# Gimbal derotated video, using clouds as the horizon

#### dimebag2

##### Active Member
Something that has been overlooked is that the clouds and artificial horizon are not aligned in the first half of Gimbal.
Looks at this derotated video, that uses artificial horizon as a reference:

The clouds have a diagonal motion for the first 15s, because they are not aligned with artificial horizon. This messes up the view we have of what's going on. This diagonal motion is absent in Sitrec and the 3D recreations, in which the clouds remains parallel with artificial horizon. The derotation mechanism is slightly off here.

Let's derotate this same video so that the clouds are flat:

Now we get something extremely interesting: the object rotates counter-clockwise (CCW), in a gradual way, before the fast CCW rotation at the end.

If anything, the pod roll model predicts no rotation, or CW rotation, in those first 20s, before the CCW pod roll at the end. If this was a glare, how could it rotate CCW in the first half of the vid like this?

This gradual CCW rotation is once again consistent with the close trajectory, that as a reminder looks like this (note that the object's rotation is not corrected for the effect I describe here, it should rotate CCW slightly along the path in the first half).

Now remember the F-18 was low in fuel, and was facing a strong 120 Knots headwind if going in the same direction than the object during the first half of the video. Those are specific conditions, so does a "standard" jet pitch (~3-4°) as used in Sitrec apply here ?

Setting the pitch to a slightly negative value (-1°) gives this prediction for pod roll :

Gradual CW in the first 20s, which would realigns clouds and artificial horizon as in the video, with the derotation mechanism correcting the offset it had at first. The gradual CW roll would create the impression of a "non-rotating glare" in the original video, as the object rotates the other way.

Also notes the model predicts a fast pod roll around 32s ... which is when the pod loses lock of the target.

Have we been fooled all along by the mismatch between artificial and real horizon?

#### DavidB66

##### Senior Member
Something that has been overlooked is that the clouds and artificial horizon are not aligned in the first half of Gimbal.
The tops of the clouds are lumpy and irregular. If you freeze the video at various points they may seem to slope either to the right (relative to the artificial horizon) or to the left. I think this is just pareidolia. The switch from white-hot to black-hot around 12 secs may contribute to this. In the black-hot footage, which includes the second half of the video, there is a faint dark band above the nearer cloud tops, which may better represent the true horizon. The corresponding light band in the white-hot footage is less noticeable, to my eyes anyway.
Supposing that the claimed difference is not pareidolia, but a real change, what does that imply? That there was a defect in the artificial horizon mechanism in the first half of the video which fortunately corrected itself in the second half?

#### dimebag2

##### Active Member
I thought that too first, but it's not slopes in the clouds, there truly is a diagonal motion that indicates cloud (real) and artificial horizons are not aligned for ~15s.

#### jplaza

##### Member
I think you are only seeing an artifact.

The plane is a complex system of sensors. Some of them are dedicated to know the plane's attitude, and one of the results is establishing where the horizon is, which is represented by the artificial horizon. The artificial horizon is where the plane's CPU thinks the horizon is. And that data is passed on to other systems, like ATFLIR.

ATFLIR needs the data to derotate the image. That is, ATFLIR uses the artificial horizon as reference to derotate the image, as it is the only way it has to know where the horizon is. So, we should also take it as reference for derotation.

You found there is a small discrepancy between the real and artificial horizon. But if you use the real horizon as reference, then you are going to find artifacts for sure, since ATFLIR is reacting to where it thinks the horizon is (i.e., uses artificial horizon to set the roll and pitch), not where it really is.

#### jarlrmai

##### Senior Member
I think you are only seeing an artifact.

The plane is a complex system of sensors. Some of them are dedicated to know the plane's attitude, and one of the results is establishing where the horizon is, which is represented by the artificial horizon. The artificial horizon is where the plane's CPU thinks the horizon is. And that data is passed on to other systems, like ATFLIR.

ATFLIR needs the data to derotate the image. That is, ATFLIR uses the artificial horizon as reference to derotate the image, as it is the only way it has to know where the horizon is. So, we should also take it as reference for derotation.

You found there is a small discrepancy between the real and artificial horizon. But if you use the real horizon as reference, then you are going to find artifacts for sure, since ATFLIR is reacting to where it thinks the horizon is (i.e., artificial horizon), not where it really is.

