The Way of the Bee

Consider this the first formal announcement of the result of my work over the past nine months.

That’s right. I’ve been missing because I’ve literally been busy. Go figure.

After months of reading through textbooks on image processing, noise generation, papers by the likes of Perlin and Worley, and procedural development techniques—along with a little on fluvial and tectonic geomorphology—I felt I was ready for an actual application of it.

The Way of the Bee!

The Way of the Bee is a game about being a beehive. You are the queen, you are the colony. You fight against real threats to bees, such as fungal and pathogenic threats, overheating or freezing, resource starvation, predators, and weather and climactic threats.

Choose your colony location carefully…

Journey as a colony swarm to your chosen location. Establish a hive, and send out sentries to find floral groves and gather nectar to bring back to it.

Things look different when they’re an existential threat to you…

Be wary of threats from birds to lizards and frogs to fish to bears, navigating a procedurally generated terrain to find the sustenance you need to grow the hive. Be mindful of getting winded, staying adequately warm, and dodging inclement weather that might otherwise cut your journey short.

Return to the hive with your bounty of nectar, and tell the other bees

Return to the hive after finding substantial groves and setting a marker, then communicate your find to the other bees with a dance. Get it right, and a swarm will depart for the nectar and automate the process… but be mindful of threats to their well-being!

Explore a vast landscape generated with Hollywood techniques

Gather enough resources before winter, and you will have enough bees to form a second colony swarm, to go to a new environment, further progressing in the game and facing new threats in doing so. If you don’t have enough, weather the winter with casualties, and begin again in the spring with your remnant bees.

Find flowers among the trees, with your eyes and ears, as a bee would use its own senses…

Along the way, you will encounter heat waves that require dedicated bees to cool the hive, frosts which will requires bees to expend calories to warm it, parasites and hive beetles, full-on attacks by wood peckers and bears, and even hornets. Learn the way bees can defend their hive, from massive stinging (at the cost of the bees that sting), overheating the enemy, and even mummifying it in propolis.

The choice, and the price to pay, is yours alone; but think quickly.

There are things more fearsome in the woods than the swarm…

Experience the fear of losing your queen, and desperately expending precious resources to hatch a new one. See badgers, skunks, and birds and with untold terror. Sacrifice everything in the name of your hive and its survival. Understand the need to sacrifice sick bees who are not recovering, and send them away to die alone; or invest calories and workers in medical care until enough of the bees recover.

If the hive survives, if the family lives on, then your honeybee life has meaning.

The colony must survive.

Coming soon, opening platforms will be Steam and Itch.io.


Announcing my Red Bubble Store!

You’ll find it here. Every now and then I’ll get a little mentally tired with whichever piece I’m working on, and I’ll just doodle for a bit. Every now and then, now and then, I’ll come up with something brilliant while doing it.

Time to turn them into t-shirts and mugs!

So I’ve got two items so far and probably a few more coming through the rest of the week—the one featured to the right of this paragraph is a Christmas tree made out of fractal constructs, which admittedly is the kind of thing you either get or you don’t… which also makes it funnier.

And to the left of this paragraph, we have my take on Spider-Man: No Way Home. (Also a play on “Magneto Was Right” / “Thanos Was Right” / “Mysterio Was Right”) I’ll give it a seven out of ten, it was a darned good movie; there were only a few things that bugged me and if we’re going to be realistic, they specifically bugged me, so I can’t complain. My wife and I had a lot of fun and I’m looking forward to Morbius later in the coming year.

Give it a look now and then!

I Canceled my Art Station Plus account today.

I was seriously expecting, a year ago, to finish my trial of their services and make it an even-more-premium Pro account at this point; but times change fast, don’t they? It was certainly worth it a few years ago.

It was fun while it lasted, sure. I got to watch a number of interesting Trojan Horse was a Unicorn presentations on animation, I learned from some top artists for some Hollywood favorites, I even got to hear about the process which many other artists went through in going from concept to product.

