ATPL General Navigation - Class 4: Convergency.
Summary
TLDRIn this educational video, Grant explores the concept of great circles, the shortest paths between two points on the Earth's surface, and their constantly changing direction. He explains how the convergence of meridians, which depends on latitude, affects the great circle track using simple geometry and the sine function. The video demonstrates how to calculate the change in direction, known as convergence, and how to switch between great circle and rhumb line tracks. It also clarifies the differences and similarities between the two, including the fact that they only match at the midpoint of a journey and that the rhumb line is always closer to the equator.
Takeaways
- đ A great circle is the shortest possible distance between two points on Earth, and its direction changes as you travel along it.
- đ The direction of a great circle can be determined by simple geometry, comparing angles at different points along the path.
- đ The change in direction of a great circle is influenced by the convergence of lines of longitude, which varies with latitude.
- đ Convergence is zero at the equator and increases towards the poles, and it can be calculated using the sine function of the latitude times the change in longitude.
- đ§ The great circle bearing can be calculated by adding the convergence to the original angle at the starting point.
- đą The average great circle track can be found by averaging the bearings at two points, which is equivalent to the rhumb line track between them.
- đ The rhumb line is a path with a constant direction, which is always closer to the equator than the great circle path.
- đ The highest latitude of a great circle route is reached when the great circle track is at 90 or 270 degrees, indicating purely east or west travel.
- âïž The conversion angle, which is half the convergence, is used to switch between a great circle and a rhumb line.
- đ°ïž The relative position of 'north' changes as you move along a great circle, which is why the direction is constantly changing.
- đ The sine function is key to understanding convergence, as it provides a mathematical model that starts at 0 at the equator and reaches 1 at the poles.
Q & A
What is a great circle and why does its direction constantly change?
-A great circle is the shortest possible path between two points on the surface of the Earth. Its direction constantly changes because as you move along the circle, the relative position of 'north' changes, necessitating a continuous adjustment in direction to maintain the shortest path.
How can the angle difference at two points on a great circle be determined?
-The angle difference at two points on a great circle can be determined using simple geometry by creating parallel lines and comparing the angles at each point. The angle at point B is found by adding the angle at point A to the change in longitude and the convergence angle.
What is convergence and how does it relate to the change in direction of a great circle?
-Convergence is the angle by which the direction of a great circle changes as it moves from one meridian to another. It depends on the latitude and the change in longitude between two points, affecting how much the direction of the great circle needs to adjust.
Why is the sine function used to calculate convergence?
-The sine function is used to calculate convergence because it provides a value that starts at 0 at the equator and increases to 1 at the poles, which corresponds to the change in convergence from 0 degrees at the equator to full convergence at the poles.
How is the average great circle track calculated?
-The average great circle track is calculated by taking the average of the initial and final great circle bearings. This average is equivalent to the rhumb line track between the two points.
What is a rhumb line and how does it differ from a great circle?
-A rhumb line is a path that crosses all meridians at a constant angle, maintaining a constant direction from start to finish. It differs from a great circle, which changes direction as it follows the shortest path between two points.
At what point are the great circle track and the rhumb line track the same?
-The great circle track and the rhumb line track are the same at the midpoint between two longitudes, where the lines are parallel.
What is the conversion angle and how is it related to convergence?
-The conversion angle is the difference between the rhumb line track and the great circle track. It is equal to 0.5 times the convergence, providing a way to convert between the two types of tracks.
How does the latitude affect the convergence of lines of longitude?
-The latitude affects the convergence of lines of longitude because the lines converge more as you move from the equator towards the poles. At the equator, there is zero convergence, while at the poles, there is full convergence.
What is the relationship between the great circle track and the change in longitude at the poles?
-At the poles, the convergence is equal to the change in longitude. This means that the angle of the great circle track at the poles is determined solely by the change in longitude between the two points.