There's also a concept called perceptual horizon

"The perceptual horizon doesn’t always agree with the actual horizon in a scene. In other words, perhaps you’re using a bubble level on top of your camera, and it says the image is completely level, but your photos still appear strongly tilted."

https://photographylife.com/why-leveling-the-horizon-isnt-easy#the-perceptual-horizon

#### DavidB66

##### Senior Member
I thought that too first, but it's not slopes in the clouds, there truly is a diagonal motion that indicates cloud (real) and artificial horizons are not aligned for ~15s.

Yes, I see your point now. With the AH set to horizontal, the clouds do consistently seem to show an upward trend as they move to the right, during at least the white-hot part of the video. Since we can't reasonably suppose the clouds are actually moving upwards, this implies an error in the AH. I had assumed that this was highly unlikely. One would expect the AH system in an F-18 to be about 'as good as it gets'! However, some sources do suggest that slight errors in the AH are to be expected during a tight 'bank and turn':

Errors in both pitch and bank indications occur during normal coordinated turns. These errors are caused by the movement of the pendulous vanes by centrifugal force, resulting in the precession of the gyro toward the inside of the turn. The error is greatest in a 180° steep turn. If, for example, a 180° steep turn is made to the right and the aircraft is rolled out to straight-and-level flight by visual references, the miniature aircraft will show a slight climb and turn to the left. This precession error, normally 3° to 5°, is quickly corrected by the erecting mechanism. At the end of a 360° turn, the precession induced during the first 180° is cancelled out by precession in the opposite direction during the second 180° of turn. The slight precession errors induced during the roll-out are corrected immediately by pendulous vane action.
Extract from 'Pilot Friend' webpage: http://www.pilotfriend.com/training/flight_training/fxd_wing/attitude.htm

I can't evaluate the implications of this for ATFLIR, as discussed by jplaza at #4 above, as this is far too complicated for me.

#### dimebag2

##### Active Member
Simply put, let me know where you disagree here:

1/ the clouds rotate CW in the first 15s.
2/ when stabilizing the clouds, the object rotates CCW relative to them.

What could cause 1, other than pod roll? And if pod roll causes 1, how can it cause 2?

#### Mick West

Staff member
This is something I looked at a while ago. I attempted to extract the angle of the clouds using two methods (red and blue) and compared them against the bank angle indicator (green)

Ignore the first large discontinuity in the blue, that's just a glitch in my tracking. You can see red and blue are essentially the same.

There's two major points here:
1) The extracted cloud tracks seem much more continuous than the angle indicator tracks. However I think that may be, at least partly, an artifact of having to smooth them a lot.
2) They appear to converge, and possible cross over (ie. go from less than, to equal, to maybe greater than) around 25-30 seconds. However the tracking at the end is too noisy.

I think the difference has to do with the fact that the bank angle is relative to the forward direction, but the cloud horizon is looking sideways. from a banked and pitched plane. It's complicated.

It does not change the fact that the glare does not rotate relative to the frame of the video, which I think is the most significant observable.

#### dimebag2

##### Active Member
If it's only a 3D perspective effect, it should be seen in Sitrec, no?

There is something off with jet pitch here.

#### Mick West

Staff member
If it's only a 3D perspective effect, it should be seen in Sitrec, no?
What angle should the horizon be at in the video? What is it as a function of Az (camera), bank, and pitch?

#### markus

##### Active Member
Setting the pitch to a slightly negative value (-1°) gives this prediction for pod roll :
A negative pitch results in the plane falling out of the sky because now the lift vector is directed downwards. It's not aerodynamically possible.

The wind doesn't matter. The effect of wind is to change the local reference frame of the airplane. The pitch angle required to fly level won't change regardless of wind speed and angle. Only the groundspeed changes. Remember, what the instruments indicate is calibrated airspeed.

#### Mendel

##### Senior Member.
A negative pitch results in the plane falling out of the sky because now the lift vector is directed downwards. It's not aerodynamically possible.
I disagree.

There are 3 vectors in play, lift, thrust, and weight. For level flight, there needs to be a horizontal component (mostly provided by thrust), and the vertical components need to cancel out. If the aircraft has excessive lift (obvious e.g. with STOL aircraft), the thrust vector may need to be somewhat down to make up for it.

The question is, is there a configuration of the F/A-18 where it can achieve level flight with negative pitch?