However, it’s been slowly degrading over the past year, and Art Station’s refusal to make an enforceable statement on “AI Art” and its presence on the platform was, if I’m honest, the only real nail the coffin even needed.

It’s not like I’ve made a lot of sales on it; realistically I’m not sure I ever made even one. That wasn’t why I was there; I was looking for an outlet for art assets like Sounds.com has been for audio. I was looking for a number of video courses on art in practice. Most of all, it used to be that I could find and start a conversation with just about any major creator, which seemed too good to be true.

Now, every single post is either part of the protest against AI Art on the platform—which I am, meagerly but proudly, a part of—or something which may very well be AI generated itself. True work is something a trained eye, with a mind knowledgeable about the profession, can see; particularly in contrast to other elements submitted to the platform.

I won’t try to tell people what their own vision of AI Art should be, but you should critically listen to both sides of the argument. The difficulty in producing a portrait, is what we admire in it. Punching letters into a prompt and making your GPU suffer to produce a passable image is not art, it does nothing for you; and it runs on the stolen work of others.

To produce such art, programs like DALL-E require pre-training systems like CLIP (Contrastive Language-Image Pre-training). This means that they need to consider a wide database of material from established artists, and the prompts associated with them; and then produce a new item matching features from that compressed database by comparing the description with the associated prompts.

Is there a use for that? Absolutely. It could do wonders for natural language processing alone, which has a hand in many wonderful tools. However, saturating an established site for human artists with material prompted by a user, with potentially no artistic background, and using it to compromise the living of the very people who brought us to this point isn’t just unseemly, it’s artless in itself. It’s losing track of the very reason the site exists.

AI use is not AI “art”, art is an act of personal creation. Feeding a prompt into a text box and receiving dozens of interpretations in a matter of minutes is not an act of creation. We have to bleed and grind through all of our bad lines and improper shading before we can ever create something truly spectacular, and the sheer reality of that feeling is what drives me to do this.

So, in summary, am I willing to pay a premium for a site which promotes artwork made for pennies on the dollar, if that? Absolutely not. It doesn’t benefit me, and it doesn’t benefit my audience in any way.

Here to stay? Probably. Make no mistake, though. This is the Visual Arts equivalent of Napster all over again. If you spend hours doodling a drawing and pouring your effort and your soul into it, you should be financially compensated for that.

Way of the Bee is approaching alpha!

After long and arduous work, and considerable help with an early preview before the Albuquerque Game Developer Group (AGDG), Way of the Bee is almost ready for early access!

Recent changes include control refurbishing and slimming down of some over-designs, and the addition of a basic pause menu. “Bee Vision” now goes from a 120° fish-eye when still to a 40° field of view when moving, to make navigation easier. Flowers now simply need contact with the bee to finish a path from the hive; and a bee can return to the hive by simply approaching it. The bee will automatically re-enter the nest.

Day/night cycles now include compositor changes to better reflect the environment. The gradual pull of gravity has been entirely removed; though the latent code is still in there in case I need to automatically adjust flight altitude for a quality-of-life improvement.

Birds no longer steal the camera on attack, but their screech is in 3D sound and they can be located by it. “Bee time” engages on an attack, though I feel it might be more useful if it engages later on in the attack; this provides a moment to sweep left or right and dodge the bird.

I will hopefully be debuting it again in front of AGDG today, in the late evening. The last demo gave me a lot of wonderful and useful feedback and exceeded my expectations, and if all goes well, I will have a very short list of adjustments and expansions before an alpha/early-access can be released to the general public!

How to get back into a project, after fighting a disease.

We’ve all been there. We’re hammering away at a project invigorated by not just mastery of resources, but raw passion for it. We have ideas for art, story, song, all of the core components of our planned piece; we have mathematical precision and the comprehension of how to use it to build the framework that will give it life.

Then, out of the blue, for perhaps the most innocent of reasons, something terrible happens to us. We become ill, we capture a pathogen from the earth and air and people around us; we become weak and oppressed by a force of nature.

In my case, it was a particularly notorious bat virus.

