finally, the ray tracer supports adding spheres that are perfectly reflective:
All that’s necessary for doing this is to calculate the exact point of impact by the ray on the sphere (again using a binary intersection search) and get the corresponding reflected velocity vector based on the normal vector on the sphere at that point. Here is a similar image, but with a textured accretion disk:
Eventually I’d like to incorporate more algorithms for different equations of motion for the rays. For example, someone else has encoded a similar algorithm for a Kerr black hole (i.e. a black hole with angular momentum), and there is even a port of it to C# already, which I was able to integrate into my ray tracer easily:
A couple more ideas:
There’s no reason the ray tracer couldn’t support different types of shapes besides spheres, or even arbitrary mesh models (e.g. STL files).
I’d also like to use this ray tracer to create some more animations or videos, but that will have to be the subject of a future post.
Some possible future work might be to create a renderer for black holes (in addition to regular spheres) which would actually curve the surrounding space and affect the direction of the rays. This would necessitate a radically more complex ray tracer, which would need to follow the path of the ray in a stepwise fashion using Boyer–Lindquist coordinates within the Kerr metric. Perhaps a project for another weekend (or two)!
As the developer of a relatively popular app, there are numerous things I have to worry about on a regular basis. One of these things, however, is something for which I was kind of unprepared, and still not sure how to deal with: the emergence of knock-off apps — apps that have a suspicious and often hilarious resemblance to the original.
There were, of course, counterfeit apps that were blatantly using my trademark in their name (“DiskDigger” is a registered trademark in the U.S.), or my graphics in their store listings, in which case the Google Play Store thankfully took them down upon request.
But then there are apps that don’t quite use my trademark, and don’t quite have the same icon and screenshots. But because they are close enough, they can take advantage of being near the top search results for this category of apps, and lead numerous unsuspecting users to install them and be greeted with a barrage of ads and spyware.
Most of these apps seem to come from Indian and Middle-Eastern developers, which makes the horribly broken English in their app verbiage even more amusing to read. A small part of me even applauds their enterprising spirit, and I don’t fault them for wanting to make a buck, but I wish they’d find ways of doing it more honestly.
Take a look! Can you tell the knock-off apps from the real ones?
The problem is that because these apps don’t explicitly violate my trademark, Google refuses to take them down, which is unfortunate because these apps do literally nothing except shove ads in the user’s face, and therefore actively harm the ecosystem of the Play Store. Shouldn’t Google care more about preventing the Play Store from becoming a cesspool of bottom-feeding cash grabbers?
In my day job, one of my responsibilities is to oversee the technical evolution of our product and to optimize its performance. I’ve even written a few guidelines that detail numerous recommendations for maximizing performance of Android apps in general.
One of the things I have always recommended is to reduce the complexity of View hierarchies, and try not to overcrowd or have too many nesting levels of your Views, since this can supposedly have a negative impact on performance. However, I made these statements based on common sentiment on the web, and based on the Android documentation, instead of on actual hard evidence. So I looked into it from an interesting perspective: I dug into View hierarchies as they are used by other major apps, and compared them with our own usage. This isn’t a totally “scientific” analysis, and it only looks at a single facet of proper View usage. Nevertheless the findings are rather surprising, and are actually challenging my insistence on View minimalism.
I looked at the Twitter, Facebook, and Slack apps, and compared each of their “feed” screens to the feed screen of our own Wikipedia app. (The reason I chose these apps is that the “performance” of their feeds is nearly perfectly smooth, especially considering that some of their content includes auto-playing videos and animations.) I used the superbly useful and little-known UI Automator Viewer tool, which is bundled with the Android SDK, to explore these view hierarchies.
For reference, the deepest nesting that I found in the feed of the Wikipedia app is seven (7) levels deep:
But get ready: The deepest nesting in the Slack app is… eighteen (18) levels deep. And yet it performs perfectly smoothly:
The deepest nesting in the Facebook app is twenty (20) levels deep. And yet it works just fine:
The deepest nesting in the Twitter app is twenty three (23) levels deep, which includes eight (8) nesting levels for each item in their ListView (it’s not even a RecyclerView!). And yet I’m able to scroll the Twitter feed infinitely without a single hiccup.
Therefore I’m compelled to reevaluate the importance we should be placing on optimizing View hierarchies, at least from the perspective of “nesting.” Indeed, this seems to be yet another case for balancing reasonable performance guidelines with more immediate product goals, or put more simply, avoiding premature optimization.