#### dimebag2

##### Active Member

This plot shows that the clouds themselves are not aligned with banking. I don't know how you can then look at the object's angle with artificial horizon and say it's fixed in the camera frame in the first 20s, with confidence. Especially given how ambiguous it already was due to the difficulty of defining object's angle.

Why would this first observable be only valid for 20s was always a question, and we have an answer here.

What angle should the horizon be at in the video? What is it as a function of Az (camera), bank, and pitch?

Shouldn't Sitrec answer this, since it includes all the equations? The only way to get same clouds angle as in the vid at the beginning is to set -8 negative pitch, and it's not even enough.

Because pitch could not be so negative, I assume derotation is off at first, and realigns with artificial horizon in the first 15s. The object is not fixed in camera frame there, it rotates CCW.

Another clue that something is off is the vertical position of the cue dot, that is relative to plane boresight. This has been pointed out several times, again recently:

#### Mick West

Staff member
This plot shows that the clouds themselves are not aligned with banking. I don't know how you can then look at the object's angle with artificial horizon and say it's fixed in the camera frame in the first 20s, with confidence.
Because it does not move relative to the camera frame, whereas the cloud horizon does.

What else are you suggesting?

#### dimebag2

##### Active Member
Because it does not move relative to the camera frame, whereas the cloud horizon does.

What else are you suggesting?

See my post #1, that there is either change in jet pitch, pod roll, or both, to explain the change in cloud angle.
If the cloud angle is simply a 3D perspective effect, why is it missing from 3D recreations with standard jet pitch/no pod roll (yours, and Edward's)?

And the object has an intrinsic CCW rotation relative to this motion, consistent with the close flight path.

#### Mick West

Staff member
And the object has an intrinsic CCW rotation relative to this motion,
If you stabilize for either the artificial or cloud horizon, then you have to rotate the frame CCW, and so this rotates the glare. You're simply following a different line on my graph.

Where's the "intrinsic" rotation? In both cases the object is not rotating in the frame when the horizon is rotating.

The real question here is why there's a difference between the two horizon angle - why is the cloud motion angle smaller than the artificial, and why does it not reflect the steps (rapid changes and then constatant portions)?

Hence my question:
What angle should the horizon be at in the video? What is it as a function of Az (camera), bank, and pitch?

#### Mick West

Staff member
Now remember the F-18 was low in fuel, and was facing a strong 120 Knots headwind if going in the same direction than the object during the first half of the video. Those are specific conditions, so does a "standard" jet pitch (~3-4°) as used in Sitrec apply here ?
What Markus said:
The wind doesn't matter. The effect of wind is to change the local reference frame of the airplane. The pitch angle required to fly level won't change regardless of wind speed and angle. Only the groundspeed changes. Remember, what the instruments indicate is calibrated airspeed
Headwind is irrelevant to pitch, as it's all about airspeed, which is the speed displayed on-screen, and what the sim uses.

#### Mendel

##### Senior Member.
Now we get something extremely interesting: the object rotates counter-clockwise (CCW), in a gradual way, before the fast CCW rotation at the end.

If anything, the pod roll model predicts no rotation, or CW rotation, in those first 20s, before the CCW pod roll at the end. If this was a glare, how could it rotate CCW in the first half of the vid like this
you yourself are rotating the image counterclockwise with your processing, which makes the UAP rotate counterclockwise

the ATFLIR has no way to track the actual horizon, so there is nothing in that opto-mechanical device that mimics the processing you are performing

this means your gradual counter-clockwise rotation is not "extremely interesting", but self-inflicted and therefore boring

it would only be interesting if it was the object that was rotating, and not just its glare on the ATFLIR pod head

#### Mick West

Staff member
Why would this first observable be only valid for 20s was always a question, and we have an answer here.
No, you don't.

The reason the observable is only valid for the first 20s is because the pod does not need to roll in the first 20 seconds, and so you can see the effects of the jet's rolls (banking). After the first pod roll (which increases in frequency as we get closer to the singularity), the pod roll obscures the subsequent jet rolls. It was never a question.

#### Mendel

##### Senior Member.
This gradual CCW rotation is once again consistent with the close trajectory,
i don't understand this claim at all
would it not be consistent with the far trajectory as well?
and wasn't this what we knew before?

if your observation could make the close trajectory straight (and level?), that'd be a revelation, but does it?

