## Floating Point Numbers and Rounding

(originally posted over at Cohost)

I was writing about how to parse C++17-style hex floating point literals, and in doing so I ended up writing a bunch about how floats work in general (and specifically how floating point rounding happens), so I opted to split it off from that post into its own, since it’s already probably way too many words as it is ?

Here we go!

## How Do Floats Work

If you don’t know, a floating-point number (At least, an IEEE 754 float, which effectively all modern hardware supports), consists of three parts:

• Sign bit – the upper bit of the float is the sign bit: 1 if the float is negative, 0 if it’s positive.
• Exponent – the next few bits (8 bits for a 32-bit float, 11 bits for a 64-bit float) contain the exponent data, which is the power of two to multiply the given hex value with. (Note that the exponent is stored in a biased way – more on that in a moment)
• Mantissa – the remaining bits (23 for 32-bit float, 52 for a 64-bit float) represent the fractional part of the float’s value.

In general (with the exception of subnormal floats and 0.0, explained in a bit) there is an implied 1 in the float: that is, if the mantissa has a value of “FC3CA0000” the actual float is 1.FC3CA0000 (the mantissa bits are all to the right of the decimal point) before the exponent is applied. Having this implied 1 gives an extra bit of precision to the value since you don’t even have to store that extra 1 bit anywhere – it’s implied! Clever.

The exponent represents the power of two involved (`Pow2(exponent)`), which has the nice property that multiplies or divides of a float by powers of two do not (usually, except at extremes) affect the precision of the number, dividing by 2 simply decrements the exponent by 1, and multiplying by 2 increments the exponent by 1.

For a double-precision (64-bit) float, the maximum representable exponent is 1023 and the minimum is -1022. These are stored in 11 bits, and they’re biased (which is to say that the stored 11 bits is `actualExponent + bias` where the bias is 1023. That means that this range of [-1022, 1023] is actually stored as [1, 2046] (00000000001 and 11111111110 in binary). This range uses all but two of the possible 11-bit values, which are used to represent two sets of special cases:

• Exponent value `00000000000b` represents a subnormal float – that is, it still has the effective exponent of -1022 (the minimum representable exponent) but it does NOT have the implied 1 – values smaller than this start to lose bits of precision for every division by 2 as it can’t decrement the exponent any farther and so ends up sliding the mantissa to the right instead.
• For this `00000000000b` exponent, if the mantissa is 0, then you have a value of 0.0 (or, in a quirk of floating point math, -0.0 if the sign bit is set).
• Exponent value `11111111111b` represents one of two things:
• If the mantissa is zero, this is infinity (either positive or negative infinity depending on the sign bit).
• If the mantissa is non-zero, it’s NaN (not a number).
• (There are two types of NaN, quiet and signaling. Those are a subject for another time, but the difference bit-wise is whether the upper bit of the mantissa is set: if 1 it’s quiet, if 0 then it’s signalling).

If you wanted to write a bit of math to calculate the value of a 64-bit float (ignoring the two special exponent cases) it would look something like this (where `bias` in this case is 1023):

``````(signBit ? -1 : 1)
* (1 + (mantissaBits / Pow2(52 + 1))
* Pow2(exponentBits - bias)``````

## Standard Float Rounding

Okay, knowing how floats are stored, clearly math in the computer isn’t done with infinite precision, so when you do an operation that drops some precision, how does the result get rounded?

When operations are done with values with mismatched exponents, the value with the lowest exponent is effectively shifted to the right by the difference to match the exponents.

For example, here’s the subtraction of two four-bit-significand (3 bits of mantissa plus the implied 1) floats:

``````  1.000 * 2^5
- 1.001 * 2^1``````

The number being subtracted has the smaller exponent, so we end up shifting it to the right to compensate (for now, doing it as if we had no limit on extra digits):

``````  1.000 0000 * 2^5
- 0.000 1001 * 2^5 // Shifted right to match exponents
------------------
0.111 0111 * 2^5
1.110 111  * 2^4 // shifted left to normalize (fix the implied 1)
1.111      * 2^4 // round up since we had more than half off the edge``````

Note that in this example, the value being subtracted shifted completely off the side of the internal mantissa bit count. Since we can’t store infinite off-the-end digits, what do we do?

Float math uses three extra bits (to the “right” of the mantissa), called the guard bit, the round bit, and the sticky bit.

As the mantissa shifts off the end, it shifts into these bits. This works basically like a normal shift right, with the exception that the moment that ANY 1 bit get shifted into the sticky bit, it stays 1 from that point on (that’s what makes it sticky).

For instance:

``````      G R S
1.001 0 0 0 * 2^1
0.100 1 0 0 * 2^2 // 1 shifts into the guard bit
0.010 0 1 0 * 2^3 // now into the round bit
0.001 0 0 1 * 2^4 // now into the sticky bit
0.000 1 0 1 * 2^5 // sticky bit stays 1 now``````

Note that the sticky bit stayed 1 on that last shift, even though in a standard right shift it would have gone off the end. Basically if you take the mantissa plus the 3 GRS bits (not to be confused with certain cough other meanings of GRS) and shift it to the right, the operation is the equivalent of:

``mantissaAndGRS = (mantissaAndGRS >> 1) | (mantissaAndGRS & 1)``

Now when determining whether to round, you can take the 3 GRS bits and treat them as GRS/8 (i.e. GRS bits of 100b are the equivalent of 0.5 (4/8), and 101b is 0.625 (5/8)), and use that as the fraction that determines whether/how you round.

The standard float rounding mode is round-to-nearest, even-on-ties (that is, if it could round either way (think 1.5, which is equally close to either 1.0 or 2.0), you round to whichever of the neighboring values is even (so 1.5 and 2.5 would both round to 2).

Using our bits, the logic, then, is this:

• If the guard bit is not set, then it rounds down (fraction is < 0.5), mantissa doesn’t change.
• If the guard bit IS set:
• If round bit or sticky bit is set, always round up (fraction is > 0.5), mantissa increases by 1.
• Otherwise, it’s a tie (exactly 0.5, could round either way), so round such that the mantissa is even (the lower bit of the mantissa is 0), mantissa increments if the lower bit was 1 (to make it even).

Okay, so if all we care about is guard bit and then <round bit OR sticky bit>, why even have three bits? Isn’t two bits enough?

Nope! Well, sometimes, but not always. Turns out, some operations (like subtraction) can require a left shift by one to normalize the result (like in the above subtraction example), which means if you only had two bits of extra-mantissa information (just, say, a round and sticky bit) you’d be left with one bit of information after the left shift and have no idea if there’s a rounding tiebreaker.

For instance, here’s an operation with the proper full guard, round, and sticky bits:

``````  1.000 * 2^5
- 1.101 * 2^1

// Shift into the GRS bits:
1.000 000 * 2^5
- 0.000 111 * 2^5 // sticky kept that lowest 1
------------------
0.111 001 * 2^5
1.110 01  * 2^4 // shift left 1, still 2 digits left
1.110     * 2^5 // Round down properly``````

If this were done with only two bits (round and sticky) we would end up with the following:

``````  1.000 * 2^5
- 1.101 * 2^1

// Shift into just RS bits:
1.000 00 * 2^5
- 0.000 11 * 2^5
------------------
0.111 01 * 2^5
1.110 1  * 2^4 // shift left 1, still 2 digits left``````

Once we shift left there, we only have one bit of data, and it’s set. We don’t know whether or not we had a fraction > 0.5 (meaning we have to round up) or < 0.5 (meaning we round to even, which is down in this case).

So the short answer is: three bits because sometimes we have to shift left and lose one bit of the information, and we need at least a “half bit” and a “tiebreaker bit” at all times. Given all the float operations, 3 suffices for this, always.

## Networking Is Hard (Part 3)

In my previous two posts, I started looking into what it would take to code the networking for my game, and came up with a first draft, before realizing that floating-point discrepancies between systems totally threw my lockstepping idea for a loop.

#### Lockstepping With Collisions

In order to solve the issue with different systems having different floating-point calculation results, I decided to somewhat revamp the network design, and really leverage the fact that I don’t care so much about cheating – you could never get away with a networking scheme like this in a competetive game.

• First, the timing of the scrolling, player bullet fire rates, enemy fire rates, etc were all modified to be integer-based instead of float-based.  There are no discrepencies in the way that integer calculations happen from machine to machine, so the timings of things like enemy spawning, level scrolling, etc. are now all perfectly in sync, frame by frame.
• Next, when the client detects a collision between entities, it sends a message to the server (which, you may recall, is running on the same system as the client – each machine gets one) notifying it of a collision.  These messages are also synced across the network.
• Thus, whenever an enemy does on any one client, it dies on both servers on the same tick (that is, the “first” client relative to the game clock to detect a collision determines when the collision took place).  This means that there are no longer any timing differences between deaths on each server (and if a collision happens to be missed by one client, but hits on the other, it will eventually reach the other client).
• Because players were already sending “I died!” flags as part of their network packets, these were already always perfectly in sync, so no change was needed there.