Not that I haven’t gotten the vaccines; my family has a long history of autoimmune diseases to begin with, and experience has shown me that I am not an exception to this. So, when everybody started dropping around me, I got that vaccine the moment it was available. I hid from public spaces for longer than I thought I could bear. I got the second shot of the vaccine. I’ve had three boosters now, one each year, for the new variants.

And let me tell you, thank God I did; because that still felt like a particularly ugly head cold. It took me about a week to get over it, during which I made only marginal progress on my swarm AI. It was a setback I would have preferred to avoid, but it’s the hand I was dealt, as they say; and now, I face the uphill battle of getting back into the zone.

It is a zone, of course; much as sports players, gamers, and any other form of skilled craftsman has a zone. It’s an ideal state of mind for the task, on the threshold of being too hard and too easy. “Too hard” has been offset for me, as I’m still recovering; it’s much less difficult to get overwhelmed now, and I am faced with a gradual recovery until I’m at my former strength and acuity.

So, in lieu of my usual update, I’m chronicling advice I have for anyone who faces the same problem.

Understand that it isn’t your fault.

Diseases happen, much as rain storms, droughts, black outs, blizzards, and taxes do. Last week, wishing I felt better would not make me feel any better. It would expend my mental resources, and time which would be better spent resting, staying hydrated, and recovering, would instead be wasted on a vain mental effort.

No one created this virus on their own. If it isn’t the coronavirus, no one gave you your genes on their own, or created the weather intentionally; certainly not you. Disease saps our strength and makes us lean into depression, particularly as we strain ourselves; but the disease is not your fault.

This is not to say that it would not have been convenient if I was at full strength over the last week; only that by realizing I was not, I was capable of addressing the pertinent problem of my illness, and could properly prepare for the moment when it passed and I was at full strength again.

Beating oneself up over a problem which could come about because of the most innocent of things never leads to anything productive. In truth, such incidents need to be planned for early in the project—they will happen—and time should be allotted to account for them. Not everyone always does this, particularly on personal endeavors, but this is an effort everyone should consider making.

Recognize that it will come to an end, and prepare your notes for your recovery.

For the coronavirus, given you are fully vaccinated, the CDC recommends about five to ten days of isolation. Once this period of isolation is over, your body will still be clearing out all of the debris from fighting the virus and attempting to stockpile strength and energy.

Once you know that you are ill, given that the disease has not hit you in force, I recommend taking notes for yourself about everything you’ve been doing. Technical notes, on the state of the project, where it needs to go, and all general strategies for getting there.

If you happen to have a change log and a UML diagram together already, great! Use them! However, it’s also good to give yourself a gentle note, no more than two or three paragraphs long, on what needs to be done next. Use simple sentences—simpler than you think that you will need—that focus on the core actions that need to be done. It’s likely that after the incident hits, you will have fewer mental resources and less patience, so write kindly and clearly.

Bring all of your work to a closing point, the closest semantically complete block you’re comfortable with; and leave comments on the purpose of anything new you write. You want to tie this up before the symptoms hit you in full.

Afterward, I recommend grabbing any supplies you will need—groceries, tissues, cold medicine, whatever it may be—and preparing to stay hydrated and comfortable for the duration of the disease. Again, go easy on yourself, and have faith that you’ll shake it and return to your former glory.

Ease yourself back into the Zone using simpler projects.

This is a big one. Time has passed, your head or stomach are clearing up, and you’re ready to get back to work; but your mind is like stone and concrete. You don’t remember what you were doing, you’re still a little weak, but you’re optimistic.

The last thing you want to do here is to pretend that you’re at the same strength you were before the sickness; you’re still gaining momentum again. I recommend easing yourself back into the project by taking on something parallel, like studying an API you’ve been curious about.

Grab something that relates to your project, but is simpler. For me, it was re-learning Qt 5. I’ve had a book on it for some time now, which is written and edited in a manner just-dodgy-enough to keep me on my toes. In the past three days, I’ve gone through four chapters, and may finish the book before the end of next week.