#### Mendel

##### Senior Member.
@Edward Current , you were tracking clouds for your precise azimuth measurement. Any thoughts?

#### dimebag2

##### Active Member
If you stabilize for either the artificial or cloud horizon, then you have to rotate the frame CCW, and so this rotates the glare. You're simply following a different line on my graph.

Where's the "intrinsic" rotation? In both cases the object is not rotating in the frame when the horizon is rotating.
If you take the clouds as the "real" horizon, it rotates CCW. It's not "non-rotating" anymore, as in the 1st observable.
Hence my question:
Like I said, if I trust Sitrec and pitch is 3-4 deg, horizon and artificial horizon should be levelled. So what's wrong?

#### logicbear

##### New Member
If you take the clouds as the "real" horizon, it rotates CCW. It's not "non-rotating" anymore, as in the 1st observable.
When the jet banks the camera, along with the entire jet, rotates relative to the horizon (without using the pod roll motor) but the derotation device is not supposed to cancel that rotation since the pilot expects the image to rotate when he banks, so the glare stays fixed in the image (hence the observable) but rotates relative to the horizon. The jet initially keeps banking more and more to the left / CCW so the horizon rotates CW in the image and this should still cause a CCW rotation of the glare relative to the horizon even if the artificial horizon indicator is a bit off and the real horizon is more closely reflected by the cloud motion. Later, as the object nears 0 degrees azimuth, the pod engages its motor to rotate CCW about its roll axis which would rotate the horizon CW but the derotation device rotates the image CCW to cancel that unwanted rotation, so the glare should rotate CCW both in the image and relative to the horizon. At the same time the jet also banks a bit to the right / CW, reducing the glare's CCW rotation relative to the horizon, but not its rotation in the image.

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#### Mick West

Staff member
If you take the clouds as the "real" horizon, it rotates CCW. It's not "non-rotating" anymore, as in the 1st observable.

It's non-rotating exactly the same regardless of which measure of horizon you use. i.e. in the frame of image (the camera's frame) the object does not rotate, but the horizon does.

If you think that's rotation "intrinsic" to the object, then you have to explain why the camera was mysterious rotating at exactly the same rate as the object (and then later, not).

#### markus

##### Active Member
I disagree.

There are 3 vectors in play, lift, thrust, and weight. For level flight, there needs to be a horizontal component (mostly provided by thrust), and the vertical components need to cancel out. If the aircraft has excessive lift (obvious e.g. with STOL aircraft), the thrust vector may need to be somewhat down to make up for it.

The question is, is there a configuration of the F/A-18 where it can achieve level flight with negative pitch?
Most civilian aircraft are designed so there's nonzero lift at zero pitch (e.g. by mounting the wings at a slight angle, giving it some camber, etc), but that's not typically a design goal for aerobatic aircraft and in particular it is not the case with the F-18. The wings are symmetric about their chord and are mounted along the fuselage symmetrically (modulo some small amount of washout). At zero pitch there's no lift; the AoA indicator is referenced to the fuselage longitudinal axis for this reason. This changes somewhat with flaps extended but flaps will also increase drag so the overall conclusion doesn't change much; in fact, the standard landing configuration (flaps and gear out, velocity vector centered on the "E" bracket on the HUD) corresponds to an AoA of 8.1 degrees.

#### markus

##### Active Member
I don't know how you can then look at the object's angle with artificial horizon and say it's fixed in the camera frame in the first 20s, with confidence.
Are you saying the instruments are wrong?

#### dimebag2

##### Active Member
It's been said the object does not rotate with banking in the first 20s. By about 7-8°, see left plot below.
But the artificial horizon is not aligned with real horizon, by about 8°, before it realigns after 20s.
So you need to detrend the plot on the left to correct for this offset from real horizon. This completely flattens the orange curve on the left.

After removing this trend, you can measure the objet angle with enough precision to see it does not rotate with plane banking, while the cloud line do? Given how ambiguous it is to define the object axis in those first 20s?

Nobody sees a problem with the misaligned clouds, and the fact that the horizon offset corresponds to the angle delta of the "non-rotating glare", for this 1st observable? Really? We should just ignore this?

EDIT: messed up, see below. The trend goes the other way, but we still get a difference of ~7-8° between bank angle and object rotation from the graph below.

And I still got no answer for why Sitrec and 3D recreations would not capture this misalignment.