As an added bonus, since all collision detections are now handled by the client (and communicated to the server), the server never has to do any collision detection calculations on its own, which eases up on the CPU load somewhat (previously, the client and server were both doing collision calculations).  All the server has to do now is apply collisions reported by either the local client or the remote client.

So now, the actual mechanism by which the game keeps in sync from system to system is set, but how does it handle the three main enemies of the network programmer?

#### Network Gaming’s Most Wanted #1 – Latency

“Lag” is one of the most dreaded words in the network gaming world.  It’s always going to be present – nothing can communicate across the internet faster than the speed of light (and, because of transmission over copper, it’s really more like a sluggish 2/3rds the speed of light!).  Routers and switches also add their own delays to the mix.  According to statistics gathered by Bungie from Halo 3’s gameplay, most gamers (roughly 90%)end up with a round-trip latency of less than (or equal to) 250ms.  That is, it takes a quarter of a second for data to go from System A to System B and back to A.  That’s a long time for a fast-action networked game!  Thankfully, because messages sent from system to system in this game’s network design are never dependent on messages from the other, nothing has to round trip, so the latencies can effectively be halved, making the system much better at handling lag (because, quite frankly, there’s just less of it!)

As discussed previously, because the client can run ahead of the server and, thus, process local player input immediately, there’s no latency in what the player presses and what actually happens on-screen.  But what about how the remote player’s actions look?  With a ping under 100ms, there are next to zero visible discrepancies on the system.  That is, low-ping games are virtually indistinguishable from locally-played games.

At around the 400ms ping mark, it does start to become obvious that things aren’t quite right – due to the interpolation of remotely-shot bullets, they accelerate up to a certain part of the screen until they reach their known location then slow down to normal speed, which is fairly noticeable (I’m still trying to smooth this out a touch).  When enemies get too close to the remote player as it fires, due to the delay, the bullets will collide with the enemy, but the enemy will live longer than it appears it should (because the local client does not reliably know that the remote bullet is actually still alive, it can’t deal actual damage to the enemy, it has to wait for the server to confirm).

Above 1-2 seconds of latency, all bets are off – the local player will find the game still perfectly playable, but the movements of the remote player will be completely erratic, and remotely fired bullets won’t act at all like they should.  But, since 90% of gamers have much lower latencies, this is not really an issue.  For the majority of gamers, the game will look and play pretty close to how it would if both players were in the same room.

#### Network Gaming’s Most Wanted #2 – Packet Loss

Latency’s lesser-known brother is packet loss, which is where data sent from one machine to another never makes it (due to routing hardware failure, power outage, NSA interception, alien abduction, etc).  On a standard internet connection, you can generally assume that about 10% of the packets that you send will get lost along the way.  Also, just because you send Packet A before Packet B doesn’t mean they’ll arrive in the right order – a machine might get a packet sent later before one that was sent earlier.

With the XNA runtime, there are four different methods that you can send packets with (obviously you can mimic these with any networking setup, but I’m using XNA so it’s my frame of reference here):

• Unreliable – the other system will get these in potentially any order, or it may not even get them at all.  The name says it all – you can’t rely on these packets.  This is probably not the best option to use.
• In-Order – These packets are for data for which you really only need the most recent data; you only care about the most recent score, for instance – not what the score was in a previous packet.  Thus, these packets contain extra version information so that the XNA runtime can ensure that packets that arrive out of order don’t reach you.  As soon as a new packet comes in, it becomes available to the game.  If a packet that’s older than the most recent one comes in, it’s discarded.  You immediately get new ones at the cost of never getting older ones.  For many games, this is a perfect scenario.
• Reliable – These packets will always arrive.  When the XNA runtime receives one of these, it sends an acknowledgement to the other system that it received it.  If the system that sent it doesn’t receive such an acknowledgement, it’ll resend (and resend and resend and…) until it finally arrives at the destination.  Packets sent reliably are not vulnerable to packet loss; if you send it, as long as the connection remains valid you know it will reach the destination eventually.  However, these packets may not arrive in the proper order (you may receive Packet C before Packets A or B).
• Reliable, In-Order – On the surface, this sounds like the best choice!  These packets always arrive in the right order, and they always arrive!  That is, you will always get Packets A, B, C, and D, in that order.  There’s a hidden downside, though:  If the game recieves Packet C, but has not yet received Packets A or B, it has to hold onto that packet until both A and B arrive, which, if they need to be resent, can really ratchet up the latency.  Any one packet that needs to be resent will hold up the whole line until it arrives.  Clearly, this type of packet should only be used when absolutely necessary – for normal gameplay, it’s better to use In-Order or Reliable on their own.

Eventually, I decided to send packets in the Reliable way, but not In-Order.  But, to minimize the amount that the game has to wait for resent packets to arrive, each packet contains eight frames worth of input/collision data.  That way, as long as one out of every string of 8 packets arrives, the server will have all of the relevant information to sync up to that point.  And if, for some reason, 8 packets in a row are all lost in transmission, they’ll be resent and make it eventually anyhow.

To handle this, the game essentially has a list of frames that it’s received data for (8 of which come in with each packet).

• For each frame that a packet contains, if the frame already been simulated by the server (a frame from the past), that frame is ignored.
• Similarly, if the frame is already in the list, it’s ignored.
• If it’s not a past frame and it isn’t already in the list, add it to the list (in order – the list is sorted from earliest to latest).
• After this is done, if the next frame that the server needs to simulate is in the list, remove it from the list and go!  Otherwise, wait until it is.

The game doesn’t care which order the frames are received in – as long as it has the next one in the list, it’ll be able to continue on.  Because of the redundancy, it rarely has to wait on a resend due to packet loss.  In fact, using XNA’s built-in packet loss simulation (thank you, XNA team!), the packet loss has to be increased to over 90% before the latency of the simulation starts to increase (hypothetically, the magic number is above 87.5% packet loss – greater than 7/8 packets lost).

The disadvantage of this system is that it does add to the bandwidth use, as each packet now contains an average of eight times as much data as it would normally, which brings us to…

#### Network Gaming’s Most Wanted #3 – Bandwidth

Ah, bandwidth.  There’s no point in having a low-latency connection between two systems if the game requires too much bandwidth for the connection to keep up.  Because the Xbox Live bandwidth requirement is 8KB/s (that’s kilobytes), that became my goal as well.

This is where I overengineered my system a bit.  I was estimating, as a worst case, an average of 10 collisions per frame.  With packet header overhead, voice headset data, and everything, with 8 frames worth of data in each packet, I expected to be just BARELY below the 8KB/s limit.

When I finally got the system up and running, it turned out the game was using less than 4KB/s.  The average amount of collisions per frame is closer to TWO than it is to 10 (unfunny side note: I had the right number, but the wrong numerical base.  My answer was perfect in binary), even with a lot of stuff going on (though an individual frame may have many more, there are usually large spaces between collisions as waves of bullets smack into enemies).  The most I’ve been able to get it up to with this system is about 5KB/s, which means the game still has a delightful 3KB/s of breathing room.  I think I’ll keep it that way!

#### Final Remarks

Hopefully this has been an informative foray into the design of a network protocol for an arcade-style shoot-em-up game.  I’m no network professional – in fact, this is the first network design I’ve ever done, so I’m sure people who do this stuff for a living are laughing at my pathetic framework.  If anyone has any suggestions as to how I might improve my network model, I’m all ears – while it works pretty well, I’m always open to ideas!

## Networking Is Hard (Part 2)

In my previous post, I weighed the advantages and disadvantages of my game vs. a standard FPS with regard to networking.  After doing so, I came up with an initial network design.

The biggest issue, personally, was dealing with the causality of the network game.  Each player gets information about the other with a delay, so neither player is ever seeing exactly the same set of circumstances (perfectionism and network lag don’t mix very well).  I think Shawn Hargreaves describes it best in one of his networking presentations: he says to treat each player’s machine as a parallel world.  They’re not exact, but the idea is to try to make them look as close as possible.  It doesn’t matter  if things don’t happen exactly the same, but you don’t ever want the following conversation to occur:

Player 1: “Wow, can you believe I killed that giant enemy at the last second?  That was amazing!”

Player 2: “…What giant enemy?”

Procyon’s Initial Design

The design I started with was a hybrid of both the peer-to-peer lockstep and client-server models.

Basically, within each system there is a client/server pair.  In addition, each server runs in lock-step with the other, so that they both always tick with the exact same player inputs for a given frame, keeping the two machines’ servers perfectly in sync.  Since the only data being sent across the network is player inputs, bandwidth use was ridiculously low.  And since nothing ever has to round-trip from the client to the server and back, the system still gets the effectively-halved-ping that a pure lockstep setup gets.  Also, because the client and the server are running on the same machine, communications between the two (mostly, the server notifying the client of events that happened based on the other player’s input) becomes very trivial – the server can just call callbacks to the client, and no sort of RPC mechanism is necessary.

Ignoring the clients for a moment, the servers are a very traditional lockstep system.  Each normal tick, the player input is polled and then sent across the network.  Once the server receives the input from the remote system for the next frame, it ticks forward.  As you can imagine, each machine’s server is a bit lagged, because it has to wait for inputs across the network.  That’s where the client comes in.

