# Do pilots adjust the aircraft's flight path to allow for the curvature of the Earth?

If an airliner is flying at approximately 500-600mph, it seems to me that there would need to be a significant adjustment for altitude so as to not fly off into space.

Wikipedia says that there's an 8 inch drop for every mile because of the Earth's curvature. But, I've never heard of any airliner adjusting for the curvature. Also, shouldn't aircraft have to adjust somehow for the Earth's rotation because it varies depending on the latitude?

• Why don't ships or cars "fly off into space"? Speed doesn't matter (well, not at such slow speeds anyway :)), but one thing ships and airplanes and cars have in common does. – Luaan May 23 '16 at 9:09
• @Pondlife A small quibble: Wikipedia does not say "that there's an 8 inch drop for every mile"; it merely quotes Samuel Rowbotham's method for calculating the Earth's curvature, which uses that formula without justification. It doesn't claim it as fact or say anything about whether it is correct or accurate. – Reinstate Monica -- notmaynard May 23 '16 at 13:54

Aircraft altitude is measured (inferred) by atmospheric pressure. The aircraft is usually flown at an altitude that maintains constant ambient pressure (by pilot or autopilot, as the case may be). Changes in local barometric pressure (provided by air traffic control) are used to recalibrate the aircraft altimeter. As long as the aircraft is flown at a constant ambient pressure (hence constant altitude), it will be following the earth's curvature (as the atmosphere is attached to the spherical earth and has same properties at same distance from the center, in an ideal case) as the altitude is measured from the surface, which is curved, and not a plane.

There is no adjustment needed as the aircraft will naturally follow the curvature of the earth without any input from the pilot. This is because the aircraft flies through the atmosphere which also follows the curvature of the earth.

• And the gravitational force works at right angles with the surface of the earth too. – SMS von der Tann May 18 '16 at 11:51
• @SMSvonderTann you might find that an "almost" is missing in your sentence (earthscience.stackexchange.com/q/7520) – Federico May 18 '16 at 11:58
• @SMSvonderTann if that were true, you'd be able to walk up cliffs. – Pete Kirkham May 18 '16 at 15:41
• @Federico, while the gravitational force does indeed act “almost” perpendicular to the Earth surface, it is irrelevant, because the force we are concerned with is the weight and that works exactly perpendicular to the geoid. – Jan Hudec May 18 '16 at 16:07
• @jpmc26, the Geoid, or “sea level” is defined as equipotential surface of weight and, since the weight field is smooth, the weight force is always perpendicular to it, exactly. The Earth also does actually assume that shape, except for terrain features, because rock is soft enough to behave as liquid (and most of it is molten inside Earth anyway). – Jan Hudec May 19 '16 at 5:08

There isn't an adjustment for altitude. An aircraft flying level at a given altitude and trimmed for level flight will stay at that altitude. That means the flight path will have a gentle nose-down curve (looking from far away from the earth) as the direction of down (towards the centre of the earth) changes.

Think about the gravitational potential energy of the aircraft. To climb (which is actually flying in a straight line when you consider the curvature of the earth), the aircraft has to gain energy. In a level flight attitude, it doesn't gain any energy, so it will stay at the same altitude. A path that doesn't gain or lose altitude is an ellipse that goes around the earth.

Another way of thinking about it is to consider how "down" changes as the aircraft travels. The weight of the aircraft always acts towards the centre of the earth, and is matched (in level flight) by the lift of the wings. Imagine if you had a model aircraft suspended on a piece of string, dangled from your hand. If you hold the string and carry the model a quarter of the way around the earth, the bottom of the model will still point down (towards the centre of the earth). The model has rotated 90 degrees, without you having to rotate it by hand.

When you trim for level flight, you do so by finding the pitch attitude where your speed and altitude remain constant (or at least stable: atmospheric conditions might make them fluctuate a lot). That attitude might be a touch more nose-down than it would be if the earth were flat, but it's imperceptible.