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#### Mick West

Staff member
It's been said the object does not rotate with banking in the first 20s. By about 7-8°, see left plot below.
But the artificial horizon is not aligned with real horizon, by about 8°, before it realigns after 20s.
So you need to detrend the plot on the left to correct for this offset from real horizon. This completely flattens the orange curve on the left.
This seems backwards. Looking at my graph, the amount of rotation of the cloud horizon (from about 16° to about 35°) is MORE than the artificial horizon (from about 23° to about 35°) over the first 20 seconds. During that time the "object" does not rotate relative to the camera.

So if you are comparing with the real horizon, then the lack of rotation is even more apparent. It's not removing a trend, it's increasing it.

#### DavidB66

##### Senior Member
Nobody sees a problem with the misaligned clouds
Surely it is the Artificial Horizon that is 'misaligned'. The source I quoted at #6 above (and there are other sources, e.g. the Wikipedia article on 'Attitude Indicator') gives some reasons for temporary errors in the AH, especially when the aircraft is making a tight turn.
I still got no answer for why Sitrec and 3D recreations would not capture this misalignment
Ideally it might be nice if Sitrec could incorporate some correction factor, but how? The extent and duration of misalignment depends in part on specific features of the instruments and their maintenance. According to 'Pilot Friend' the causes, among others, include:
Errors in the indications presented on the attitude indicator will result from any factor that prevents the vacuum system from operating within the design suction limits, or from any force that disturbs the free rotation of the gyro at design speed. Some errors are attributable to manufacturing and maintenance. These include poorly balanced components, clogged filters, improperly adjusted valves, and pump malfunction. Such errors can be minimized by proper installation and inspection.
Most of these factors are unknown. In the Gimbal case an ad hoc adjustment might be made using the clouds as an indicator of the true horizon, but various people have argued above that there is no need for this. [Edit: this was written before seeing #28 above.]

#### dimebag2

##### Active Member
This seems backwards. Looking at my graph, the amount of rotation of the cloud horizon (from about 16° to about 35°) is MORE than the artificial horizon (from about 23° to about 35°) over the first 20 seconds. During that time the "object" does not rotate relative to the camera.

So if you are comparing with the real horizon, then the lack of rotation is even more apparent. It's not removing a trend, it's increasing it.
Yes sorry, got things mixed up here, that's why I was saying above the object rotates CCW relative to clouds. Correcting for misalignment makes the object rotates CCW, instead of non-rotating. So it's moving opposite to the clouds, in other words it's rotating in the image frame.

#### Mick West

Staff member
So it's moving opposite to the clouds, in other words it's rotating in the image frame.
The "image frame" is the sides of the image, not the horizon.

#### dimebag2

##### Active Member
The "image frame" is the sides of the image, not the horizon.
Rotating in the scene, if you will.

#### Mick West

Staff member
Surely it is the Artificial Horizon that is 'misaligned'. The source I quoted at #6 above (and there are other sources, e.g. the Wikipedia article on 'Attitude Indicator') gives some reasons for temporary errors in the AH, especially when the aircraft is making a tight turn.
The artificial horizon angle is a measure of the bank angle of the plane, it would match the real horion when looking forward, in level flight.

If the pilot is banked 45° left, and looks 90° to the left, they would not expect the horizon to be tilted 45°, but if they look forward they would.

It's an interesting issue, but I'm a bit busy right now.

#### markus

##### Active Member
It's been said the object does not rotate with banking in the first 20s. By about 7-8°, see left plot below.
But the artificial horizon is not aligned with real horizon, by about 8°, before it realigns after 20s.
So you need to detrend the plot on the left to correct for this offset from real horizon. This completely flattens the orange curve on the left.

After removing this trend, you can measure the objet angle with enough precision to see it does not rotate with plane banking, while the cloud line do? Given how ambiguous it is to define the object axis in those first 20s?

Nobody sees a problem with the misaligned clouds, and the fact that the horizon offset corresponds to the angle delta of the "non-rotating glare", for this 1st observable? Really? We should just ignore this?

And I still got no answer for why Sitrec and 3D recreations would not capture this misalignment.
Look:

Horizon tilts. Artificial horizon tilts. Glare remains locked in place. No sophisticated analysis is required to see this. Just look at it.