Each system also runs a client, which runs ahead of the server.  This client always ticks every (in Procyon’s case) 60th of a second, processing the local player’s inputs right away, and running prediction on the remote player based on the last set of inputs and the last known position that the server knows about.  That way, the local player’s ship always reacts right away to player inputs (on the client, which is represented on-screen).

Essentially:

1. The current local player’s input is polled and sent to the server.
2. The input is also sent immediately across the network to the other machine.
3. The client ticks using this input – the position and actions of the remote player are predicted based on the last-known input and position from the server.
4. If the server has the remote input for the next frame that it has to do (which is always an eariler frame than the client), it also ticks (no prediction necessary on the server, as it has up-to-date information about both players for the given frame it’s simulating).  Again, there is a server on both machines, but they’ll both end up with the exact same simulation.

#### (Mostly-) Deterministic Enemies

Here’s where one of the advantages of the game comes into play.  In my last post, I mentioned that enemies are deterministic in their movements.  This is actually a key part of the networking.  What it means is, there is no prediction required for simulating enemy behaviors ahead of the server.  For instance, say that the client is currently 5 frames ahead of the server.  While the client is ticking frame 450, the server is only on 445 (because it hasn’t received network input for frame 446 from the other machine yet).  Even though the server hasn’t simulated enemy movements yet, their movements are very strictly defined, so the client doesn’t have to guess – it knows exactly where an enemy is on frame 450.  Consequently, the client running ahead of the server is not a problem with regards to enemy positions – when you see an enemy in a specific place on the screen, you know that’s exactly where it will be on the server when the server finally simulates that same frame.

Unless the other player kills it first.

That’s right – there is exactly one case in which an enemy’s behavior is non-deterministic, and it’s most-easily described as follows:  your local client is currently simulating frame 450, the server has simulated frame 445.  You see a large cannon ship start to charge up its beam weapon for a massive attack.  However, the server gets remote player input from frame 446 (the next frame it has to simulate), and when simulating it, realizes that the other player dealt the killing shot to the ship.  Suddenly, the client’s view of that enemy is wrong – it should have died four frames prior.

In essence, the only way that a client’s view of an enemy is ever wrong is if the other player has killed it in the past, and the server hasn’t caught up.  This is a very important property: the only time, ever, that an enemy is not where you see it as is when it’s not anywhere at all (because it already died).  However, this is where it starts to get tricky.

Most of the time, it’s not a big issue.  The client kills the enemy as soon as the server tells the client that the enemy should be dead, and it just dies a few frames too late.  But when the enemy has fired bullets (or any other type of weapon), suddenly there’s a problem.  What has to happen in that case is that the client has to remove the bullets that shouldn’t really have been spawned.  With reasonable pings, the bullets will generally be close enough to the enemy’s death explosion that they won’t really be noticeable, they’ll blink up in the middle of the explosion and disappear by the time it’s done.  With larger lag times, though, bullets for dying enemies might spontaneously disappear.  Ah, lag.  How networked games love you so.

The real problem, though, is when, using the frame numbers above, you crash your ship into an enemy on tick 450 that actually died on tick 446.  What then?  You crashed into something that shouldn’t even have been there!  After much internal debate, I decided there’s a really simple way to arbitrate this: on your screen, you crashed, so you still get the consequences.  Even though you hit something (either a should-have-been-dead enemy or a bullet spawned from such an enemy) that shouldn’t have even been there, you still ran into something on your screen, and still pay the price.  So, in addition to player inputs that get sent to the server (and across the network), the client also sends a flag that says “I died!”

As an added bonus, the server no longer needs to calculate collisions against either player – when a player dies, the client signals it (This plays into another advantage of my game – in a competetive multiplayer game, you would never, ever trust a client machine to tell you whether or not its player got hit by something).

#### Random Number Generation

One issue has to do with random number generation.  Obviously, when connecting the two machines across the network, the machines have to agree on a random number seed so that random numbers generated on each are the same.  In fact, not only do the machines on either side of the network have to have the same random number seeds, but the client and server also have to start with the same seeds, or they’ll get out of sync (a machine getting out of sync with itself is always funny).

However, what happens when an enemy (Smallship 5) shoots a bullet in a random direction, and then dies “in the past” based on a server correction?  On the server(s), this random number generation never happened.  On the client, however, a bullet was shot, and a random number was created.  With a single random number generator, that means all of the random numbers from that point on on the client are going to be incorrect.  Thankfully, there’s an easy way around that.

Give every enemy its own random number generator!

I found a very lightweight (and very high-quality) random number generator: Marsaglia’s mutliply-with-carry generator (or, if you prefer, the infinitely more difficult-to-read Wikipedia version).  This generator only requires two unsigned integers for each generator to do its magic, so there wasn’t much overhead to hand one of these off to each enemy.  So, the initial random number seed is decided upon before the level begins to load.  As the level begins to load, and enemies are created, a new generator is created for each entity, using random seeds generated from the original generator.  This way, each enemy gets its own generator, and if an enemy ever generates any extra random numbers before it dies, it doesn’t affect any of the other enemies at all!  Problem solved.

#### Remote Player Bullets

Again, pretend the client is ticking frame 450.  When the server reaches 445, it notices that the remote player fired a bullet.  So it notifies the client: “Hey, 5 frames ago, the other player fired a shot!.”

• The client knows that it needs to spawn a new remote player bullet.
• It also knows exactly where that bullet will be at the current tick (since it was spawned at tick 445, but it’s now tick 450, it can tick the bullet forward 5 frames).
• It does not, however, know that the bullet will actually survive until tick 450.  It’s possible that anywhere from tick 446 until 449, that it might hit something.

Even though it knows exactly where the bullet should be, it doesn’t display it there.  It actually starts the bullet in front of its current estimate of where the remote player is (the predicted position), and interpolates it into the correct spot.  That way, remote player’s bullets don’t seem to appear way out in front of the remote player, they still start right where they seem like they should.

The client knows exactly where the bullet should be, but not that it’s actually there (because it might have died on, say, tick 448).  So while it does collision detection against enemies, if the remote bullet hits an enemy, it doesn’t actually do any damage on the client – it just removes the bullet.  If it turns out it dealt damage after all, the server will eventually send a correction to the client.

However, eventually, the server reaches frame 450 (when the client first learned about the bullet).  If the bullet is still alive on the server, then we know that it never hit anything from frame 445 until then, so it was a live bullet when the client found out about it.  Also, if it’s still alive on the client (it didn’t collide with anything while the server caught up to frame 450), that means the client knows that the bullet is still alive.

Now that it knows that it has a bullet that is guaranteed to still be alive, and is at the exact position that it’s supposed to be, the bullet can flip from being treated as a remote bullet to being treated exactly like a locally-fired bullet.  Basically, this bullet now will deal damage and act exactly like a bullet fired by this machine’s own player!  It’s one less thing that will act different on the other machines, further strengthening the illusion that both players are seeing the same thing.

#### Floating Points Are Sharper Than Expected

However, there was one big issue with this entire design: floating-point errors.  Due to minute differences in the way floating point numbers are calculated on different systems (due to different CPUs, code optimizations, quantum entanglement, etc), the collision code on each system acted slightly differently.  Consequently, the two servers running in lockstep weren’t actually in sync.  This was causing all sorts of issues – slight timing differences on enemy deaths caused discrepencies in scores and energy amounts…and the real kicker is that it was possible that two objects would collide on one system and never collide at all on the other simulation (a grazing collision on one might not trigger as a collision on the other system).  Added bonus when an enemy died on one server just after shooting a ton of bullets, but just before shooting them on the other server.  Suddenly, one player’s screen would be full of enemy bullets, and the other’s would be clear as the summer sky.

This became a real problem, and it took a while to solve it (again, perfectionism and networking don’t mix)…and next time, I’ll present the solution.

## Networking Is Hard (Part 1)

I haven’t had much development time in the last (nearly two) month(s), but I have had time to nearly finish up the networking code for the game, so I thought I would describe some of the work that went into the design of the system.

#### Resources

I tried to find some references on how most shmups (shoot-em-ups) such as my current game handle their networking, but I didn’t really find anything.  There are a lot of resources on how RTS games do it (such as this great article about Age of Empires’ networking), and even more articles about how to write networking for an FPS game (the best of which, in my opinion, are the articles about the Source engine’s networking and the Unreal engine’s networking).  There’s also a great series of articles on the Gaffer on Games site.  And no list of references would be complete without the Gamedev.net Multiplayer and Networking forum.

One of the most seemingly-relevant things to my game that I read was that, in current system emulators (for instance, SNES emulators) that have network play, they work in lock step with each other…that is, each system sends the other system(s) its inputs, and when all inputs from all systems have arrived, they can tick the simulation forward.  I did an early experiment to see how well this would work with my game, and it turns out, not so well – the latency introduced with even a moderate ping (100ms) is rather prohibitive.  The player would press up on the gamepad and have to wait for the ship to start moving…and it would only get worse with higher latencies!  Obviously, this wouldn’t work.