Outlines
đ Understanding Great Circle Navigation
The script introduces the concept of a great circle, which is the shortest path between two points on the Earth's surface, and its constantly changing direction. The presenter, Grant, explains how to determine the change in direction using a top-down diagram of the Earth centered at the North Pole. He illustrates that the angle between meridians changes as one moves away from the equator towards the poles. By using simple geometry, the script demonstrates how to calculate the difference in angle at two points on a great circle route. The explanation includes creating additional lines for clarity and using the z-angle rules to establish that the change in longitude plus the original angle gives the new great circle direction. The concept of convergence, which depends on latitude, is introduced, with the sine function proposed as a way to calculate it. The script concludes with a theoretical equation for convergence and an example to test the theory.
đ Calculating Convergency and Great Circle Bearings
This paragraph delves into the specifics of calculating the convergency and great circle bearings. It begins with an example where the great circle bearing from point A to point B is given, and the task is to find the new bearing at point B. The script explains the use of the sine function to calculate convergency based on latitude and change in longitude. An example calculation is performed, resulting in a convergency of 25 degrees, which is then used to adjust the original bearing to find the new great circle bearing at point B, which is 65 degrees true. The concept of the average great circle track is introduced, which is equivalent to the rhumb line between two points. The script also explains the relationship between the great circle track and the rhumb line, noting that they are equal at the midpoint between two longitudes, and that the rhumb line always curves towards the equator. The paragraph concludes with an explanation of the conversion angle, which is half of the convergency and represents the difference between the rhumb line and the great circle track.
đ« Great Circle Tracks and Convergency at Different Latitudes
The script continues with an example of an aircraft following a great circle track from point A to point B, and the task is to determine the great circle track on departure from A. The explanation involves drawing a rhumb line and understanding that it will be at a 90-degree angle to the change in longitude since the track is from west to east. The convergency is calculated using the sine of the latitude (45 degrees) and the change in longitude (60 degrees), resulting in a convergency of approximately 42 degrees. The conversion angle, which is half of the convergency, is then used to adjust the rhumb line to find the great circle track. The script emphasizes that great circle tracks are constantly changing due to the changing relative position of north, and at the poles, convergency equals the change in longitude. At the equator, there is no convergency since the lines of longitude are parallel. The sine function is used to calculate convergency, which is essential for determining the conversion angle. The paragraph concludes by summarizing the differences between rhumb lines and great circle tracks, noting that they only coincide at the midpoint between two points and that the average track of the great circle is equal to the rhumb line track.
Mindmap
Keywords
đĄGreat Circle
đĄTrack Direction
đĄConvergence
đĄLatitude
đĄLongitude
đĄRun Line
đĄConvergency Equation
đĄConversion Angle
đĄEquator
đĄPoles
đĄSine Function
Highlights
A great circle is the shortest possible distance between two points on the Earth's surface.
The direction of a great circle route constantly changes due to the curvature of the Earth.
The angle difference at two points on a great circle can be determined using simple geometry.
Convergency of lines of longitude depends on latitude, with zero convergence at the equator and full convergence at the poles.
The sine function is used to calculate the convergency based on latitude and change in longitude.
Convergency equals zero at the equator and equals the change in longitude at the poles.
An example demonstrates calculating the new great circle bearing at point B using convergency.
The average great circle track can be found by averaging the bearings at two points.
The run line track is equivalent to the average great circle track at the midpoint between two longitudes.
The run line always curves towards the equator, unlike the great circle.
The highest latitude or vertex of a great circle is reached at tracks of 90 or 270 degrees.
The conversion angle, which converts from a great circle to a run line, is equal to 0.5 times the convergency.
An aircraft's great circle track on departure can be calculated using the run line and convergency.
Convergency at the poles is equal to the change in longitude, affecting the great circle track.
At the equator, there is no convergency because lines of longitude are parallel.
The run line follows a constant track, while the great circle track is constantly changing.
The great circle and run line tracks are the same at the mid-longitude between two points.
The difference between the great circle and run line tracks is the conversion angle.