Meanwhile, I touch on something involving the swarm AI I was working on before every day. This way, the project becomes fresh in my mind again. By spending a couple of hours every morning simply reviewing, and experiencing, some of the latent bugs, I become familiar with the code again.

Think of it like returning to a gym after a two month break. You need to ease yourself back into the routine, with steady progression and confrontation of the tasks. It’s all too easy to simply quit and walk away forever; but then we would never see our projects bear fruit.

If there is a vaccine, get vaccinated. Prepare for infection.

There’s been a disappointing, and alarming, amount of dissent on this. I can tell you for a fact that the mRNA vaccines are the result of several centuries of research into combating disease, and they work. In most parts of the world, your government will give it to you for free. It will be all of fifteen to twenty minutes; less, if you lose patience and sneak out early after the shot.

It’s really no big deal. However, that shot can make the difference between something truly life threatening which may haunt you for months even if you survive it; and something that only takes a few days to get over. Familiarize your immune system with the infection, before you get infected.

Then, have some supplies on hand. If you can’t cook, have some canned soup around; soup helps keep you hydrated and nourished, and tends to taste pretty reasonable even in the midst of influenza. Have tissues, keep a water bottle around, and maybe some cough drops and cough suppressant.

Don’t lose faith, don’t lose vision.

That awesome idea you had, whether it’s for a game, or an animation, or anything else, will be a reality. If you’re committed, then you have nothing to worry about. The trick is to stay committed.

The truth is, we feel like hell after disease. We’re tired, lethargic, and injured. There’s a decent chance you won’t want to think about the math, or even the art, at all; and that’s OK, sometimes you’re just knowing yourself and you’re right. Maybe you do need time to recover.

However, you should never lose the dream that’s fueling this effort. It’s a question of persistence and gradual perturbation of the project, until your health is back and its familiarity has returned to you.

This week’s work has primarily been about getting in-game swarms to follow paths set by the scout bee. It’s going well. That said, I look back on it and feel like the past week could have been a couple of days.

Maybe. However, I was also sick over the past week, and that’s just a part of life. Over the coming handful of days, I intend to recover my momentum and get back into pounding this game out. I still intend to have a releasable preview available before the end of the year.

This is how I approach my return to that state of mind and body, and I hope it helps those among you who worry, perhaps rightly, that you too may come down with a malady. Keep on building!

Streaming Media on Non-Windows OS

In this particular instance, I’m dealing with a flaw on Disney+, which, as a die-hard Marvel fan, I have been a member at for some time now. I love those shows, want them to continue, and do not want to have to restart my entire desktop and lose all of my work just so I can watch them. (Not to mention the sheer span of classic animation available.) As of recently, a flaw in their page is preventing videos from loading for me, due to a dependence on specific user agent strings. It’s inconvenient, but no biggie; I just need to override a few things.

Whenever this happens, it’s important to remember that it wasn’t personal. Sometimes a design flaw, often involving the User Agent string from your browser, will keep a service like (but not limited to) Disney+ from functioning on your OS. This is absurd, as it’s a central point of web browser security to isolate everything from the OS. Otherwise it would be dangerous! However, some new designers just don’t know better, and they’ll test the User Agent string for the browser but keep the part about the OS.

I still insist that that operating system part never should have been there in the first place, but I’ll come back to that after my instructions.

Similar actions are probably possible in other browsers like Chrome, but I’ll be detailing Firefox as an example.

Setting Up a Custom Profile

The problem with changing a user agent string is that it will, generally, cause other sites to malfunction or perform improperly. (We’ll come back to how to set that string in the next section.) To do this, from Firefox, open about:profiles.

Here, you will be presented with existing profiles, which are collections of custom settings. You may already have a few available, but we’re creating a new one. Click on “Create A New Profile”.

Firefox has a wizard for this (whatever our opinion of wizards may be), but it is rather straightforward. You will need a memorable name (I recommend against using spaces or characters like “/”, “\”, “:”, or any other that may have specific meaning in your terminal). The root and local directories don’t need to be changed in most situations. Once you have it, switch to that profile by clicking on “Launch Profile in New Browser”.