Since there were many articles about the FPS  model of networking, I opted to use it as a starting point, as my game is also a fast-action game.  First off, I decided to make a (partial) list of the advantages and disadvantages of my game type vs. the FPS model:

#### Advantages of Procyon vs. an FPS

• Only two players are supported, which greatly simplifies the bandwidth restrictions and the communication setup.
• There’s no need to support joining/quitting in the middle of a game (again, simplifying game communications and state management)
• Since it’s cooperative, I’m not really worried about players cheating, so I can allow clients to be authoritative about their actions moreso than the average FPS can.
• Players can’t collide with each other, so there’s no need to worry about such things (in fact, a player basically can’t affect the other player at all – a player can only affect enemies)
• Enemies are deterministic in their movements – none of my enemy designs have movement variation based on the player’s actions, so collision vs. enemies in a given frame is always accurate (unless the enemy was dead “in the past” due to the actions of the other player, which is discussed later).

#### Disadvantages

• In an FPS, the view is more limited – it’s rare that a player sees all of the action at once due to the limited field of view.  However, in Procyon, everything that is happening in the game is completely visible on both screens, meaning the game generally has to be less lenient about discrepancies.
• Most FPS games are “instant hit.”  For instance, when you fire a pistol in a FPS game, there’s no visible bullet that streaks across the screen, so it’s easier to fudge the results on the server.  In Procyon, however, all of the weapons are on-screen and visible, so fudging them in a minimally-noticeable way becomes very difficult.
• In many FPS games (though not all), there are WAY less entities active at a time than there are in a shoot-em-up.
• This falls into the realm of “not a lot of references for this type of game,” but many of the “cheats” employed by FPS game programmers are really well-known, but very domain-specific; with a shmup, I’d have to make up my own network fakery to hide latency.

#### Network Strategies

There are two main network strategies used in games (at least, in those that I researched): lockstep and client-server.

Lockstep keeps all systems involved perfectly in sync, though at the cost of making it more susceptible to latency, because the game doesn’t step until the other player’s inputs have traversed the network.  Generally, every so often (say, every 1/30th of a second), the current input is polled and then sent across the network.  Then, once both/all inputs have reached the local machine, it ticks the game.  This is the type of networking used by emulators (out of necessity – there’s no reliable way to do any sort of prediction with an emulator anyway) and some RTS games (such as Age of Empires, linked above).  RTS games can hide the latency a bit by causing all commands to execute with a bit of a delay.  One interesting property of lockstep networking is that nothing ever has to round-trip across the network.  That is, nothing that server B ever sends back relies on something that Server A sent previously.  Because of this, the game’s latency is effectively halved, as the meaningful measurement is one-way latency, not a full there-and-back ping.  Lockstep games are frequently done peer-to-peer: each system sends its inputs to each other system…there’s no central entity in charge of all communications.

Conversely, a client-server architecture (which is used by an overwhelming majority of FPS games) puts one system in charge and generally gets authority over the activities of all players.  In this case, the client sends its information to the server (inputs, positions, etc), and the server sends back any corrections that need to be made (locations/motion of the other players, shots fired, things hit/killed, etc).  The client can run ahead of the server based on a prediction model, where it gives its best guess as to where things are currently based on the last communication from the server.  This makes game lag much less apparent (At least, in the player’s own movements), as the player’s inputs can be processed immediately on the local machine.

In the next installment, I’ll talk through Procyon’s initial network design, which is a bit of a hybrid between the client-server and lockstep architectures…stay tuned!

## .NET Reflection and State Machines

The .NET framework’s reflection setup can be amazingly useful.  At a basic level, it allows you to get just about any information you could want about any type, method, assembly, etc that you could want.  In addition, you can programmatically access type constructors and methods, and invoke them directly, allowing you to do all sorts of neat stuff.

One useful application of this is in the creation of state machines.  Imagine an entity in a game that flies around in a pattern for a bit, then stops to shoot some bullets, then returns to flying.  Such an entity would have two states, “Flying” and “Shooting.”

#### A First Attempt

You might decide to create a state machine that could handle this entity, but also be usable for all entities that need a state structure.  What should a state machine like that need to be able to do?

One such set of possibilities is as follows:

• Be able to assign a state based on a string state name.
• Each state has three functions:
• Begin – This function is called whenever the state is transitioned into, allowing it to initialize anything that needs it.
• Tick – This is the meat of a state, it is called once per game tick while the state is active.
• End – This is called when the state is transitioned out of, allowing any clean up.  Note that this is called before the next state’s Begin function is called.
• However, a state shouldn’t have to specify functions that it doesn’t need.  If a state has no need for the Begin function, it doesn’t need to implement it.
• If the state changes during a Tick call, the new state will tick as soon as it happens.  This allows many states to fall through in rapid succession.  For instance, a player character in a 2D Mario-like puzzle might be in the middle of the “Jumping” state, while holding right on the controller.  When the character hits the ground, it would transition into the “Standing” state, but from there, it would then transition into “Walking” because the player is holding right.  That way, the character would land and immediately start walking, which is what one would expect when playing the game.

So what would a first draft version of such a machine need to look like?  First, we’d need a place to store delegates to the three functions needed for a state:

```public delegate void StateDelegate();

class StateInfo
{
public StateDelegate OnBegin { get; set; }
public StateDelegate OnTick { get; set; }
public StateDelegate OnEnd { get; set; }

public void Begin()
{
if (OnBegin != null)
OnBegin();
}

public void Tick()
{
if (OnTick != null)
OnTick();
}

public void End()
{
if (OnEnd != null)
OnEnd();
}

public StateInfo()
{
OnBegin = null;
OnTick = null;
OnEnd = null;
}
}
```

Given that, we can now store them in a dictionary, indexed by by the state’s name (a string).  To initialize it, we would just call AddState with the state name and three delegates (which may be null, to specify that a given function is unnecessary), and it would add that state to the list.  We also need to be able to Tick the state machine and change the state.  The whole contraption would look something like the following:

```public class StateMachine
{
Dictionary<string, StateInfo> states = new Dictionary<string, StateInfo>();
bool ticking = false;
StateInfo currentStateInfo = null;
string currentStateName = null;

public void AddState(string stateName, StateDelegate begin, StateDelegate tick, StateDelegate end)
{
StateInfo info = new StateInfo();
info.OnBegin = begin;
info.OnTick = tick;
info.OnEnd = end;

states.Add(stateName, info);

// If no states have been set yet, this state becomes the initial state.
if(currentStateName == null)
State = stateName;
}

public string State
{
get { return currentStateName; }
set
{
// End the previous state.
if (currentStateName != null)
currentStateInfo.End();

// Set the new state
currentStateName = value;
currentStateInfo = states[currentStateName];

// Initialize it.
currentStateInfo.Begin();

// If we're in the middle of the Tick function, go ahead and tick the new
// state now, as well.
if (ticking)
currentStateInfo.Tick();
}
}

public void Tick()
{
// Set ticking so that we know that we're in the middle of ticking this machine.
// That way, if the state changes in the middle of this, we know to run the next
// state's Tick as well.
ticking = true;
currentStateInfo.Tick();
ticking = false;
}
}
```

As you can see, it’s fairly straightforward.  An entity can call AddState (probably during its constructor) on the state machine to add a new state.  It can assign a new state by using State = “newState”, which will end the current state and begin the new state.  Finally, the entity just needs to call Tick to run the current state.

In practice, this might look something like:

```public class FlyShooter : EntityBase
{
/* Entity vars Go here */
StateMachine machine = new StateMachine();

public FlyShooter()
{
/* Initialize stuff */

// Flying doesn't need an end
machine.AddState("Flying", FlyingBegin, FlyingTick, null);

// Shooting doesn't need Begin or End.
machine.AddState("Shooting", null, ShootingTick, null);
}

void FlyingBegin()
{
/* Begin flying state */
}

void FlyingTick()
{
if( IsTimeToShoot )
{
machine.State = "Shooting";
return;
}
/* Tick flying */
}

void ShootingTick()
{
if( DoneShooting )
{
machine.State = "Flying";
return;
}
/* Tick shooting */
}

public override void Tick()
{
machine.Tick();
/* Also tick anything that happens in all states */
}
}
```

There’s not much to it.

One thing to note is that the way that the state fall-through works (that is, changing state in the middle of another state’s Tick function) will still cause the rest of the current function to execute (after it’s already partially run another state), so it’s best to either change states at the very end of the function or to return after setting the new state, as is done above).

Though this works pretty well as-is, say you want to add new states to an entity afterwards.  What if the enemy now needs a special “Dying” state.  Not only do you have to add new functions (DyingBegin, DyingTick, and DyingEnd), you also have to remember to call AddState at object startup.  Wouldn’t it be nice if you could just add the new functions and it would just work?  As it turns out, using .NET’s reflection functionality, you can do exactly that!

#### Mirror, Mirror On the Wall, What Are the States That I Can Call?

There are a number of ways to implement a state machine system using reflection.  The one that was settled upon for Procyon was the design with the least redundancy (that is, “don’t repeat yourself”).  State names only appear once in a given entity’s codebase (excepting when states are set), and it takes a small number of characters, right where the state code is defined, to mark a state.  At the point that it’s added to the code, it’s already ready to use, there’s no additional list of states to update.

