Benjamin Høyer: Cartography

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Azimuthal Projections

Azimuthal projections are projected onto a plane from a defined projection point, for example from the Earth's centre through the surface onto a plane. In this respect, they are the most intuitive and simple projections. With this in mind, they are effectively what would be captured with a camera, taking a photo of the Earth from, say, the moon. In fact, azimuthal projections are commonly used for mapping the moon. The moon is a particularly well-suited candidate, since (to a good approximation), the same half-side of the moon always faces us.

The only two parameters in defining a projection done in way are where the projection lines originate from and where the projection plane is located. The projection origin is defined in multiples of R, with certain values giving the projection certain useful properties. Setting the projection source 1.35 R, as in Clarke's 'minimum error projection', is used for showing the general shape of continents. The location of the tangent point can be polar (normal), equatorial (transverse) or somewhere in between (oblique). The projection can always be considered as touching the sphere only once -- any movement of the projection plane along the axis of projection will only affect the final scale.

Figure 7

tan θ = d / r , tan θ = d ' / αr , d / r = d ' / αr, d ' = αd      (3.1)

Distance of the plane is increased from r to α r, i.e. by (α - 1) r. The scale of the projection changes linearly with the distance of projection plane from centre.

Gnomonic Projection

The characteristic perspective point in this case is at R, the centre of the sphere. All great circles are projected to straight lines, making them incredibly useful for sea or air navigation. In the simplest case, the polar case, parallels of latitude project to concentric circles about the pole (centre of projection). Meridians project to radial lines, crossing each line of latitude at right angles. The Equator -- a great circle -- projects to a circle of infinite radius, and so cannot be drawn: only (less than) a hemisphere can be represented with this projection. (Figure 6)

The polar case is particularly easy to describe mathematically. The longitudinal extent of a point is translated directly to the angular component of a polar plot; the latitude adjusted as below:

Figure 8

(φ, λ) → (r, θ) as follows: λθ      R tan(90° - φ) → r      (3.2)

In Cartesian coordinates, using the transformation below. Note, this is a top-down view onto the projection plane with the X axis aligned antiparallel to the prime meridian. Also note tan(90° - φ) = 1 / tan φ = cot( φ .

Figure 9

X = -r cos λ = -R cot φ cos λ  Y = r sin λ = R cot φ sin λ       (3.3)
X = -R cos λ cosφ / sin φ Y = R sin λ cosφ / sin φ       (3.4)

Noting that the longitude coordinate λ remains unchanged in the transformation, it becomes clear that (straight) lines of constant latitude on the sphere will project to (straight) lines of constant θ on the projection plane. Since a line of constant latitude is a great circle, any straight line drawn from a point to the centre of the projected map will give the shortest route. In the normal polar case, this will also be of constant bearing, or azimuth. Any oblique gnomonic projection will also display straight great circle lines through the centre (since the location of the pole is arbitrary, and only defines the origins of the coordinate system), although they won't be lines of constant bearing (since bearing is relative to the coordinate system).

Area preservation can be investigated by considering the parallel and meridian scales separately. The parallels are not evenly spaced: their separation increases as longitude decreases (approaches the equator). Unless the meridian scale reduces in accordance with the parallel scale to keep their product unity, area cannot be preserved.

First consider the parallel scale. Each parallel represents a small circle of increasing radius as the longitude decreases. The circumference of these can easily be found and compared to produce an expression for parallel scale.

Figure 10

r = R cos φ  r ' = R cot φr / r ' = cot φ / cos φ = 1 / sin φ
and so the exaggeration of parallel scale is 1/ sin φ       (3.5)

Then consider the meridian scale. Point A is located at a slightly higher latitude than P, but so close that the arc AP can be considered a straight line. The line A'R is perpendicular to OP'. Angle XP'O is equal to EOP', λ, by corresponding angles.

Figure 11

The meridian scale exaggeration is given by:

A'P' / AP = A'P' / A'R × A'R / AP and A'P' / A'R = 1 / sin φ while A'R / AP = A'O / AO by similar triangles
and so XO / BO = 1 / sin φ and the exaggeration is 1 / sin φ × 1 / sin φ = 1 / sin2 φ       (3.6)

Even at reasonably high latitudes, near the projection origin, area exaggeration is significant. For example, at a 45° elevation, the parallel scale error is 1/sin(45°) = 1.414 and the meridian error 1/sin2(45°) = 2. That is 41% and 100% increase respectively. The area increase is given by the product of the two orthogonal scales, 1/sin3 φ. So at 45°, area distortion gives an increase of about 183%. That being said, errors in area representation are kept reasonably small, less than 30%, for latitudes from 90° to about 66°.

If 1 / sin3 φ = 1.3 then φ = 66° approximately.        (3.7)

Given that this projection is most useful in the vicinity of the tangent point, a polar gnomonic is rarely used. However, the projection can relatively easily be calculated to place the tangent point anywhere on the globe. Since any point on the sphere has the same relation to the whole as any other, the facts already shown apply to any point. The transverse (equatorial) case is more common, but the general (oblique) can usefully be derived.