It’s helpful to check and see that your typical browser profile is still the default. Click on the “Set as Default Profile” button if you have to.

Note that the two browser profiles effectively function as separate programs.

Setting a Custom User Agent String

Once you have your browser open, I suggest giving it a default color different from your typical browser, so you can tell them apart. The theme color of the site you’re trying to access is usually a good bet, it’s unwise to use this for any other site anyway.

We’ll now be adding a custom property to this profile. It is very important that you perform the next steps using your new profile, not your old one.

From the browser launched with your new profile, open about:config. This is how you’ll set configuration preferences for your profile. As of at least Firefox 106.0.2, you’ll have a search bar at the top of the screen.

This is a little more than a search bar, as it also allows you to set new, custom properties which can be accessed from scripts. Look up this property:


It isn’t typical to have it set, but if you’re using your new browser profile, there shouldn’t be any harm in overwriting it. Press the “+” button next to the string to do so, after verifying your spelling and ensuring you are adding precisely that property.

The user agent was once heavily used to determine the idiosyncrasies of different browsers, and to ensure that a page (along with its style sheets and scripts) loads properly on that browser. Sometimes it still is, though the browsers aren’t locked in the same kind of war they were in the 1990s.

Sometimes, someone will accidentally use this as a security check to ensure that a non-custom-browser is not being used, which is unwise on so many levels; but we’ve all been naive once and it’s no reason to give them flak, it’s just something we need to work around.

Modern user agent strings specify both the browser kernel and the operating system it’s built for, which really should never have entered the question, but I digress. The truth is, we didn’t know what we were doing back then, or where it would go. By overriding this string, you can fake your browser and OS. You can request pages that might typically only be visible to Chrome from Firefox, or see how a page would look on MacOS from Linux… given that your browser is capable of properly rendering the page.

Today, we’ll be using a specific custom string to convince the site at the other end that we’re running on Windows.

Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/42.0.2311.135 Safari/537.36 Edge/12.246

Once you have added this string to your profile, sites will be convinced that you are running your browser from the Windows operating system (or at least, using something based on Windows NT 10.0). It should work immediately for the profile you wrote it into.

This is of course likely to change with time; and more modern user agent strings can be found on a number of sites, for me most typically www.WhatIsMyBrowser.com. Feel free to experiment with them! Just keep it in that custom profile.

If your user agent string is correct, your browser will receive the exact same data as your chosen operating system. This is, with good reason for both sides, as much as the server ever knows about your specific computer. If it knew more, it could potentially execute arbitrary code that affects your host environment, and the internet as we know it, streaming media and all, would collapse on itself.

Creating a Custom Launcher

Now that we have this custom profile, we’re doing good; we can access our subscription service and everything. However, it would be nice to just double-click an icon on the desktop and immediately get to it, wouldn’t it?

I’ll walk you through this, again using my example of Disney+.

Create a launcher on your desktop, or if this option is unavailable to you, a shell script. Name it appropriately (I suggest the name of the service you’re accessing), and give it the following launch command:

firefox -p <your profile name> www.disneyplus.com

Where <your profile name> is the name of the new profile which you provided. firefox -p allows you to run Firefox with the profile name specified in the next argument. Here, I assume you want to start the launcher at Disney+, but you can replace that string with the address of the host you’re attempting to reach, or simply remove it and navigate manually. After all, this could (and will) happen to anyone.

If you would like a custom icon to your launcher, I suggest using a site like icons8.com and I personally prefer SVG over PNG; but that’s up to you.

Using that launcher will take you to your streaming service, ready to go!

Advice for Web Developers

It’s admittedly been a long time since I was a web developer, and the web has changed quite a bit since then. That said, user agent strings, back in the day, were used to work with incompatibilities between browsers, with the assumption that the operating system would play a part in that.

There were a few mistakes in that assumption. For starters, desktop environment ≠ operating system. On this machine alone I regularly switch between KDE Plasma (like living in a penthouse) and XFCE (like living economy). They both handle things entirely differently, but the user agent string will betray you, as in either case it will say “Linux”.