Most reflection code works by looking for .NET Attributes.  Attributes are class objects that can be added as metadata to a class, a method, a property, or any number of other pieces of your code.  Also, you can create your own attributes – program-specific metadata that you can later find.

In the case of the state machine, a very simple attribute is needed.

```using System.Reflection;

[AttributeUsage(AttributeTargets.Method)]
public class State : Attribute
{
}
```

It doesn’t need any sort of custom data, so there’s not much to it.  The AttributeUsage at the top (which is, itself, an attribute) specifies that the “State” attribute being defined can only be applied to methods.  Attempting to add it to anything else (say, a class) would generate a compiler error.

With this attribute, you’ll be able to mark methods as states.  In this case, each method will take a StateInfo as a parameter (meaning that the StateInfo class, which was described above, has to be a public class now).  The method will then fill in the state information with delegates.  The most elegant way to handle this is using anonymous delegates.  Here’s that enemy class again, using the new method:

```public class FlyShooter : EntityBase
{
/* Entity vars Go here */
StateMachine machine;

public FlyShooter()
{
/* Initialize stuff */

// Create the new state machine with "Flying" as the initial state.
machine = new StateMachine(this, "Flying");

// Note - no states are being explicitly initialized here.
}

[State]
void Flying(StateInfo info)
{
info.OnBegin = delegate()
{
/* Begin flying state */
};

info.OnTick = delegate()
{
if (IsTimeToShoot)
{
machine.State = "Shooting";
return;
}
/* Tick flying */
};

// Don't need to set info.End
}

[State]
void Shooting(StateInfo info)
{
info.OnTick = delegate()
{
if (DoneShooting)
{
machine.State = "Flying";
return;
}
/* Tick shooting */
};
}

public override void Tick()
{
machine.Tick();
/* Also tick anything that happens in all states */
}
}
```

As you can see, the [State] attribute is there.  Each of the (up to) three functions for a state are initialized inside of the state function, cleanly grouping the three functions together into one unit.  The state function’s name is the state’s name, so it’s only specified once.  And, of course, there’s no manual list manipulation; to add a new state, simply add the code for the new state and make sure that it’s marked by a [State] attribute…the runtime will take care of the rest.

To get this whole thing to work, though, there’ll obviously have to be some code to find methods using it.  When creating your state machine, you’ll now want to pass in a pointer to the object that the state machine belongs to.  From that, it will be able to get the type of object for which you’re creating the state machine.  Automatically adding states that are marked with your state attribute goes something like this:

• Scan through each method in the type of the object you passed in (if you want to be able to grab private methods from parent classes of the current object’s type, you’ll need to scan through the classes in the heirarchy one by one)
• If it doesn’t have the State attribute, continue on to the next method.
• If it does have the attribute, get the name of the method (this becomes the state’s name).
• Once you have the name of the method, call the method and let it populate a StateInfo that you hand to it.
• Once that’s done, add the method into the state dictionary, using the state method’s name as the key (and thus, the state name).

The AddState function can go away – it’s no longer necessary, as the state list is now update completely automatically.  The StateMachine class’ constructor needs to be updated a bit, to add all of this reflection magic into it.  It will now look something like this:

```/// <summary>
/// Create a new state machine using the given object - it will automatically
/// search for State attribute-marked methods and use those to generate its
/// state table at creation time.
/// </summary>
/// <param name="param">The object that the state machine belongs to</param>
/// <param name="initialState">The name of the initial state of the machine.</param>
public StateMachine(object param, string initialState)
{
Type paramType = param.GetType();

// Loop through parent types to make sure to pick up any private states.
for (Type currentType = paramType; currentType != typeof(object); currentType = currentType.BaseType)
{
// Now loop through each method in this type (and this type only, no parent methods).
foreach (var method in currentType.GetMethods(BindingFlags.Instance | BindingFlags.DeclaredOnly |
BindingFlags.NonPublic | BindingFlags.Public))
{
// See if this method is a State.  If it's not, go on to the next method.
object[] attributes = method.GetCustomAttributes(typeof(State), true);
if (attributes.Length == 0)
continue;

// Get the method name, ensure it's not already in the list.
string methodName = method.Name;
if (states.ContainsKey(methodName))
continue;

// Call into this method to populate the state info...
StateInfo info = new StateInfo();
method.Invoke(param, new object[] { info });

// ...and add it into the state list.
states.Add(methodName, info);
}
}

if (initialState != null)
State = initialState;
}
```

None of the rest of the state machine class’ code needs to change; it all works exactly as it did before.  The new constructor means that the list gets created at state machine creation time, but the rest of the internal data remains exactly the same.

#### Possible Improvements

It works pretty well as-is, but there are ways it could be improved.  For instance:

• You could add a type of “Sleep” method to the state machine, to make it delay a certain number of ticks before continuing to tick the current state
• You could add a list of transitions to the StateInfo, so that all transitions to different states can be declared up-front, in an easy-to-read manner.  That removes that extra state-setting clutter from your tick routine (and even means that a state may not need a Tick routine at all!)
• If you need multiple sets of states for objects (say the player needs a collection of states for movement and a separate collection of states for weapons fire), you could modify the State attribute to provide a collection name (like [State(“Movement”)]), and specify a state collection when creating your state machine.

This is a small list of possible changes.  If you have other ideas, let me know!

## Lightning Bolts

You’re flying your ship down a cavern, dodging and weaving through enemy fire.  It’s becoming rapidly apparent, however, that you’re outmatched.  So, desperate to survive, you flip The Switch.  Yes, that switch.  The one that you reserve for those…special occasions.  Your ship charges up and releases bolt after deadly bolt of lightning into your opponents, devastating the entire enemy fleet.

At least, that’s the plan.

But how do you, the game developer, RENDER such an effect?

#### Lightning Is Fractally Right

As it turns out, generating lightning between two endpoints can be a deceptively simple thing to generate.  It can be generated as an L-System (with some randomization per generation).  Some simple pseudo-code follows: (note that this code, and really everything in this article, is geared towards generating 2D bolts; in general, that’s all you should need…in 3D, simply generate a bolt such that it’s offset relative to the camera’s view plane.  Or you can do the offsets in the full three dimensions, it’s your choice)

```segmentList.Add(new Segment(startPoint, endPoint));
offsetAmount = maximumOffset; // the maximum amount to offset a lightning vertex.
for each generation (some number of generations)
for each segment that was in segmentList when this generation started
segmentList.Remove(segment); // This segment is no longer necessary.

midPoint = Average(startpoint, endPoint);
// Offset the midpoint by a random amount along the normal.
midPoint += Perpendicular(Normalize(endPoint-startPoint))*RandomFloat(-offsetAmount,offsetAmount);

// Create two new segments that span from the start point to the end point,
// but with the new (randomly-offset) midpoint.
segmentList.Add(new Segment(startPoint, midPoint));
segmentList.Add(new Segment(midPoint, endPoint));
end for
offsetAmount /= 2; // Each subsequent generation offsets at max half as much as the generation before.
end for```

Essentially, on each generation, subdivide each line segment into two, and offset the new point a little bit.  Each generation has half of the offset that the previous had.

So, for 5 generations, you would get:

That’s not bad.  Already, it looks at least kinda like lightning.  It has about the right shape.  However, lightning frequently has branches: offshoots that go off in other directions.

To do this, occasionally when you split a bolt segment, instead of just adding two segments (one for each side of the split), you actually add three.  The third segment just continues in roughly the first segment’s direction (with some randomization thrown in)

```direction = midPoint - startPoint;
splitEnd = Rotate(direction, randomSmallAngle)*lengthScale + midPoint; // lengthScale is, for best results, < 1.  0.7 is a good value.
segmentList.Add(new Segment(midPoint, splitEnd));```

Then, in subsequent generations, this, too, will get divided.  It’s also a good idea to make these splits dimmer.  Only the main lightning bolt should look fully-bright, as it’s the only one that actually connects to the target.

Using the same divisions as above (and using every other division), it looks like this:

Now that looks a little more like lightning!  Well..at least the shape of it.  But what about the rest?

#### Adding Some Glow

Initially, the system designed for Procyon used rounded beams.  Each segment of the lightning bolt was rendered using three quads, each with a glow texture applied (to make it look like a rounded-off line).  The rounded edges overlapped, creating joints.  This looked pretty good:

..but as you can see, it tended to get quite bright.  It only got brighter, too, as the bolt got smaller (and the overlaps got closer together).  Trying to draw it dimmer presented additional problems: the overlaps became suddenly VERY noticeable, as little dots along the length of the bolt.  Obviously, this just wouldn’t do.  If you have the luxury of rendering the lightning to an offscreen buffer, you can render the bolts using max blending (D3DBLENDOP_MAX) to the offscreen buffer, then just blend that onto the main scene to avoid this problem.  If you don’t have this luxury, you can create a vertex strip out of the lightning bolt by creating two vertices for each generated lighting point, and moving each of them along the 2D vertex normals (normals are perpendicular to the average of the directions two line segments that meet at the current vertex).

