libinput's touchpad acceleration is the cause for a few bugs and outcry from a quite vocal (maj|in)ority. A common suggestion is "make it like the synaptics driver". So I spent a few hours going through the pointer acceleration code to figure out what xf86-input-synaptics actually does (I don't think anyone knows at this point) .
If you just want the TLDR: synaptics doesn't use physical distances but works in device units coupled with a few magic factors, also based on device units. That pretty much tells you all that's needed.
Also a disclaimer: the last time some serious work was done on acceleration was in 2008/2009. A lot of things have changed since and since the server is effectively un-testable, we ended up with the mess below that seems to make little sense. It probably made sense 8 years ago and given that most or all of the patches have my signed-off-by it must've made sense to me back then. But now we live in the glorious future and holy cow it's awful and confusing.
Synaptics has three options to configure speed: MinSpeed, MaxSpeed and AccelFactor. The first two are not explained beyond "speed factor" but given how accel usually works let's assume they all somewhoe should work as a multiplication on the delta (so a factor of 2 on a delta of dx/dy gives you 2dx/2dy). AccelFactor is documented as "acceleration factor for normal pointer movements", so clearly the documentation isn't going to help clear any confusion.
I'll skip the fact that synaptics also has a pressure-based motion factor with four configuration options because oh my god what have we done. Also, that one is disabled by default and has no effect unless set by the user. And I'll also only handle default values here, I'm not going to get into examples with configured values.
Also note: synaptics has a device-specific acceleration profile (the only driver that does) and thus the acceleration handling is split between the server and the driver.
Ok, let's get started. MinSpeed and MaxSpeed default to 0.4 and 0.7. The MinSpeed is used to set constant acceleration (1/min_speed) so we always apply a 2.5 constant acceleration multiplier to deltas from the touchpad. Of course, if you set constant acceleration in the xorg.conf, then it overwrites the calculated one.
MinSpeed and MaxSpeed are mangled during setup so that MaxSpeed is actually MaxSpeed/MinSpeed and MinSpeed is always 1.0. I'm not 100% why but the later clipping to the min/max speed range ensures that we never go below a 1.0 acceleration factor (and thus never decelerate).
The AccelFactor default is 200/diagonal-in-device-coordinates. On my T440s it's thus 0.04 (and will be roughly the same for most PS/2 Synaptics touchpads). But on a Cyapa with a different axis range it is 0.125. On a T450s it's 0.035 when booted into PS2 and 0.09 when booted into RMI4. Admittedly, the resolution halfs under RMI4 so this possibly maybe makes sense. Doesn't quite make as much sense when you consider the x220t which also has a factor of 0.04 but the touchpad is only half the size of the T440s.
There's also a magic constant "corr_mul" which is set as:
/* synaptics seems to report 80 packet/s, but dix scales for * 100 packet/s by default. */ pVel->corr_mul = 12.5f; /*1000[ms]/80[/s] = 12.5 */It's correct that the frequency is roughly 80Hz but I honestly don't know what the 100packet/s reference refers to. Either way, it means that we always apply a factor of 12.5, regardless of the timing of the events. Ironically, this one is hardcoded and not configurable unless you happen to know that it's the X server option VelocityScale or ExpectedRate (both of them set the same variable).
Ok, so we have three factors. 2.5 as a function of MaxSpeed, 12.5 because of 80Hz (??) and 0.04 for the diagonal.
When the synaptics driver calculates a delta, it does so in device coordinates and ignores the device resolution (because this code pre-dates devices having resolutions). That's great until you have a device with uneven resolutions like the x220t. That one has 75 and 129 units/mm for x and y, so for any physical movement you're going to get almost twice as many units for y than for x. Which means that if you move 5mm to the right you end up with a different motion vector (and thus acceleration) than when you move 5mm south.
The core X protocol actually defines who acceleration is supposed to be handled. Look up the man page for XChangePointerControl(), it sets a threshold and an accel factor:
The XChangePointerControl function defines how the pointing device moves. The acceleration, expressed as a fraction, is a multiplier for movement. For example, specifying 3/1 means the pointer moves three times as fast as normal. The fraction may be rounded arbitrarily by the X server. Acceleration only takes effect if the pointer moves more than threshold pixels at once and only applies to the amount beyond the value in the threshold argument.Of course, "at once" is a bit of a blurry definition outside of maybe theoretical physics. Consider the definition of "at once" for a gaming mouse with 500Hz sampling rate vs. a touchpad with 80Hz (let us fondly remember the 12.5 multiplier here) and the above description quickly dissolves into ambiguity.
Anyway, moving on. Let's say the server just received a delta from the synaptics driver. The pointer accel code in the server calculates the velocity over time, basically by doing a hypot(dx, dy)/dtime-to-last-event. Time in the server is always in ms, so our velocity is thus in device-units/ms (not adjusted for device resolution).
