Benjamin Høyer: Cartography

Conical Projections

This final family of projection uses a cone as its projection surface. Often the apex of the cone is aligned with the polar axis of the sphere, resulting in a polar projection. The results in relatively simple to generate maps with good levels of accuracy, particularly in mid to high latitude regions. Figure 26 shows the principle of conic projections. As with the cylindrical types, the cone can have either one latitude of intersection, or two.

Figure 26

Conical Projection with One Standard Parallel

This projection preserves meridian distances. Points are projected the correct distance from the intersection line, O.

Figure 27

The point P is mapped so that meridian position does not alter (i.e. meridians are straight lines) and the distance OP = OP' (i.e. the meridians are the correct length). Although the meridians are straight lines converging to a single point, this point is not the pole, since OA is always greater than ON, the distance to the pole. Hence, the meridians converge to the apex of the cone and the pole projects to an arc of finite length. Since the pole is projected to an line, there is clearly a parallels scale expansion occurring. Also, because the projection plane is entirely outside the sphere and the projection lines radiate outwards from the centre, the parallels must always be expanded. Thus, we can deduce that the projection is neither area nor shape preserving,

s_φ = 1    and   s_λ > 1    ⇒    s_φ ≠ s_λ   and   s_φ s_λ ≠ 1     (5.1)

The parallel scale can be calculated with help from the following diagram. The half-angle at the apex of the cone is φ0 and is equal to the latitude of the contact point with the sphere, since ∠ AOC is right-angled.

Figure 28

The scaling of the parallels depends on how the true parallel length, 2π R cos φ₁ as usual, is represented on the cone. The circumference of each parallel on the cone is defined by

2π AP' sin φ₀       (5.2)

The distance AP is found by adding AO and OP'.

AP = AO - OP'      (5.3)

(Note: OP' as defined in (5.5) will be negative, thus the result does make sense.)

sin φ₀ = DO/AO by Δ ADO    AO = DO / sin φ₀ = R cos φ₀ / sin φ₀ by Δ ODC      (5.4)
AO = R cos φ₀    OP' = R (φ₁ - φ₀)      (5.5)

Hence, substituting (5.4) and (5.5) into (5.3),

AP = R cot φ₀ - R (φ₁ - φ₀) = R (cot φ₀ - φ₁+ φ₀)      (5.6)

Then the expanded parallel circumference is found using (5.2) to be

2π R (cot φ₀ - φ₁ + φ₀) = 2πR (cos φ₀ - (φ₁ + φ₀) sin φ₀)      (5.7)

The scale factor is the ratio of the projected length (5.7) to true length 2π R cos φ₁, as

scale factor: s_λ(φ₀, φ₁) = [cos φ₀ + (φ₀ - φ₁) sin φ₀] / cos φ₁     (5.8)

Minimising this we should find that it never falls below 1. We can adjust φ₁ and φ₀ separately: it is easiest to start with φ₁. S_λ is minimised when cos φ₁ is maximised. Hence, set φ₁ = 0. The other φ₁ term has a smaller effect, so losing it does not affect the result. Now

s_λ(φ₀, φ₁) = cos φ₀ + φ₀ sin φ₀ = s_λ(φ0)      (5.9)

Graphing this we see that, as expected in (4.12), the parallel scale is indeed always more than one: scale is always exaggerated along parallels. Negative values of φ₀ produce the same curve since the function is symmetric, i.e. s_λ(φ₀) = s_λ(-φ₀). (This is not the case when φ₁ ≠ 0, but the general result still stands.)

Figure 29

We can confirm that scale and area are both conserved along the standard parallel, that is, when φ₀ = φ₁. This corrects the inequalities in (5.1).

s_λ(φ₀, φ₁ = φ₀) = [cos φ₀ + (φ₀ - φ₁) sin φ₀] / cos φ₁
s_λ(φ₀) = cos φ₀/cos φ₀ + 0 cos φ₀/cos φ₀ = 1        (5.10)
s_φ =1 = s_λ and s_φ × s_λ =1       (5.11)

Setting φ₁ = 45° (a typical conical projection) the scale becomes

s_λ(φ₀, π/4) = [cos φ₀ + (φ₀- π/4) sin φ₀] / cos π/4 = √2 (cos φ₀ + (φ₀- π/4) sin φ₀)   (5.12)

Conic Projection with Two Standard Parallels

The added parallel decreases the overall distance from the surface of the sphere to the projection cone, which acts to reduce scale deformation. Since part of the cone passes inside the sphere, the result is in some areas a compressions. Areas north of the higher standard and south of the lower will be enlarged (on the northern hemisphere) while areas in between are compressed.

This projection bears many similarities to the single standard version on first glance. The pole is again projected to an arc, and parallels are concentric circles centred on the cone's vertex. The meridians converge towards to cone's vertex, stopping at the polar arc (where φ= 90°). Drawing out the graticule with equal angular spacings in both directions we see that all parallels cross at right angles. Meridian scale is again conserved, but since the pole if projected to an arc, the parallel scale cannot possibly be constant (or everywhere finite). Therefore, by the same steps as in the single standard case, neither area nor shape is conserved on the map as a whole. However, as usual, both are conserved along the lines of contact, each of the standard parallels.

The parallel scale will vary as follows (where φ₁ > φ₂),

s_φ > 1 if φ > φ₁
s_φ < 1 if φ₁> φ > φ₂
s_φ > 1 if φ < φ₂          (5.13)

Bonne's Projection

This projection has only one standard parallel but manages to increase its accuracy considerably by applying convenient stretches to the parallels. This projection is anal sinusoidal projection mentioned briefly before.

A central meridian is chosen and drawn to the right length as a straight line. Since the scale is correct along this, the parallels are spaced equally and are kept at their true length, which means that the other meridians must be curved to fit. Since the parallels are correctly represented, the pole is drawn as a single point, as is correct.

The Sanson-Flamsteed cylindrical project of the cone is take to be at infinity, producing a cylinder rather th not projected to straight lines, this is not strictly speaking a perspective conical projection. The graticule of this projection forms a network as in Figure 30.

Figure 30
http://www.progonos.com/furuti/MapProj/Normal/ProjPCon/projPCon.html

It turns out that this projection is equal-area in some sense: the total area of the projection is preserved, but this is only an average. The scale along the parallels is true (by definition), and along the central meridian. To either side of the central meridian, the meridian scale variation leads to great shape and area distortions at large extents of longitude. Due to the correct parallel scale everywhere, the projection performs reasonably well along large extents of latitude. For this reason, Bonne's projection is commonly used to display long, narrow countries of limited extent, where the central meridian can be aligned through the centre of the country. It is in fact also popularly used for maps of South America and other areas not near to the equator.