It’s just not enough to be helpful.

So, rather than depending on this poorly placed identifier, I suggest sanitizing it with a regular expression, and completely ignoring the operating system. It is the goal of any browser to behave the same on every OS that supports it, and there are a heck of a lot of them out there. Believe me, you would not like your job better if you had to build your pages and services to check for every single Windows Update or personalization setting that changed the desktop environment.


Let’s take a moment to remember that all policies exist for a reason, and it is important to ensure that whatever you are doing is not ultimately in violation of your contract. (Not that I intend to tell you how to live your life.) For legal reasons, it is important to consider that while a browser setting is within your control, this should not be used in any manner that would violate intellectual property rights.

That seems unlikely here, but it bears repeating that animators and film makers are still trying to make a living and deserve our respect.

I would like to make it clear that I do not endorse theft, piracy, unauthorized access of a streaming service, or anything in violation of contract or law; this is purely advice on customizing your browser settings to enjoy an optimal experience on your choice of operating system and desktop environment.

If you must, contact your streaming service to find out what their policy is on making such changes. Or, you know, don’t; but if there’s trouble of some kind, don’t come back to me with it. I’m an animator myself, after all; and it isn’t that expensive. Pay people for their hard work, they did it for you!


Sometimes it isn’t possible to get a service to work via user agent alone; perhaps some other detail is in the way. Maybe it depends on a plugin that has not been made, or does not have an equivalent, for your operating system—super rare these days, sure, but not impossible.

On occasions like this, I recommend one of two things: a compatibility layer, like WINE for Linux, BSD, Macintosh, and others for emulating Windows; or a virtual machine like VMWare or VirtualBox.

Hardware virtualization has progressed by leaps and bounds over the years. A CPU is now capable of furnishing two separate operating systems—effectively different machines—with 64-bit processing when virtualization is on. (For MSI motherboards like mine, that’s IOMMU and SVM enabled in the BIOS/UEFI settings; other motherboards have similar settings you’ll have to look up.)

This allows programs like VirtualBox to create virtual hard drive files, of dynamic size if you would like, and run 64-bit Windows 10/11 on it, or any other operating system you can get an install disk and a license for. This is a major component in server technology at this point, and we are a long way away from the days of simply treating a virtual machine as “one computer pretending to be another”.

I’m not going to delve into the details of these here, but in the instance of running Linux, you can install something like Chrome or Firefox for Windows with WINE and run the service through that; or you can install Windows 10/11 as a whole on a VM, and just watch it through Edge in that.

Hopefully this post has given you a number of inspirations on how to semi-permanently get any service available to you on your chosen work operating system.

Way of the Bee, Preview!

Way of the Bee

I’ve got a prototype trailer up for Way of the Bee, and it’s been a ride making it.

From the terrain generator, to the art, to the custom AI, this has been all-me; which made it a work out. Of course, I have bottomless thanks to the people on the KDE team, the Blender team, the Unity developers, and many more for making the environment which I used to produce this, as while it’s my dream, it would be a much smaller one without them.

Additional shout outs to Perlin, Worley, Peachy, Musgrave, & Ebert for “Texturing and Modeling: A Procedural Approach”, which has opened up many new facets out of the world of procedural generation for me; Toady One for showing me that a dream can become an icon with enough effort and time; the Microsoft .NET team for starting with an overt Java knock-off but turning it into an incredibly respectable and efficient language to build this with, Ton Roosendaal for establishing a standard for 3D modeling that isn’t just influential, but open; and so many others.

Well, I’m getting back to it. It should be purchasable by the end of the year, as a minimum viable game—complete enough to make it through, and preferably enjoyable. Then, we start dressing it up!

Bird Demo! (Way of the Bee)

Basic Bird Predator for Way of the Bee!

After considerable deliberation, arguing, researching, pounding out, arguing, bending, prying, polishing, and welding, I have a navmesh-free situation-aware basic AI going for my bird predators. I’m finally satisfied with it.