That is, you get something like this:

#### Animation

This is the fun part.  How do you animate such a beast?

As with many things in computer graphics, it requires a lot of tweaking.  What I found to be useful is as follows:

Each bolt is actually TWO bolts at a time.  In this case every 1/3rd of a second, one of the bolts expires, but each bolt’s cycle is 1/6th of a second off.  That is, at 60 frames per second:

• Frame 0: Bolt1 generated at full brightness
• Frame 10: Bolt1 is now at half brightness, Bolt2 is generated at full brightness
• Frame 20: A new Bolt1 is generated at full, Bolt2 is now at half brightness
• Frame 30: A new Bolt2 is generated at full, Bolt1 is now at half brightness
• Frame 40: A new Bolt1 is generated at full, Bolt2 is now at half brightness
• Etc…

Basically, they alternate.  Of course, just having static bolts fading out doesn’t work very well, so every frame it can be useful to jitter each point just a tiny bit (it looks fairly cool to jitter the split endpoints even more than that, it makes the whole thing look more dynamic).  This gives:

And, of course, you can move the endpoints around…say, if you happen to have your lightning targetting some moving enemies:

So that’s it!  Lightning isn’t terribly difficult to render, and it can look super-cool when it’s all complete.

## Collision Detection Performance (Volume 2)

Yikes, I’m getting backlogged on the stuff I want to write about!  <Insert generic complaint about how work is eating up most of my time here>

Anyway, this will probably be way shorter than it deserves, but my memory on the subject is about 3 months old.  Basically, this will be more or less a short chronicle of the dumb story of the mesh vs. mesh tests.

#### Appearances Can Be And Are Frequently Deceiving

When I started work on the collision detection observation, there was one surprising fact: the mesh vs. mesh code (used to determine whether the player was intersecting enemy ships) was working at full speed!  I didn’t seem to have to do any optimization at all on it to get it working.

I did, however, opt to go ahead and change the functions to not be recursive (as the current implementation was, of course, recursing into both meshes’ sphere trees).  When I finished that work, suddenly, the routine was much, much slower.  Was, in this case, the recursion overhead better than what it took to handle the double tree recursion in a non-recursive way?

#### The Answer May Surprise You

While I was working out the kinks in the non-recursive version, I had added some visual feedback to display unambiguously when the meshes were colliding.  While experimenting, I reverted back to the old (recursive) method, and found out something surprising: the collision results weren’t accurate!  It was reporting collision in cases where the player was inside of the object’s space, but nowhere near any edges!

Turns out, there was an erroneous fall-through case, and it was falling through to a “true” result, causing the entire collision test to short-circuit with an invalid collision result.  Sure the original routine was fast, but it didn’t work at ALL.  Now that I’d fixed that and had an accurate version, it was super, super slow.  Now it became an imperative to optimize.

#### Double Dribble

Through the course of optimizing the routine, I tried all sorts of crazy stuff.  I eliminated the recursion, I tried AABB vs. Rotated AABB tests, I tried Sphere vs. AABB tests, etc.  There was nothing that I could do.  As I was stepping through the code, however, I started noticing something odd: there were duplicate triangles all over the place.  I would see the same triangle tens of times in a given collision test.

As it turns out, the way I was dividing up the mesh in the tree creation routine was boneheaded.  What I would do was as follows:

1. If the current set of triangles has less than a specified number of entries, stop splitting the mesh, we’ve reached a leaf node.
2. Figure out the bounding box of the current set of triangles
3. Find the axis on which it was best to split the set of triangles (where best was determined as the most balanced split, i.e. where the difference between triangle counts on the left and right of the split was lowest)
4. For each triangle
1. If the triangle has a vertex to the left of the split, add it to the left side.
2. If the triangle has a vertex to the right of the split, add it to the right side.
5. Once the mesh is divided into the two sets, recurse into each set to split it further as necessary

Now, a cursory analysis of this algorithm should tell you that step 4 is horribly broken.  If a triangle straddles the split, it would get added into both sides, thus expanding the bounding boxes of each sub element to encompass all triangles that straddle the split, causing there to be tons of duplicates of the same triangles throughout the code.

The correct way to do it would be:

1. If the current set of triangles has less than a specified number of entries, stop splitting the mesh, we’ve reached a leaf node.
2. Figure out the bounding box of the current set of triangles
3. Find the axis on which it was best to split the set of triangles (where best was determined as the most balanced split, i.e. where the difference between triangle counts on the left and right of the split was lowest)
4. For each triangle
1. Calculate the centroid of the triangle (i.e. average the 3 vertices).
2. If the centroid is less than the split, add the triangle to the left, OTHERWISE add it to the right.
5. Once the mesh is divided into the two sets, recurse into each set to split it further as necessary

There, now a triangle only gets added to the side that it’s MOSTLY on, and there are no duplicates.  This change greatly boosted the speed of the collision, however there were cases where it would still cause a framerate dip.  Curses and drat!

#### All the King’s Horses…

Try as I might, I could not get the mesh vs. mesh intersection test to perform well.  So what happened when you get into a situation like that?  You do what every game developer does: you cheat!

What existed at the time was a wicked fast sphere vs. mesh collision test.  So rather than treat the player ship as a mesh, I opted to treat it as a representative collection of spheres (There are 29 spheres total, and it’s a pretty good approximation of the player ship – in fact, it’s probably better than necessary).

Suddenly, player vs. enemy collision performance was great!  Since this was the only place that I had needed a mesh vs. mesh, I could get rid of that slow beast.

#### ~fin~

All of the collision code is complete, now, so next (give or take a post or two) I’ll talk about something with a bit more of a visual element – Bolts of lightning!

## Collision Detection Performance (Volume 1)

I have been hard at work on my game (in my ridiculously limited spare time) for the last month and a half. One major hurdle that I’ve had to overcome was collision detection code. Specifically, my collision detection performed great on my PC, but when running it on the Xbox 360, everything would slow to a crawl (in certain situations).

The types of collision detection I have to deal with are varied, due to the weird way that I handle certain classes of obstacle (like walls):

• Player bullets vs. Enemy – Player bullets are, for simplicity, treated as spheres, so sphere/mesh testing works here.
• Enemy bullets vs. Player – Same as above.
• Player vs. Wall – Because the game’s playing field is 2D, the walls in-game are treated as 2D polygons, so it boils down to a 2D mesh vs. polygon test.
• Player vs. Enemy – Mesh vs. Mesh here
• Beam vs. Enemy – The player has a bendy beam weapon.  I divide the curved beam up into line segments, and do ray/mesh tests.

The worst performance offender was, surprisingly, the sphere vs. mesh test, which will be the subject of this article.  Before optimizing, when I’d shoot a ton of bullets in a certain set of circumstances, the framerate would drop well into the single digits, because the bullet vs. mesh (sphere vs. mesh) collision couldn’t keep up.  Here are the things that I changed to get this test working much, much faster.

#### When using Value Types, Consider Passing As References

One thing that was slowing my code down was all of the value type copying that my code was doing.  Take the following function:

```public static bool SphereVsSphere(Vector3 centerA, float radiusA, Vector3 centerB, float radiusB)
{
float dist = radiusA+radiusB;

Vector3 diff = centerB-centerA;
return diff.LengthSquared() < dist*dist;
}
```

Simple, yes?  This function, however, falls prey to reference type copying.  You see, “centerA” and “centerB” are both passed in by value, which means that a copy of the data is made.  It’s not an issue when done infrequently, but with the number of SphereVsSphere calls that were happening during a given frame, the copies really started to add up.

There’s also a hidden set of copies:  the line “Vector3 diff = centerB-centerA” also contains a copy, as it passes centerB and centerA into the Vector3 subtraction operator overload, and they get passed in by value.  Also, a new Vector3 gets created inside of the operator then returned, which, I believe, also copies the data into diff.

To eliminate these issues, you should pass all of your non-basic value types (that is, anything that’s not an int, bool, float, anything like that) by reference instead of by value.  This eliminates all of the excess copies.  It does come at a price, though: in my opinion, it does make the code considerably uglier.