Transcripts
a great circle is a line across the
earth with a constantly changing track
direction but by how much does it change
let's find out
[Music]
hi i'm grant and welcome to the fourth
class in the gnab series today we're
going to be taking a look at the
constantly changing track of great
circles and then also how to switch
between a great circle and a run line
so we established in the previous class
that a great circle is the shortest
possible distance between two points but
the direction is constantly changing
if we look at a top-down diagram right
at the north pole you can see that if we
move from
this meridian here
the angle is quite small to this
meridian here the angle is clearly
larger
so if we take point a and point b on
this map and we think of the angle at
point a we can find out the difference
in angle at point b
using some simple geometry
so if we take a line parallel
to
point a
and pop it at point b
we can see that this angle in here let's
call it angle
um
x
is equal to this angle in here x
so we basically have to find the
difference that is added on
in this angle
and how do we do that
well we just create some
more lines to help us
so if we come
in here
and we take a parallel line going up we
can see that this angle in here
let's call that y degrees
using simple z-angle rules we can see
that this angle in here
would also be equal to y
and what is the difference between this
point and this point
or this line and this line
we're on different lines of
longitude so it's going to be the change
in longitude
plus our original angle is going to give
us our new great circle direction and
that's true at all points along here
so you can see even at this
midway point i've created this angle in
here is going to be x and then this
small angle in here let's call it z
and the reflective z angles
gives you
the change in longitude plus the
original angle
so as lines of longitude are curved they
go from being perfectly parallel at the
equator
all the way up to a single point at the
poles
this means that the convergency of the
lines of longitude must depend
on how close we are to either of these
points
so there's zero convergence at the
equator and there'd be full convergency
up at the poles
basically it must depend on the latitude
that we are at
so if we think about the equator as
being zero degrees and the poles as
being 90 degrees
we're looking for some sort of function
that gives us zero at zero and the full
convergency at 90.
so we have a function for that and that
is the sine function
so we can come up for a theory
and an equation for how much lines
converge
based on this
sine wave
so if we say
sine of the latitude
times by the change in longitude
then we should get the convergency
and let's just test that theory
so we know that we have zero convergency
at the equator so sine of zero
times whatever the change in longitude
is doesn't matter
is going to be
zero because sine zero is zero
so
convergency
equals zero
parallel lines makes sense
then we'll go all the way up to
sine of 90
times by the change in longitude
sine of 90 is one
and that's what we saw in the previous
example of that top down view at the
north pole we know that the convergence
of those lines is
the change in latitude between them
so there you go we've got a theory and
an equation for convergency
let's take a look at a wee example of
how we would use that
so here we go the great circle bearing
of b from position a
is 0 4 0 degrees true what is the new
great circle bearing at b
so first things first we draw the effing
picture
so we have point b
which is over here north 30
east 60 and point a is further sorry
point a is less east than point b so
point a is going to be over here and
it's at the same uh north 30.
we have north up this direction
and north up this direction
and we have a great circle going between
both of them
we have this angle in here as being 0 4
0 degrees
and we're looking for this angle in here
cool now we've got it all down on the
page
let's just use our equation and figure
this out so the convergency
equals sine lat
times the change in the longitude
so sine of the latitude is 30 sine 30
degrees
times with the change in longitude east
10 to e60 that's going to be 50.