• "downward" is a little abstract in this usage. The airplane will stay level relative to the earth. Relative to some fixed point beyond earth's reference (a distance satellite?) the airplane will change orientation, but there will no longer be a concept of "up" and "down" from that perspective. – abelenky May 18 '16 at 17:08
• @abelenky I think I've found what was unclear and tried to make it a bit clearer. Do you think it's better now? – Dan Hulme May 18 '16 at 17:59
• @DanHulme Hey Dan. Why an ellipse? For practical purposes, we assume that the Earth is a perfect sphere. Surely the path would therefore be a circle? I do like the model aeroplane analogy. – Simon May 18 '16 at 19:53
• @Simon Earth is 'fatter' at the equator than across the poles. The difference in diameter is about 26.5 miles. That is, you're 13 miles closer to the center of Earth if standing at sea-level at a pole vs. standing at a sea-level point on the equator. – reirab May 18 '16 at 20:31
• @A.Danischewski I'm sorry if that last analogy is confusing: it seems like you haven't understood what I'm trying to say. The string in the analogy is not about gravity, it's just a way of seeing that the model is balanced and level. As you say, when the aircraft flies, gravity now pulls it backwards, so the balance will change (viewed from outside), keeping the fore-and-aft line perpendicular to the line of action of the weight. – Dan Hulme May 20 '16 at 8:58

This is more of a physics question rather than an aviation question. While other answers have addressed the question from the aerodynamics point of view, let me try answering it from a physics perspective: frame of reference.

## Frame: where are you going?

How do you know an object is moving? The answer is you don't - there is nothing like a "fixed absolute coordinate in space". Speed is measured by referencing another object. If you think of it, every time we mention an object's speed, we always mean it's speed relative to something. The car's speed is 50mph relative to the ground below it, although we seldom say it explicitly in our daily conversations.

## Curvature of the Earth

Most of our daily experiences of motion are related to one of the four fundamental forces of nature: gravity. Gravity is spherical - if we pick a random point on the planet, the gravitational force should be the same as any other random point (let's assume the Earth is a perfect sphere for now).

Consider me walking on the surface of the Earth. I would be unaware that the Earth is spinning without external reference points like stars and the sun. If the Earth stands still, I would walk just the same, because when I walk I am interacting with the ground below me, which is part of the Earth; I am not interacting with Saturn or any other planet.

## How do planes fly?

Well, it happens that planes fly by interacting with the atmosphere. The atmosphere is affected by the Earth's gravity like the ground - the ground+atmosphere spins round and around together. Airplanes are not interacting with other stars or planets or satellites in any way - its engines produce thrust against the atmosphere (which moves together with the ground).

You chose your frame of reference as an arbitrary point in space, which is why it leads to incorrect conclusions. If you are travelling on a train and you want to move around, you don't have to factor in how fast the train is moving - you are interacting with the train, not the rail which the train travels on. Otherwise, as the train moves forward following the curvature of the Earth, you will rise higher and higher in the cabin, eventually hitting the ceiling. You are following gravity, which is spherical. Same for planes.

• "If the Earth stands still, I would walk just the same." Well, excepting for the effects of the earthquakes or similarly-destructive geological events that would result from that. :) The extra 13 miles of radius at the equator would suddenly become a problem. The weather could get interesting, as well. – reirab May 18 '16 at 20:57
• @reirab - The first chapter of the What If? book (which is widely excerpted online) discusses exactly that. – Bobson May 19 '16 at 3:31
• @Bobson Oh, I was thinking if the atmosphere stopped, too. Yeah, leaving the atmosphere as is while stopping the surface would be even worse. Also, xkcd is wonderful. – reirab May 19 '16 at 12:57
• Be careful - "engines produce thrust against the atmosphere" - no they don't. They produce thrust by accelerating an expanding mass of air backwards from the exhaust or by driving a prop which accelerates air backwards. Newton's Third Law. If you could somehow supply air to the intake of a jet engine, it would work perfectly well in a vacuum. Of course, a prop wouldn't work since it would have no air to accelerate backwards. – Simon Jul 14 '16 at 9:54
• @Simon well, we usually don't carry liquid oxygen and inject it into an jet engine (-: – kevin Jul 14 '16 at 11:23

Is a plane fast enough to get to space? Well, unfortunately for space exploration, no, it isn’t. Let’s check with a few calculation.

## Assumptions and definitions

First, let’s make an assumption: our plane P is flying in the void: there is no air slowing it down. Of course, this assumption is very wrong, but we only care about speed here. Air friction only slows P down. So, if P doesn’t go into space without an atmosphere, it won’t either if we add an atmosphere.