Here’s the updated routine:

```public static bool SphereVsSphere(ref Vector3 centerA, float radiusA, ref Vector3 centerB, float radiusB)
{
Vector3 diff;
Vector3.Subtract(ref centerA, ref centerB, out diff);
float dist = radiusA+radiusB;

return diff.LengthSquared() < dist*dist;
}```

Instead of having a nice-looking overloaded subtraction, now there’s a call to Vector3.Subtract.  While it’s not so bad in the case of a simple subtraction, when you have a more complicated equation, they pile up pretty quickly.  However, given the speed boost just making this change can give you, it’s totally worth it.

#### Use Hierarchical Collision Detection (But Use a Good Bounding Volume)

Heirarchical collision detection is a good thing.

For those of you that DON’T know, basically instead of testing your collider against every triangle in a mesh, you have a tree in which each node has a bounding volume, and the leaves contain the actual triangles.  The idea is that, by doing a much simpler collider vs. bounding volume test, you can elminiate large amoungs of triangles before you ever have to test them.

In this case, I was using a sphere tree, where each node in the tree has a bounding sphere, and the leaves of the tree contain actual mesh triangles.  I used spheres instead of AABBs (Axis-aligned bounding boxes) because transforming AABBs is expensive (and they become Oriented bounding boxes after the transform).  Transforming a sphere is easy, however.  None of my object transforms have scale data, so it’s a simple matter of transforming the sphere’s center.

However, the use of bounding spheres has a dark side.  Unless all of your heirarchy levels is roughly sphere-shaped, a sphere is a terribly inefficient bounding volume.  They’re usually much larger than the geometry that they contain, so there are more recursions into lower levels of the tree (think of there as being more dead space around the geometry when using spheres than bounding boxes).

By also adding bounding boxes to the data, I could use them where I’m not having to transform them.  For instance, because this is sphere vs. mesh, and the entire mesh is rigid, I can take the mesh’s world 4×4 matrix, and transform the sphere by the INVERSE of it.  This way, the sphere is in model space, and I can use the bounding volumes without having to do any transformations at lower levels.

But now I needed a sphere vs. AABB test.  However, I didn’t much care if it was exact or not, so instead I used a simple test where I expand the box by the radius of the sphere, then test whether the sphere is inside of the box or not.  Near the corners (surely this is where the term “corner case” comes from), this can give false positives, but it will never say the sphere DOESN’T intersect the box when it should say it does.  This is an acceptable trade-off.

```public static bool SphereVsAABBApproximate(ref Vector3 sphereCenter, float sphereRadius, ref Vector3 boxCenter, ref Vector3 boxExtent)
{
Vector3 relativeSphereCenter;
Vector3.Subtract(ref sphereCenter, ref boxCenter, out relativeSphereCenter); // Get the sphere center relative to the box's center.
Vector3Helper.Abs(ref relativeSphereCenter); // Per-component absolute value.

return (relativeSphereCenter.X <= boxExtent.X+sphereRadius && relativeSphereCenter.Y <= boxExtent.Y+sphereRadius && relativeSphereCenter.Z <= boxExtent.Z+sphereRadius);
}```

Simple, but effective.  Converting from using a sphere bounding volume to AABBs cut down the number of recursions (and triangle comparisons) being done dramatically, since the AABBs are a much tighter fit to the geometry.

#### Recursion Is Weird

One suggestion I got, also, was to eliminate recursion.  The heirarchical nature of the algorithm meant that my test was recursive.  Here was the test as originally written:

```public static bool SphereVsAABBTree(ref Vector3 sphereCenter, float sphereRadius, CollisionTreeNode node)
{
if (!SphereVsAABBApproximate(ref sphereCenter, sphereRadius, ref node.BoxCenter, ref node.BoxExtent))
return false; // No collision with this node, return false

if (node.Left != null)
{
// This means that there are Left and Right children (either they're both null, or both set).
if(SphereVsAABBTree(ref sphereCenter, sphereRadius, node.Left))
return true;
return SphereVsAABBTree(ref sphereCenter, sphereRadius, node.Right); // A node with child nodes can't have triangles, so just return this result.
}

// This node has triangles, so test against them.  If any of them intersects the sphere, return success.
for(int i = 0; i < node.Indices.Length; i+=3)
{
if(SphereVsTriangle(ref sphereCenter, sphereRadius, ref node.Vertices[node.Indices[i+0]], ref node.Vertices[node.Indices[i+1]], ref node.Vertices[node.Indices[i+2]]))
return true;
}
return false;
}```

As you can see, it recurses into child nodes until it either gets a false test out of both of them, or it reaches triangles.  But how do you eliminate recursion in a tree such as this?  More specifically, how do you do it while using a constant (non-node-count dependent) amount of memory?

The trick is as follows:  Assuming your nodes contain Parent pointers in addition to Left and Right pointers (where the Parent of the trunk of the tree is null), you can do it with no issue.  You track the node that you’re currently visiting (“cur”), and the node that you previously visited (“prev”, initialized to null).  When you reach a node, test as follows:

• if you came to it via its parent (that is, prev == cur.Parent), you’ve never visited it before.
• At this point, you should do your collision tests.  It’s a newly-visited node.
• prev = cur
• cur = cur.Left — This is basically “recursing” into the Left node.
• If you arrived from its Left child, you visited it before and recursed into its left side
• Since this node has already been visited, do not do the collision tests.  They’ve already been shown to be successful.
• prev = cur
• cur = cur.Right  — Since we recursed into Left last time we were here, recurse into Right this time.
• If you arrived via its Right child, you’ve visited both of its children, so you are done with this node.
• Again, this node has already been visited, so do not run the collision tests.
• prev = cur
• cur = cur.Parent — We’re done with this node, so go back to its parent.
• When doing the collision tests, if a Sphere vs. AABB bounding volume test ever fails, we don’t have to “recurse” so go back to its parent.
• prev = cur
• cur = cur.Parent
• Finally, if you do a Sphere vs. Triangle collision test and it succeeds, we can immediately return, as we have a guaranteed collision, and no more triangles or nodes need to be tested.

Doing all of this makes the routine bigger, but no recursion is necessary, so there’s no additional stack space generated per node visited (and no function call overhead, either).  The finished code is as follows (note that I also added a quick sphere vs. sphere test right at the outset, because it’s a very quick early out if the sphere is nowhere near the mesh):

```public static bool SphereVsAABBTree(ref Vector3 sphereCenter, float sphereRadius, CollisionTreeNode node)
{
CollisionTreeNode prev = null, cur = node;

// At the top level, just do a sphere/sphere test for a super-quick out.
if (!SphereVsSphere(ref sphereCenter, sphereRadius, ref node.Sphere.Center, node.Sphere.Radius))
return false;

while(cur != null)
{
if(prev == cur.Parent) // Only do the tests if we JUST got here.
{
if (!SphereVsAABBApproximate(ref sphereCenter, sphereRadius, ref cur.BoxCenter, ref cur.BoxExtent))
{
// No intersection?  Go ahead and just back out of this node now.
prev = cur;
cur = cur.Parent;
continue; // By continuing, we bypass the rest of this code and re-visit the parent immediately.
}

for(int i = 0; i < cur.Indices.Length; i+=3)
{
if(SphereVsTriangle(ref sphereCenter, sphereRadius, ref cur.Vertices[cur.Indices[i+0]], ref cur.Vertices[cur.Indices[i+1]], ref cur.Vertices[cur.Indices[i+2]]))
return true;
}
}

// "Recurse"
if (cur.Left != null)
{
if (prev == cur.Parent) // If this is the first visit to the node, recurse left.
{
prev = cur;
cur = cur.Left;
continue;
}
if (prev == cur.Left) // If this is the second visit, recurse right.
{
prev = cur;
cur = cur.Right;
continue;
}
}

// If there are no child nodes or prev == cur.Right, return to the parent.
prev = cur;
cur = cur.Parent;
}
return false;
}```

#### Mission Complete

After making that set of changes, the sphere vs. mesh tests no longer bog down on the Xbox, even in a degenerate case such as when there are tens or hundreds of bullets well inside of the mesh’s area.

Getting the sphere vs. mesh test working was a great accomplishment, but as much as I thought it was already working well, it turns out that mesh vs. mesh testing was a much bigger problem.  However, that’s another story for another day.

## Understanding Half-Pixel and Half-Texel Offsets

For those of you not using Direct3D 9 or XNA, you can safely ignore this post (OpenGL and Direct3D 10 are immune to this particular oddity).  However, if you are, it’s likely that you’ve had to deal with the dreaded half-texel offset.  Today, after I don’t know how many years of using Direct3D, I came to realize that I really didn’t understand what the source of the issue was.  Now that I’ve sort of gotten a handle on it, I figured I’d post it to my super new journal.  Consider it a test run.

#### Coordinate Spaces

The first thing to note is the basic coordinate space.  I’m going to be referring to texture space and clip space a lot, so I thought I’d just do a quick refresher here on what I mean (mostly to make sure you’re thinking with the same terminology that I’m using).

• Clip space – the post-projection half-cube of space where X,Y in [-1..1] and Z in [0..1]
• X = -1 is the far left edge of the screen
• X =  1 is the far right edge
• Y = -1 is the bottom edge
• Y = 1 is the top
• Z = 0 is near (the near plane)
• Z = 1 is far (the far plane)
• Texture space – the area where u,v in [0..1] on a texture map making up a single end-to-end repeat of the texture.
• 0,0 represents the upper-left coordinate on the texture
• 1,1 is the lower-right.

For simplification, we can ignore the Z direction in clip space, since this article is really about the screen-space aspects of it.

#### Texture Sampling

Texture sampling is pretty straightforward.