Birds follow a relatively smooth path to their opponent, avoid obstacles like the terrain, do not turn on a dime (unless I tell them they can), and circle around gradually to face targets which aren’t within an angle window in front of them. Physics and animation are whole different modules from path finding, so the animation should always match the behavior.

I remind you that in this game, you are the 🐝. The birds are not your friends… But, having a player-vs-player mode in which one player does control a bird sounds like a lot of fun. So I may come back to that!

Most navigation uses a clever concept called a navigation mesh (or in some cases, a navigation volume); which divides traversable terrain into a set of linked convex polytopes. Each pair of connected polytopes can be walked across, and as long as an agent is bound to it, they will behave quite normally.

This is usually great, and is supported by freakin’ everything! However, it isn’t always enough. As an example, for my bird here, we need to worry about altitude, too. One way to do that would be a navigation volume (using polygons instead of edges for meeting points between convex polyhedra), but I would be building that from scratch, it would take a while, and I want this playable and enjoyable before the end of December. So instead, I fell back on a few other navigation techniques and tuned the amount of error I’m comfortable with it having.

Most of this stuff actually comes from robotics, where collision detection (or hopefully anticipation!) is really essential. You can theoretically make a game that doesn’t need collisions, but nobody wants to drive their robot into a wall. The bird’s doing pretty well in that department!

Bézier Curves!

A lot of people are ready to talk about Bézier curves, as they’re a pretty fundamental part of solid geometry. Let’s be honest, if you’ve ever used so much as MS Paint and wanted a smooth curve, you’ve likely been introduced to these.

I’m working with them right now in Way of the Bee, as a matter of fact, in an effort to make by bird flights look a little more realistic and less obviously bound to 3D vectors. Grabbing a little bit of back knowledge—since it can be mildly dangerous to assume that you’re too much of an expert to learn—I noticed a lot of mysticism about them.

You have four points (at least for a cubic Bézier), and it forms a curve, efficiently, between the first point and the last, but what the heck are these other two points? Shifting them adjusts the curve, but how so? Can you make a circle with it?

I’m going to address all of this here; but let’s start with a little history.

A Little History

Béziers were around before Pierre Bézier, but he did popularize them. They’re what’s called Bernstein polynomials, which have been well documented since the early twentieth century. What the heck is a Bernstein polynomial? They’re linear combinations—for Unity users, LERPs—of basis polynomials.

Technically, a solid year before Bézier documented their use in computer graphics, they were used by another mathematician, Paul de Casteljau. (de Casetjau’s algorithm is still frequently referred to when arbitrarily splitting a Bézier curve.) de Casteljau established IP protections in France, but not internationally, as Bézier unknowingly did a year later; there was never any evidence of hostility between the two.

Bézier himself was working for Renault as an automotive engineer, and needed to find a way to accurately describe the surface of a car without reducing it to a flat polyhedron. This is exceptionally important for automotive building, as flat planes can introduce terrible aerodynamic resistances.

He was no stranger to mathematics himself and was an absolutely brilliant solid geometry engineer, but he noticed that Bernstein polynomials offered a way to arbitrarily specify a curve at any point without reducing down to jagged polygons. He had a winner, he shared it, and it’s used in virtually all forms of industrial design, from blenders to airplanes, to this day.


If it’s been a while since you’ve been in a math class, a polynomial is an equation which consists of several terms added together.

An exemplary polynomial

Linear interpolation, commonly Lerping, is interpolating in a linear way between one vector and another.

Basic Linear Interpolation Equation

By linear I technically mean “without any bumps, curve, or mathematical shenanigans” in the curve. And yes, if you look at it, you’ll note that between zero and one it covers a line between vector one and vector two; but it doesn’t have to.