Taking the limit approaching the equator we see that along the circumference (parallel scale), we project to an infinite circle: this shows that only one hemisphere can be mapped using a gnomonic projection. Similarly the area term also explodes going towards the equator, and does so far quicker than the circumference.

As φ → 0    1 / sin φ → ∞       (3.8)

Showing that neither area nor shape are conserved using this projection can be done directly by
considering orthogonal scales using (2.1) and (2.2).

Area: 1/sin φ × 1/ sin2φ ≠ 1   Shape: 1/sin φ ≠ 1/ sin2φ       (3.9)

I will not show the general cases of all the projections, as the purpose of this report is to show the general conserved properties in a number of projections, which are independent of coordinate system (and so can be shown using the simplest choice of coordinate). However, to give you an idea for the method, I will show the derivation of the mapping equations for the oblique gnomonic projection.

Figure 12
Map Projections, P Richardus and R K Andler (North-Holland, 1972) p58

Here, the projection plane falls tangent to the sphere at O, which lies at (φ0, λ0). Being a gnomonic projection, the projection point lies at the centre point of the sphere, C. We define a general point P on the sphere located at (φ, λ0+dλ) which is projected to P' on our map. Note that CO=CP=R and let ∠ COP = ∠ COP' = δ. Locate P' on the projection plane with cartesian coordinates as shown. In the diagram on the right right above, X is aligned with north and Y with the great circle through O. Given this, we see:

NOP = 90° - B   NO = 90° - φ0 (arc)   NP = 90° - φ (arc)       (3.10)

Looking at Δ OCP' we see:

OP' = R tan δ       (3.11)

Using both diagrams, we see (by applying tan δ = sin δ/cos δ):

X = EP' - OP' = OP' = R tan δ sin B = R sin δ sin B/ cos δ
Y = DP' = OP' cos B = R tan δ cos B = R sin δ cos B/ cos δ      (3.12)

Using the spherical sine rule (sin A/sin a = sin B/sin b = sin C/sin c) on the spherical triangle Δ OPN, we write

sin dλ / sin δ = sin (90° - B) / sin (90 - φ) = cos B /cos φ    (3.13)

sin δ cos B = sin dλ cos φ      (3.14)

For the same triangle, applying the spherical cosine rule (cos c = cos a cos b + sin a sin b cos C) gives

sin δ sin B = cos φ0 sin φ - sin φ0 cos φ cos dλ      (3.15)
cos δ = sin φ0 sin φ + cos φ0 cos φ cos dλ      (3.16)

Now substituting (3.14), (3.15) and (3.16) into (3.12) gives the finished mapping equations

We can now confirm result (3.4), the mapping equations in the polar case. Set φ0 = 90° and dλ = λ evaluate (3.17) to agree with (3.4):

X = -R cos λ cos φ / sin φ      Y = R sin λ cosφ / sin φ       (3.18)

Stereographic Projection

The stereographic projection is produced in precisely the same way as the gnomonic, but the projection source is located at S, that is to say at tangent point's antipode, 2R from the projection point. The has the effect of halving the angle at the projection source. The following diagram applies to the polar stereographic case. However, preserved quantities on this projection will be valid for any other tangent point, O, and this allows for easier calculation.

Figure 13

φ + θ = 90° right angle (complimentary angles)    ½ θ = δ (inscribed angles) and so δ = ½ (90° - φ)       (3.19)

The second equation uses the fact that Δ OCP and Δ OSP subtend the same arc OP. ∠ OCP is at the centre and ∠ OSP is on the circumference, hence the relation by inscribed angles.

Again for this projection, longitude translates directly to the angular component in polar coordinates on the projection plane. The radial component is easily found by expressing the distance OP' in terms of φ through analysis of Δ OSP'.

(φ, λ)→ (r, θ) as follows:      λθ      2R tan (½ (90° - φ))→ r      (3.20)

This gives the mapping equations in the same way as (3.3) and (3.4), although note that tan (½ (90° - φ)) cannot be simplified to cot (½φ) .

X = -r cos λ = -2R tan (½ (90° - φ)) cos λ
Y = -r sin λ = -2R tan (½ (90° - φ)) sin λ       (3.21)

For parallel scale analysis, consider Figure 13 and triangles Δ SAP and Δ SPO. In this case, since the angle at source and centre (δ and φ) differ, it is clearer to analyse using properties of similar triangles. The scale enlargement is given by AP/OP' .

SP/SA = 1 / cos δ from Δ SAP and SO/OSP = 1 / cos δ from Δ SPO.
So parallel scale factor = 1 / cos2δ = 1 / cos2 (½ (90° - φ))       (3.22)

For the meridian scale, again the similar triangle approach yields the result quickest. The scale enlargement is given by PQ/P'Q' , as defined in Figure 14.

Figure 14
PQ projects onto P'Q'. OPn is a straight line. P'm' and Pn are parallel and perpendicular to P'S. Q'P'm and nPQ' both equal δ.