Let’s consider a reference object O orbiting the Earth at the same altitude r as our plane. Of course, we can neglect the mass of P and O, which is several orders of magnitude lesser than the mass of the Earth.

Let vO be the orbital speed of O and vP be the speed of P. If vP > vO, then P will spiral away from Earth and end up in space. If, on the other hand, vP < vO, then P will spiral down until it crashes.

## Computing vO

Because O is orbiting on a circular trajectory, vO and r are correlated by the following formula:

vO = √(GM/r)

Where G is the gravitational constant and M the mass of the object we’re orbiting around.

In our case, we are orbiting around the Earth and

GM = 3.99×1014 m3s-2

By using the SI units and approximating 3.99 with 4, we end up with the following relation:

vO = 2×107 r-1/2

The radius of the Earth is included in r, so we must add 6 371 km to go from altitude to r. 6 371 km is actually two order of magnitude higher than any altitude you could fly at, so let’s round it to 6,400 km and be happy with it.

So, in order to orbit at 6 400 km = 6 400 000 m from the center of Earth, we would need to go at

vO = 2×107 r-1/2 = 2×107 (6 400 000)-1/2 m/s = 2×107/2530 m/s = 7.9×103 m/s = 1.8×104 mph

That’s the speed relative to the center of the Earth, not to its surface.

## So, does our poorly modeled plane fly?

Our plane P, is flying at 600 mph relative to the surface. If we suppose we’re flying above the equator, we can reach a speed of 1600 mph relative to the center; that still is an order of magnitude below the speed we need to reach to orbit at such a low altitude.

vP << vO

Our plane would crash. Fast.

## Thank goodness, planes are safer than that. But how come?

Well, as we’ve just seen, the speed of the plane is not what keeps it flying. But we made a very strong assumption in our opening section: we neglected the atmosphere.

Air certainly slow our plane down. But we’ve also built it so that, when moving through air, our plane tends to go up thanks to lift.

So we have two opposing effects, here: the plane isn’t going fast enough to escape the Earth attraction; but the lift pulls up the plane hard enough that it keeps flying. But lift depends, among other things, on the density of air around the plane: the higher the planes fly, the weaker lift is. So, all in all, the plane flies in a layer of air at the same pressure, which in turn follows approximately the surface of the Earth.

• "several orders of magnitude lesser" understatement of the year ;) – Simon May 18 '16 at 18:17
• @Simon Haha, yeah, it's approximately 19 orders of magnitude... for an A380 at MTOW. – reirab May 18 '16 at 20:36
• For your information; You somewhere claimed that an airplane orbits 6400 m from the center of Earth. That should have been 6400 km of course, so I suggested an edit to fix it. – wythagoras May 19 '16 at 6:00
• Orbiting is not necessary to be "in space" or to "leave" Earth attraction. In theory, you can go to Mars vertically, climbing 1 mm per day if you have the fuel required for that. Orbiting speed is useful only for unpowered spacecraft (this is always the case in practical). – mins May 19 '16 at 6:55
• @wythagoras Ah! I knew I screwed up somewhere when I went from thinking “it’s nearly a perfect square, too bad there’s an extra 10 factor” to “wait, it is a perfect square”. – Édouard May 19 '16 at 10:05

A tangent (pun intended) answer: if adjusting for curvature of earth was required, it infers that you could accidentally fly into space.

If you could accidentally fixed-wing fly into space, we wouldn't need sophisticated and powerful rockets.

• And anyway you can't fly into space, as flying requires air to create lift. No air, no lift, no climb. – mins May 19 '16 at 0:05
• Cute, but this argument doesn't actually work. The consequence of not adjusting for the Earth's curvature could be something other than accidentally flying off into space. E.g., at high enough altitude, your wings won't generate enough lift and your jet engines won't have enough oxygen to burn fuel with. Maybe you'd need too much fuel to get to that altitude anyway. Maybe the low pressure would burst the hull. It's possible that planes adjust for curvature to avoid these catastrophes, not to avoid flying into space. (Of course, they don't, but the question is premised on now knowing that.) – David Richerby Oct 18 '17 at 15:29

It's no different than driving. If you drive absolutely straight you'll eventually leave the road. The road lines are meant to be followed.