#### Mendel

##### Senior Member.
Nobody sees a problem with the misaligned clouds, and the fact that the horizon offset corresponds to the angle delta of the "non-rotating glare", for this 1st observable? Really? We should just ignore this?
Yes. It's been explained above (several times) why.

You stand on your incredulity when you ought to understand the answers.
This behaviour is completely in line with the "glare" explanation.

I would like to hear some more on how you determine where the "cloud horizon" is. Do you equate it with the direction of the apparent motion of the clouds?

Have you seen the reconstruction where the clouds, stitched together into a panoramic image, appear curved?

#### Mendel

##### Senior Member.
Have you seen the reconstruction where the clouds, stitched together into a panoramic image, appear curved?

This has turned out to be a pretty tough nut to crack. I completed XXL's pan, and picked out four frames 10 seconds apart. I then lined up these frames on the pan to see the distance traveled between the frames over time.

I also interpolated the line of bearing for each frame based on the number of frames that showed the same frame angle.
It's obvious with this stitch (kept purposely straight) that the attitude indicator is misaligned with the apparent cloud surface in the first part of the video.

#### dimebag2

##### Active Member
The artificial horizon angle is a measure of the bank angle of the plane, it would match the real horion when looking forward, in level flight.

If the pilot is banked 45° left, and looks 90° to the left, they would not expect the horizon to be tilted 45°, but if they look forward they would.

It's an interesting issue, but I'm a bit busy right now.

How could this effect be missing in a 3D model with observed bank angle, Az, jet pitch? This sounds like a pretty basic 3D perspective effect. This effect changes with jet pitch, as expected, in your sim.

Anyway, I'm not going to argue with a wall. Those who want to see things from a different perspective will see how the objects move relative to the real horizon, i.e. the clouds. If you don't put everything the pilots have to say to the trash, it's entirely consistent with the close flight path, as the object slowly rotates CCW along the climbing path, before final rotation.

My goal with this work has always be to check if what the pilots describe can be verified, and that's the case, in terms of flight path, motion on SA (very important), and now rotation along flight path. I just want to share here because I'm sure some people will find it interesting.

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#### markus

##### Active Member
How could this effect be missing in a 3D model with observed bank angle, Az, jet pitch? This sounds like a pretty basic 3D perspective effect. This effect changes with jet pitch, as expected, in your sim.

Anyway, I'm not going to argue with a wall. Those who want to see things from a different perspective will see how the objects move relative to the real horizon, i.e. the clouds. If you don't put everything the pilots have to say to the trash, it's entirely consistent with the close flight path, as the object slowly rotates CCW along the climbing path, before final rotation.

My goal with this work has always be to check if what the pilots describe can be verified, and that's the case, in terms of flight path, motion on SA (very important), and now rotation along flight path. I just want to share here because I'm sure some people will find it interesting.
Why do you continue to ignore the clear video evidence that shows your assumption is wrong?

#### Mendel

##### Senior Member.
If you don't put everything the pilots have to say to the trash, it's entirely consistent with the close flight path
as it was without your observation, it's just a weird path

i don't understand this claim at all
would it not be consistent with the far trajectory as well?
and wasn't this what we knew before?

if your observation could make the close trajectory straight (and level?), that'd be a revelation, but does it?
you ignoring me feels like I'm posting to a wall

#### Mick West

Staff member
I did a little poking around, and I found that if the jet roll's component is gradually reduced as you look more to the left or right, then you get the effect of the different angles with the cloud motion horizon and the artificial horizon.

Like I said:
The artificial horizon angle is a measure of the bank angle of the plane, it would match the real horion when looking forward, in level flight.

If the pilot is banked 45° left, and looks 90° to the left, they would not expect the horizon to be tilted 45°, but if they look forward they would.

The question is what should this be? It seems like when looking forward you get all the jet roll, and then when looking at 90 degrees you get zero (or rather, the jet's pitch, or 3.6° here), but what happens, say at 45°?

45° is at frame 208, according to my graph, at that time the bank angle is 26.4, and the clouds horizon is approximately 21.5 (hard to say, as it's so noisy), which isn't half of 26.4, it's about 0.81 of it.

But it is about 26.4*cos(45 degrees) + 3.6*sin(45 degrees). = 21.2

i.e. a spherical interpolation between jet bank (looking forward) and jet pitch (looking sideways)

More investigation needed. Might be a while. But looks quite promising.

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