In this diagram (and future diagrams), I’m using a 4×4 texture.  The [0..1] range in both X and Y map exactly to the range of the texture.  Note, here, that the center of the upper-left texel does not lie at 0,0; instead, 0,0 refers to the upper-left corner of that texel.  This is an important thing: if you are using bilinear filtering and you want to sample color data from a single texel, you need to sample from the center of the texel instead of the corner.  To do that, you’d sample from a half-texel in (that is, for this 4×4 texel, you’d want to sample from (0.125, 0.125), because a texel’s width in texture space is .25 (each texel is 1/4 of texture space), and half of that is 0.125).

A side effect of this is that, if you have bilinear sampling and wrapping on, sampling along the left edge gives you exactly the same data as sampling along the right edge at the same y coordinate.  That is, in both cases, you get a perfect blend of the texels on either end of the texture.  This is true for the borders between texels in general: sampling where the “dot” is on the image above is the only way to get a pure sample of a texel with bilinear filtering.  Anywhere else will be a blend between two.  Edges (the lines) are where the two texels neighboring the edge are weighted evenly.

The upshot of this is that the official center of a texel is located where the center SHOULD be: half of a texel from the edge.  However, as we’ll see in a moment, the rule for sampling texels does not apply to pixels.

#### Pixel Coordinates Are Weird And Unintuitive

When referring to “pixel coordinates” vs. “texel coordinates,” pixel coordinates are, for lack of a better way to explain it, the coordinates used when rendering to the screen (or a render target).  The pixels are the destination coordinates.

This is where clip space comes in: When you render your geometry, it goes through the gauntlet of matrices, ending with the projection matrices, and ends up in clip space (with x and y both being in [-1..1] for the visible area of the screen).

This diagram is the way that clip space maps to pixels.  note that the upper-left clip space coordinate (-1, +1) is actually RIGHT ON the pixel center.  This is the root cause of the whole offset problem.

Basically, say you have a screen-sized texture.  if you draw it to the screen using a full-screen quad (clip space from upper-left (-1,1) to lower-right (1, -1)), the upper-left pixel would sample the very upper-left corner of your quad, which would give you texture coordinates of (0,0).  But remember: that’s not the center of the texel!  It’s actually the upper-left corner of the texel.  With bilinear filtering, instead of getting a pure sample of your texture, you have a perfect blend of all 4 surrounding texels.

In simpler terms: the half-pixel/half-texel offset exists because there’s a discrepancy between how the centers of pixels and the centers of texels are computed.

To fix this, you could simply offset your texture coordinates by a half-texel before sampling (that is, when your shader gets its uv coordinate, you’d add a half texel ( (.125, .125) for our 4×4 example) before sampling, which would then give you a perfectly lined-up texture sample, and you’d get a great 1:1 mapping of pixels to texels.

However, there’s another problem.  What happens when you’re using multisample anti-aliasing (MSAA)?

#### Weird Pixel Centers and MSAA

So, we now know that the center of the origin pixel is actually the upper-left coordinate of clip space.  This causes problems when MSAA is turned on.  With MSAA, the geometry is sampled multiple times per pixel, using a set of points that exist within the pixel’s square (while a pixel is not really a square, but a point sample within a square, MSAA effectively treats a pixel as multiple samples in a square).

As an (important) aside: with MSAA on, textures are still only sampled once, from the pixel center.  MSAA does not affect the UV coordinates you get for a given pixel, it merely affects geometry coverage.

Continuing with our example, here’s a hypothetical, simple MSAA strategy wherein a pixel is sampled four times (4x MSAA) in an aligned grid:

This diagram illustrates the issue with the standard [-1..1] full screen quad, D3D’s choice of pixel center, and MSAA.  Even with a full-clip-space (and, thus, traditionally full-screen) quad, along the top and left edges, some samples no longer touch the quad, and thus the edges are not entirely filled in (a pure white quad drawn to a render target that had been cleared to black would have grey edges along the top and left).

You’ve likely seen this in a ton of games: when you turn on full-screen anti-aliasing, there’s a weird line down the side of the screen during fadeouts and the like.  This is the pixel offset problem rearing its head.

How do you fix this?  Turns out, it’s simple.  When drawing a full-screen quad, instead of adding a half-texel to the uv coordinates, you should instead SUBTRACT a half-pixel from the position (not the uv coordinates).  This shifts the quad so that it lies perfectly within the grid in the diagram, so that every MSAA sample hits the geometry.  As an added bonus, it means that the UV samples that you get will already be in the texel centers; no more adding a half-texel required!

Note, that when I say “subtract a half-pixel from position,” what I mean is “move the position of the quad one half-pixel towards the upper-left.  To do this, you actually add (-1/width, +1/height).  The reason for the signs (-, +) is that you’re moving left and up.  In post-projection space, -x is left, and +y is up.  The reason that it’s 1/width instead of 0.5/width is because, in clip space, the coordinates range from -1 to 1 (width 2), and not from 0 to 1 (width 1) like they do in textures, so you need to double the movement to account for that.

Of course, if you’re drawing world geometry and using its screen coordinates to index into a texture map (like, for instance, if you’re using light pre-pass rendering), you’ll still want to add the half-texel instead.  In fact, here is a great reference on how to handle this case.

#### Take Us Home, Article

The half-texel/half-pixel offset is a bizarre feature in Direct3D 9 (and, by extension, XNA).  In order to properly handle it when using full-screen-sized textures:

• When rendering a full-screen (or otherwise screen-aligned quad), subtract a half-pixel’s size from the output vertex position from your vertex shader.  This will ensure both that the texel centers line up with the pixel centers (for proper texture sampling) AND that the quad will play nice along the left and top edges of the screen with MSAA.
• When rendering normal geometry, the geometry is already in the proper place (i.e. it already plays fine with MSAA).  Consequently, you should add a half-texel to the uv coordinates for your full-screen sample.   This will allow you to sample from the texel centers as desired.

While this article refers mostly to full-screen effects, this information is generally more useful when downsampling textures, as you need to know where your sample points on the source texture will actually hit when rendering (and you’ll want bilinear filtering for, for instance, a 4-tap 4×4 average downsample).

Hopefully, if you were as confused by the half-texel offset as I was, this helps clear things up.

Additional references from MSDN:

Directly Mapping Texels to Pixels (Direct3D 9)

Coordinate Systems (Direct3D 10)

## Scrollathon 2008

This previous weekend, I was able to accomplish another major milestone in game development: The Scrolling Background (TM) (C) (R) (BBQ).

Click to enlarge

Requisite Scrolling Video (Xvid AVI, ~2MB)

#### The Skinny On Scrolling

The interesting thing about the scrolling method that I settled on is that it’s not based on any sort of overall world coordinate system. World coordinates don’t actually exist, the only true coordinate system is the screen coordinate system (with coordinates ranging from -16,-9 to +16,+9, for a delicious [and integer-tastic] 32:18 [2x 16:9] visible area).

So how does it work? Each level will be built out of tiles, in order. Each tile has the following data:

• A model to render
• Collision Data
• The Camera Path

The camera path for a tile is currently just an input position and an output position. That is, the position at which the camera ENTERS the tile, and the position at which the camera EXITS the tile.

Now, here’s the trick: Say you have two of the same tile next to each other. Each has an input coordinate of (0,1) and an output coordinate of (4, 0). What the system does is it moves the second one so that its input coordinate is in the same spot as the first one’s output coordinate. (that is, the second one’s input coordinate becomes effectively (4,0) like the first’s output coordinate and, relative to that, the second’s output coordinate becomes (8, -1)).

However, actual world coordinates aren’t strictly necessary, so whichever tile the camera is currently in is considered the “origin” tile. That is, it is used as the basis by which all other visible tiles get their on-screen positioning.

Thus, the setup is easy: figure out where on-screen (given the camera’s position in the tile) the tile should display, then make all of the visible tiles to the left and right relative to that.

This is nice for a few reasons:

First off, if, for some reason, a level were RIDICULOUSLY long, I would never have to worry about accumulating floating point round-off error.

The big thing is this allows me to have what is essentially a staple of the shoot-em-up game (and is actually quite visible in the video posted above): an endless loop of background.

These loops are especially useful for when fighting bosses. Say you’re zooming down a metallic corridor while scrapping with a boss that happens to be flying along with you. Rather than have to hope that the player finishes the fight before the camera hits the level’s end, you can just rely on the fact that the corridor will keep on looping until something triggers the loop’s end, signaling that the level should keep going (or end, assuming that there’s no more to the level).

This triggering system is not yet implemented, and I hope to get it done this weekend (though I have a ton of other, smaller items on the to-do list, so it may have to wait for the NEXT weekend).

#### Proximity Alert

One design element that was tricky was signaling to the player that the ship is too close to a wall. The obvious metric is, of course, a shadow. However, standard shadows only cast in one direction, which would be great if all we cared about was distance to the floor. However, we really need “distance to any object.” This looks like a job for the existing lighting system!

A new type of “light” was designed: essentially a black light, which has a center, a length, and a radius (thus, the actual light is more like a line light than a point light). Consequently, the fakey shadow from the ship will “cast” onto any surrounding objects.

Click to enlarge

And, once again, that’s all we have time for on this week’s episode of “What Did Drilian Do Last Weekend”. Stay tuned next week, same Bat-Time, same Bat-Channel!