Side-note: the velocity is calculated across several delta events so it gets more accurate. There are some checks though so we don't calculate across random movements: anything older than 300ms is discarded, anything not in the same octant of movement is discarded (so we don't get a velocity of 0 for moving back/forth). And there's two calculations to make sure we only calculate while the velocity is roughly the same and don't average between fast and slow movements. I have my doubts about these, but until I have some more concrete data let's just say this is accurate (altough since the whole lot is in device units, it probably isn't).
Anyway. The velocity is multiplied with the constant acceleration (2.5, see above) and our 12.5 magic value. I'm starting to think that this is just broken and would only make sense if we used a delta of "event count" rather than milliseconds.
It is then passed to the synaptics driver for the actual acceleration profile. The first thing the driver does is remove the constant acceleration again, so our velocity is now just v * 12.5. According to the comment this brings it back into "device-coordinate based velocity" but this seems wrong or misguided since we never changed into any other coordinate system.
The driver applies the accel factor (0.04, see above) and then clips the whole lot into the MinSpeed/MaxSpeed range (which is adjusted to move MinSpeed to 1.0 and scale up MaxSpeed accordingly, remember?). After the clipping, the pressure motion factor is calculated and applied. I skipped this above but it's basically: the harder you press the higher the acceleration factor. Based on some config options. Amusingly, pressure motion has the potential to exceed the MinSpeed/MaxSpeed options. Who knows what the reason for that is...
Oh, and btw: the clipping is actually done based on the accel factor set by XChangePointerControl into the acceleration function here. The code is
double acc = factor from XChangePointerControl(); double factor = the magic 0.04 based on the diagonal; accel_factor = velocity * accel_factor; if (accel_factor > MaxSpeed * acc) accel_factor = MaxSpeed * acc;So we have a factor set by XChangePointerControl() but it's only used to determine the maximum factor we may have, and then we clip to that. I'm missing some cross-dependency here because this is what the GUI acceleration config bits hook into. Somewhere this sets things and changes the acceleration by some amount but it wasn't obvious to me.
Alrighty. We have a factor now that's returned to the server and we're back in normal pointer acceleration land (i.e. not synaptics-specific). Woohoo. That factor is averaged across 4 events using the simpson's rule to smooth out aprupt changes. Not sure this really does much, I don't think we've ever done any evaluation on that. But it looks good on paper (we have that in libinput as well).
Now the constant accel factor is applied to the deltas. So far we've added the factor, removed it (in synaptics), and now we're adding it again. Which also makes me wonder whether we're applying the factor twice to all other devices but right now I'm past the point where I really want to find out . With all the above, our acceleration factor is, more or less:
f = units/ms * 12.5 * (200/diagonal) * (1.0/MinSpeed)and the deltas we end up using in the server are
(dx, dy) = f * (dx, dy)But remember, we're still in device units here (not adjusted for resolution).
Anyway. You think we're finished? Oh no, the real fun bits start now. And if you haven't headdesked in a while, now is a good time.
After acceleration, the server does some scaling because synaptics is an absolute device (with axis ranges) in relative mode . Absolute devices are mapped into the whole screen by default but when they're sending relative events, you still want a 45 degree line on the device to map into 45 degree cursor movement on the screen. The server does this by adjusting dy in-line with the device-to-screen-ratio (taking device resolution into account too). On my T440s this means:
touchpad x:y is 1:1.45 (16:11) screen is 1920:1080 is 1:177 (16:9) dy scaling is thus: (16:11)/(16:9) = 9:11 -> y * 11/9dx is left as-is. Now you have the delta that's actually applied to the cursor. Except that we're in device coordinates, so we map the current cursor position to device coordinates, then apply the delta, then map back into screen coordinates (i.e. pixels). You may have spotted the flaw here: when the screen size changes, the dy scaling changes and thus the pointer feel. Plug in another monitor, and touchpad acceleration changes. Also: the same touchpad feels different on laptops when their screen hardware differs.
Ok, let's wrap this up. Figuring out what the synaptics driver does is... "tricky". It seems much like a glorified random number scheme. I'm not planning to implement "exactly the same acceleration as synaptics" in libinput because this would be insane and despite my best efforts, I'm not that yet. Collecting data from synaptics users is almost meaningless, because no two devices really employ the same acceleration profile (touchpad axis ranges + screen size) and besides, there are 11 configuration options that all influence each other.
What I do plan though is collect more motion data from a variety of touchpads and see if I can augment the server enough that I can get a clear picture of how motion maps to the velocity. If nothing else, this should give us some picture on how different the various touchpads actually behave.
But regardless, please don't ask me to "just copy the synaptics code".
 fwiw, I had this really great idea of trying to get behind all this,
with diagrams and everything. But then I was printing json data from the X
server into the journal to be scooped up by sed and python script to print
velocity data. And I questioned some of my life choices.
 why the hell do we do this? because synaptics at some point became a device that announce the axis ranges (seemed to make sense at the time, 2008) and then other things started depending on it and with all the fixes to the server to handle absolute devices in relative mode (for tablets) we painted ourselves into a corner. Synaptics should switch back to being a relative device, but last I tried it breaks pointer acceleration and that a) makes the internets upset and b) restoring the "correct" behaviour is, well, you read the article so far, right?