So what is equivalent to the road lines in flight? Air pressure. The air pressure reduces the further you are from the surface. Pilots and autopilots follow the air pressure gradient, trying to keep the plane at a set air pressure. This pressure is used rather than GPS or "straight flight" because it's one of the many factors that affects flight efficiency - speed vs air resistance vs load bearing capacity of the aircraft. They are flying, or attempting to fly, in a pressure range that is going to cost the least to accomplish the various goals of the airline.

The air pressure varies according to many factors, but the main factor is height from sea level, and so by flying inside a specific range of pressures, they maintain a reasonably constant height from sea level. Since "sea level" pressure is curved along with the earth, then what you find is that they automatically follow the curvature of the earth. In other words, the planes are flying slightly down all the time.

• @mins correct, air pressure varies according to many factors. – Adam Davis May 19 '16 at 1:50

I think there are a lot of good answers in regards to the curvature of the Earth, but I didn't see anyone addressing the spin of the Earth.

The atmosphere is naturally dragged along with the surface of the planet. Wind speed is measured relative to a fixed location on the surface. Therefore differences between the movement and the Earth's surface are reflected as wind. Pilots do adjust for the effect of wind on the flight path.

If the pilot is using basic navigation techniques, he would calculate the heading he had to fly with the predicted crosswind to end up at his destination. If there is any crosswind component, the flown heading will be different than the direct heading on the ground or great circle heading.

With navigation equipment these adjustments can be done in real time. for example if you want to fly a specific GPS heading, but keep drifting to the right you would turn to a more left heading to regain and keep the desired heading.

Planes don't stay perfectly level!

That's the long and short of it. As these 600 mph machines go that 1 mile forward, both gravity and air pressure force the plane down those 8 inches. More, actually; these massive metal monstrosities have to put up a fight to be able stay up in the air! That's what the elevators, rudders, engines, airfoils, etc. are all there for.

Of course, if it were going sufficiently fast (at least 25,020 mph), it would overpower gravity, leave the atmosphere, and enter orbit. I hope it has engines that don't rely on air, and a body made of reinforced carbon-carbon!

• It's not necessary to reach escape velocity to enter orbit. In fact, it's necessary to have a smaller velocity, because if you're traveling at escape velocity, the force pulling you back toward the earth will be too small to make you reverse direction. – phoog May 25 '16 at 0:30
• @phoog somehow the comments on this answer were wiped. In them, I said that the original value I'd written was low-earth orbit velocity (17,400 mph), but just to be sure the plane would leave the atmosphere and enter orbit, I bumped that up to escape velocity. – Ben C. R. Leggiero May 25 '16 at 14:50

There have been good answers, but all seem to miss one aspect.
The aircraft does not maintain a constant distance to the surface of the earth. It simply maintains a level, where the atmospheric pressure is at a set target.
To clarify:
Autopilot (or the pilot) wants to maintain a contstant reading on the altimeter. This reading is merely a calibrated difference to a reference level, which normally is set to 1013,25hPa (or 29,92inHg) on cruise flight. This means that maintaining an altitude of about 30000ft, you want to maintain a pressure level of 300hPa. I say about 30000ft, because the true altitude of this level varies greatly, affected by airmass temperature and air pressure.
Jan Hudec was close to this on his answer, though I'd like to make a few corrections. First of all, the levels aren't isobars. Isobars is by definition "an imaginary line or a line on a map or chart connecting or marking places of equal barometric pressure" source. Second of all, there's no need to over-complicate things: the (auto)pilot monitors the altimeter, and does the necessary corrections. It would do so if the aircraft was out of trim also.

• That's true, but the reason I didn't mention it (I can't speak for anyone else) is that it doesn't really help answer the question. It just makes things a little harder to understand. Some student pilots get quite stuck with the idea that flight levels aren't levels of constant altitude (neither by QNH nor by true altitude), so I thought it would be easier to simplify and assume altitude. After all, there's no reason you couldn't fly at constant true altitude. – Dan Hulme May 18 '16 at 18:03
• He obviously meant isobaric surface. – hpekristiansen May 19 '16 at 0:25

An important flight instrument is the Attitude Indicator, which has an artificial horizon line. This instrument tells the pilot if the wings are level and whether the nose is pointing above or below the horizon.

Rather than never taking the curvature of the earth into account, when a pilot is using this instrument to keep his flight level, he is actually constantly adjusting for the curvature of the earth.