sine 30 is a half times 50. that means
our convergency is going to be 25
degrees
and from our picture we can clearly see
that b
is a bigger angle
than the angle at point a
so we have to add this
onto our original angle
for an answer the great circle bearing
at b
is going to be 4a plus 25 is going to be
65 degrees true
and if we were to take an average of
our zero four zero and our zero six five
we'd find the average great circle track
so in this case it would be
52 and a half for the average great
circle track if it asked us that it's
very simple take one take the other and
divide by two
and that average great circle track
would be equivalent to the rum line
between the two of them
so when i draw a diagram of both the
great circle track from one point to
other and the rum line on it we can see
a few points
first at the midpoint between these two
longitudes
the rum line track and the great circle
track are equal because the lines are
parallel
second the rum line is always closer to
the equator
this case we're looking at the south
pole north is still up there the equator
is going to be up here somewhere and the
rum line curves towards the equator
always
and the third point
is that the highest
latitude or the vertex of the great
circle
is reached at the point where the great
circle track is 90 or 270 degrees we're
going either purely east or purely west
and the very final point is you can see
there's clearly a difference between
our starting
rum line track
so our starting great circle track and
our starting
rum line track
and the difference is this angle in here
this angle in here is known as the
conversion angle
because it converts us from a great
circle to a run line
and the conversion angle
is equal to
0.5 the convergency
so if we know the great circle track we
can work out the run line and vice versa
if we know the convergency and the
conversion angle
the conversion angle is always 0.5 times
the convergency but convergency the
example we saw
um our equation for sine lap times
change of long equals the convergency
isn't always true it depends on what
type of chart we're using but for now
think of this as sine
latitude times the change in longitude
and conversion angle is 0.5 conversion
angle always being half of the
convergency always
so let's take a look at another example
an aircraft follows a great circle track
from a to b
what is the great circle track on
departure from a
so let's draw the effing picture
great circle track from a to b what is
the great circle track on the part from
a
point a is west 50 point b is east 10.
so a is going to be over here
at
uh south 45
west 50 degrees
and this is on the same
latitude
south 45
and it's east 10.
if these were different latitudes and
sort of angled like this
to find the convergence you would take
the average of the two
but this is not the case for this one
but we are going to cross over the
equator right here
so we have south poles going to be down
here
north going up in two directions like
this
and we know that we're looking for the
great circle which is the straight line
between the two
and yeah that's all the information we
have
and it initially looks like we don't
have enough information to find out the
great circle track
in here this is what we're looking for
but we do because we know
that if we follow a line of latitude
that's a rum line
so we can draw on our rum line
roughly like that
and we know that if we're following a
line of latitude it's going to be either
90 degrees or 270 degrees
in this case we're going from point a
which is west to the east
so it's going to be 90 degrees
this angle in here
is going to be 90 degrees between this
rum line
and then we can find out the conversion
angle in here add that onto our great
circle
sorry add that onto our run line to find
the value for the great circle
so let's pop in
our
equation
sine lat
times the change in longitude
is the convergency
so sine 45
multiplied by the change in longitude
from west 50 to east 10 we're passing
through the equator at
sorry the
greenwich meridian so we've got to add
them two together
so it's 60 degree change
and sine 45 times 60 745 is about 70
percent
so we'll say that convergency is 42
degrees
and our conversion angle
which is our difference between the
runway and the great circle
is going to be half of that 21 degrees
okay and then if we look at the picture
we can clearly see that the great circle
is going to be bigger than the rum line
so our value for
x our great circle track
is going to be 90
plus the 21
or
111 degrees
in summary then great circle tracks are
constantly changing because we're
constantly referencing everything to
north and as we move the relative
position of north to us
changes like this
the
at the poles the convergency is equal to
the change in longitude you can see that
the convergency of these lines
means that we get an angle in here
which is equal to the change in
longitude and then we add that onto our
original
uh track
and that would give us our new track for
that great circle at this second point
so convergency at the poles is
the change in longitude but then at the
equator the lines are perfectly parallel
so there is no convergency
and we use the sine function because
that starts at 0 at 0 degrees and goes
up to 1
at 90 degrees
to give us a
equation for the convergency which is
convergency is the sign of the latitude
times the change in longitude
convergency
is very useful for getting the
conversion angle
which is the difference between the rum
line and the great circle
and it is 0.5 times the convergency here
some other points about the great circle
and rum line
the rum line follows a constant track
from point a to point b
and the great circle track is constantly
changing
the only point that these two values for
track will be the same is at the mid
longitude between two points
so you can say that the
average track of the great circle is
equal to the run line track
and as we stated before the difference
between the two will be the conversion
angle
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