P'Q' / PQ = P'Q' / P'm × P'm/PQ
P'Q' / P'm = 1 / cos δ and P'm / PQ = 1/cos δ since P'm / Pn = P'S / PS by similar triangles.
So the meridional scale factor P'Q' / PQ = 1 / cos2 δ = 1 / cos2 (½ (90° - φ))      (3.23)

Now checking area and shape preservation using (2.1) and (2.2) we see that shape is conserved, but at the expense of increased area deformation, compared to the gnomonic in (3.9).

Area: 1 / cos2 (½ (90° - φ)) × 1 / cos2 (½ (90° - φ)) ≠ 1     Shape: 1 / cos2 (½ (90° - φ)) = 1 / cos2 (½ (90° - φ))      (3.24)

Orthographic Projection

The final of the azimuthal projections I will look at is the orthographic case, where the projection source is placed at infinity. In this case, the imaginary rays of light come in perpendicular to the plane of projection. It is easy to see that, as in the gnomonic projection, only half the sphere can be mapped at a time. It is also easy to see that the reason for this is quite different. In the gnomonic case, the equator mapped to an infinitely large circle: the parallel scale exploded. In the orthographic case, the equator is mapped to a circle radius R, but the meridian scale collapses to zero. Already now we can assert that shape cannot possibly be conserved. Also, the area cannot possibly be preserved. The total area of the final map is well-defined: a circle of radius R, πR2. This area must represent the surface of a hemisphere, 2πR2. Clearly area is not preserved. (3.25)

If we consider the light rays which cast the shadow onto our projection plane in reverse, we note that this projection in fact draws the sphere as if seen a very long way away, for example, looking at planets or stars in a telescope. For this reason it is of particular interest for astronomers, as it maps the surface as we see it.

The projection is set up as in Figure 15.

Figure 15

Again, this projection is most easily defined in terms of a mapping equation in polar coordinates.

(φ, λ) → (r, θ) since λθ and R cos φr      (3.26)

Straight away we can consider the scale factors in each direction. First, parallel OP' / Cr :

OP' = Cr = R cos φ and so the parallel scale factor = OP' / Cr = 1      (3.27)

The meridian scale factor is found differently to the previous two cases. We are looking for an expression for P'Q' / PQ.

Figure 16

P'Q' = R cos φ2 - R cos φ1
PQ = R(φ1 - φ2) ⇒ P'Q'/PQ = (cos φ2 - cos φ1)/ (φ1 - φ2)      (3.28)

Then applying the cosine sum-to-product formula

cos φ2 - cos φ1 =
-2 sin (½ (cos φ2 + cos φ1)) sin (½ (cos φ2 - cos φ1))      (3.29)

we get

P'Q'/PQ = (cos φ2 - cos φ1)/ (φ1 - φ2) =
-2 sin (½ (cos φ2 + cos φ1)) sin (½ (cos φ2 - cos φ1))/(φ1 - φ2)      (3.30)

Inspecting the limit as Q approaches P, i.e. φ2φ1 (which it does from below since φ2< φ1),

The first factor → -2 sin φ1 and the second factor → -½ since sin(x) / x → 1 as x → 0. (3.33)

So letting φ2φ1 and setting φ1 to φ the scale factor = P'Q' / PQ = sin φ      (3.34)

On inspection we see result is an area reduction, different to the previous cases:

2 sin φ < 1 for all 0 ≤ φ ≤ 90°      (3.35)

Having shown this, we can check the area and shape conservation, again using (2.1) and (2.2).

Area: 1 × 2 sin φ ≠ 1      Shape: 1 ≠ 2 sin φ      (3.36)

Thus, neither area nor shape is conserved. We can confirm the assertion in (3.25), that this area reduction indeed decreases the area by factor of 2. By integration, the total area expansion over all φ is 2 as in (3.37).

∫ 2 sin φ dφ = 2 where the integral is over [0, ½ π]      (3.37)

Comments on other Azimuthal Projections

In addition to the three azimuthal projections described here there are countless non-perspective projections. These are created mathematically to have a desirable characteristic, for example shape or area preservation or correct distance, or a beneficial trade-off between the two or three.

Azimuthal Equidistant Projection

In this case, a decision is made that all distances from the centre must be true. That is to say points must lie a distance from the centre on the plane equal the distance along the great circle arc on the globe. While the meridian scale is truly represented, the parallel scale must suffer accordingly. The resulting parallel scaling factor is given by:

projected length / true length = (2πR (90° - Rφ) / 2πR cos φ = (½ π - φ)/cos φ      (3.38)

This projection is in fact quite accurate, and even at latitudes as low as 45° (π/4 rad) the parallel length error (also area error) only amounts to about 11%.

Common Application of Azimuthal Projections

Since the azimuth from the centre is true for all azimuthal projections, they are commonly used to show directions from a point. In modern times you are most likely to have encountered them in airline magazines. They often display a map centred on their hub, with flight paths radiating outwards to their destinations.

Previous chapter
(Mathematical foreword)
Contents Next chapter
(Cylindrical Projections )

B Høyer June 2009