EDIT: The following link is to a question on how attitude indicators are kept accurate. A particularly acrobatic flight might cause it to tumble and become useless until it is reset. However it easily maintains accuracy during normal flights, including sustained turns.

How are attitude indicators kept accurate

Short version: Yes, a gyroscope would have problems with earth curvature, but another part of this instrument constantly maintains a "local down" for the instrument.

• I'm not for sure, but over long flights, doesn't the gyro in an attitude indicator have to be adjusted, in part because it does not follow the curvature of the earth naturally? I thought there was question about this a few weeks back, but can't find it right now. – OSUZorba May 18 '16 at 15:42
• This seems to be the perfect counter-example, if we exclude its drift, a gyroscope keeps a fixed orientation "in space", so following its plane of rotation will change the aircraft altitude, as the OP assumed. – mins May 19 '16 at 6:21
• @OSUZorba Yes, there is an erection mechanism that keeps it aligned with gravity (and that may give you false readings for instance if you keep flying in circles for a long time). If the artificial horizon was a perfect inertial platform, it would indicate inverted flight after a sufficiently long time flying along the Earth's curvature! – DarioP May 19 '16 at 13:01
• I added details on the auto-correction of this instrument. – Michael Richardson May 19 '16 at 13:59

I am not really qualified to answer, but did so on account of the many answers implying that some form of pilot control is necessary to keep an airplane following the curvature of the earth. Input from a real aerodynamicist would be appreciated.

Several answers state that when an airplane is flown to a constant altitude, it implicitly follows that curve. True enough, but the question asks whether pilot control is necessary. Are there physical constraints relevant to the question?

If an airplane were to follow a tangent, rather than the curvature of the earth, it would experience at least two effects: the direction of the earth's gravitational field would change, causing its weight vector to rotate rearwards, and the local air density would decrease as the aircraft's distance to the earth increases.

As the weight vector rotates backwards, it increasingly retards the airplane, and if it is longitudinally stable, it will, as a result, tend to pitch downwards. The effect of decreasing density seems more complicated, on account of its effects on power as well as aerodynamic forces, but there seems to be a general tendency for a trimmed, longitudinally stable airplane to be stable in density altitude - see What does it mean for a plane to be aerodynamically stable?. It is certainly true that every airplane has a maximum achievable altitude, determined by physics.

Both of these effects (longitudinal stability with respect to the local gravity field, and stability in density altitude) will tend to guide the airplane to follow the curvature of the Earth without pilot input.

Your question is basically "What makes the flight change its direction continuously?"

Let's consider the case of a plane flying overhead right now. If we are told that the plane is in level flight, the vertical, downward pull of gravity is equal to the vertical, upward lift. Further both forces are perpendicular to the fuselage and cancel out.
Imagine what happens a very very short time later when the plane has moved a very small distance ahead in a straight line. Since the plane has moved in a straight line, the lift is still pointing upwards and perpendicular to the the fuselage, but gravity is pointing towards the center of the Earth. So now only a part of the lift is countering the gravity, and there is a resultant gravitational force that will pull the aircraft down towards the center of the Earth.

This is very similar to how you can tie a stone to a rope and whirl it. The stone will go around in circles. The force of the string (tension) keeps the direction of the stone changing continuously without affecting the speed.

I'm missing one thing mentioned in the answers. The point is: They say that the gravitational force and the lift force's vertical component have to be equal to make the plane fly levelled. This is not entirely true. The plane is also turning, which needs some force. Let's see how much it is, for a plane at $10\,{\rm km}$ flying $900\,\rm{km\,h^{-1}}$:

• Gravity: $-mg = -m\cdot9.8\,{\rm m\, s^{-2}}$
• Force needed for turning: $-mv^2/r = -m\cdot (250\,{\rm m\,s^{-1})^2} / 6388000\,{\rm m}=-m\cdot0.00978\,{\rm m\, s^{-2}}$

($m$ is the weight, $g$ is the gravitational acceleration, $v$ is the speed and $r$ is the turning radius)

So the force needed for turning is $1/1000$ of the gravitational force. This is negligible in the planes power and other settings, and also much smaller than any irregularities such as varying air density, wind speed etc.

Conclusions: the adjustment for "turning down" is so little that it's negligible.