I would like you to consider that vectors are also effectively polynomials. It’s just a question of expression. If we have the point <4, 5, 7>, we can just as easily express it as:

Example Vector Polynomial


So, what does it mean to create a Bernstein polynomial using these vectors as a basis? If we just had the first point and the last, it would again be a line; but if we have a curve then we’re departing from lines in a specific way—we aren’t just interpolating between two points, we’re also interpolating between two directions. The control points define the vectors, and magnitudes, of these directions.

This is why moving either control point will not change the initial or terminal points of the curve, but will totally change how they get there. We are interpolating between both P1 and P4, and the directions of P1P2 and P3P4.

Exemplary Bezier Curve

The best way to understand any algorithm used in construction is to think about its constraints! So, to reiterate, we want it at a certain place, facing a certain direction, at the beginning; at the end, we want it at a different place, facing a different direction.

The first step in evaluating a Bézier curve is to find the line between P1 and P2, and the line between P3 and P4, as equations of t. This is a basic linear interpolation. We have two lines, now, which carry our initial and final directions.

We want to shift between the two, so we’ll be interpolating between them, too. How do you do that? Well, I’m going to introduce yet another line, but this one is a bit more flexible. As we increase t from zero to one, we’ll be describing points on P1P2 and on P4P3, let’s call them Q1 and Q2.

Bézier example, with interpolation points shown

We’ll define L as a line between Q1 and Q2.

Bezier example, with secondary interpolation

As we watch L evolve with the increase of t, we might notice something from this graph. It intersects the Bézier curve perfectly, begins at P1 facing along P1P2, and end at P4 facing along P4P3. We’ll call that interpolation, along the shifting line L, by value t, the point QL.

That’s exactly how it’s done. QL perfectly traces out the constraints of the Bézier curve.

In the instance of a non-cubic but quadratic Bézier curve, where there are only three points, you simply use P1P2 as your first vector, and P2P3 as your second.

Like all Bernstein polynomials, Bézier curves lend themselves to tesselation, or division into simpler subsets, very well. However, they don’t have to do so destructively; the information of the curve at each point is still well defined.

So Why Can’t it be a Proper Circle?

I’m going to rough-and-tumble this one.

That can be proved quickly, with just a little math—though you should note that Béziers can get damned close to a circle. The equation of a circle is well known, but let’s start with that.

Equation of a circle about the origin

Here, x and y are coordinates, and r is the radius of the circle. In order to be identical to a circle, a Bézier curve would have to be reducible, for some parameters, to this. So, for the first time in this article, let’s consider the completely-unfolded double-linear-interpolation of the cubic Bézier curve.

Completely broken down Cubic Bezier Curve

As you can see, it’s called cubic because it has a cube in it. The equation of a circle, along with any of its forms, does not have a cube in it. To some this will be obvious, but we don’t all speak math that well; but there are no parameters that reduce any part of this to a consistent square. You can get very close, but never identical; they two equations occupy different numerical spaces.

To push this further, quadratic Bézier curves describe parabolas, not circles. Cubics describe parabolas between parabolas, in a sense. They are incredibly versatile, and they are (at least for cubics and quadratics) computationally efficient, but they are not universal.

Close enough for Rock & Roll, though.

Hive Update (Part 2)

Development is coming along swimmingly!

I’ve turned down the number of grass blades from 16-per-unit to 2-per-unit, for the sake of everyone’s graphics cards and frame rate; it actually looks a little bit better to me that way.

As you can see, the hive develops as nectar is delivered to it right now; that won’t always be the case. Way of the Bee is largely about economics; nectar is converted into honey for consumption and storage. It drives the growth of larvae, which become bees. The bee count drives the creation of wax and building of the colony; while different defenses will require various resources to perform. Cooking a hornet to death, as an example, will tap into honey storage because of the sheer amount of calories necessary.

It’s going to be an interesting fusion between an adventure game and a real-time-strategy game.

We’re all loving that satisfying crunch toward the end, which actually came from a vegetable. In any case it’s definitely not a sound you would want your body to be making…

The current goal is to reach a minimum viable game, which is enjoyable to play. Then, we’ll have it available for early access on Itch, and later on Steam. Later targets could include any number of platforms, but only after the game has reached a more mature state.