• and your point is? are you answering the question, commenting or asking something else altogether? – Federico May 20 '16 at 8:05
• My point is, no, they do not adjust for it since the necessary adjustment is negligible. – yo' May 20 '16 at 8:43

For reasons that are explained in several other answers, there is no need for an aircraft pilot to make any special calculations or adjustments of the controls in order to "drop" the aircraft eight inches per mile and thereby follow the Earth's curvature. Instead what pilots actually do is to keep the aircraft at a constant pressure altitude, which (approximately) follows the curvature of the earth. The path followed by an aircraft in this fashion actually can rise or fall quite a bit due to weather-induced variations in pressure, so it's not guaranteed to drop 8 inches each and every time the aircraft flies a mile, but it will follow the curve of the Earth much more closely than it follows a straight line over any long-distance flight.

So the answer to the first part of your question is, yes, pilots do adjust the aircraft's flight path to allow for the curvature of the Earth, and this is how they do it. There is no explicit "adjustment for curvature" term in the pilots' (or autopilots') calculations, however, because they keep the aircraft on the desired (curved) path by observing the pressure altitude, much like the way you would keep an automobile in its lane on a highway by observing the lane markers, which does not require you to know anything about the radius of curvature of each curve the highway engineers laid out. If the curve is gentle enough you might not even notice that the road is curved, yet you would still be following it.

A notable difference between highway lanes and the altitude of an aircraft, however, is that the aircraft does not have the ability to fly above a certain altitude at a certain setting of the controls, due to the low density of air at higher altitudes--so at higher altitudes the aircraft will tend to descend--whereas at much lower altitudes the aircraft's engines will generate more than enough power to keep the aircraft aloft and so the aircraft will tend to climb. The physical properties of the atmosphere and the aircraft therefore tend to deflect the aircraft along a path that (approximately) follows the curve of the Earth, and they absolutely prevent the aircraft from continuing along a straight tangent line into outer space. A large part (one might say the most important part) of how pilots make the aircraft stay at a given pressure altitude is by adjusting the "trim" of the aircraft (including throttle) in order to make it tend to fly at the desired altitude.

As for the Earth's rotation, since the atmosphere generally rotates along with the rest of the Earth, and aircraft fly in the atmosphere, the aircraft do not have to fly faster or slower to overcome the Earth's rotation, any more than you have to be able to run 500 mph in order to make your way from the aft restroom forward to your seat. (Some bits of the atmosphere do sometimes go a little faster than the Earth's rotation, sometimes a little slower, depending on which way the wind is blowing at each place; pilots do to account for the wind in order to fly the desired path over the ground and to know how long it will take to arrive.)

On the other hand, pilots very much do have to take the curvature of the earth into account when planning the lateral direction in which to fly. For example, to fly from New York to London, the shortest path (fewest miles over the ground or water) heads out of New York on a course of about 51.4 degrees. The return flight starts out toward New York on a course of about 288 degrees. Similarly, from London to Tenerife is a course of 213.5 degrees, 23.1 returning, and from Tenerife to New York is 300.5 degrees, 85.8 returning. If you take those three flight paths as the three sides of a triangle, it has angles of 34.4 degrees at New York, 74.5 degrees at London, and 82.6 degrees at Tenerife, which adds up to 191.5 degrees, which is impossible for any triangle plotted on a flat plane. Pilots and other people who plan the routes of aircraft do so by calculations (nowadays generally done in software) that take into account the approximately spherical shape of the Earth to do so (and nowadays even take into account the few miles' difference between the diameter of the Earth measured from pole to pole and the diameter measured across the equator).

Lots of good answers, but nobody has addressed the simple mathematical aspect of the question.

Assuming the “8 inches per mile” is accurate, that means you’d have to “nose down” to a pitch of -1” per 7920” flown, or a pitch of -0.007 degrees. That’s pretty much level flight. The pilot nor autopilot would even be able to discern the difference. You can’t even eyeball 1 degree let alone 7/1000ths of a degree.

Yes, they do because the autopilot (or pilot) monitors the altitude and keeps it constant if so planned. This includes adjustment due the Earth curvature, among many other factors that may be more significant.

If the air density would remain constant regardless of the distance from the planet, the aircraft would not follow the planet curvature automatically and could fly into space no problem. Altitude is limited by the air density.

Artificial satellites follow the planet curvature in space, but unlike planes they move in ballistic trajectories.