(Cylindrical Projections )
(Closing Comments )
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.
Conical Projection with One Standard Parallel
This projection preserves meridian distances. Points are projected the correct distance from the intersection line, O.
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,
sφ = 1 and sλ > 1 ⇒ sφ ≠ sλ and sφ sλ ≠ 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.
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: sλ(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. Sλ 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
sλ(φ0, φ1) = cos φ0 + φ0 sin φ0 = sλ(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. sλ(φ0) = sλ(-φ0). (This is not the case when φ1 ≠ 0, but the general result still stands.)
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).
sλ(φ0, φ1 = φ0) = [cos φ0 + (φ0 - φ1) sin φ0] / cos φ1
sλ(φ0) = cos φ0/cos φ0 + 0 cos φ0/cos φ0 = 1 (5.10)
sφ =1 = sλ and sφ × sλ =1 (5.11)
Setting φ1 = 45° (a typical conical projection) the scale becomes
sλ(φ0, π/4) = [cos φ0 + (φ0- π/4) sin φ0] / cos π/4 = √2 (cos φ0 + (φ0- π/4) sin φ0) (5.12)
|Latitude||Length and Area increase |
|0° 41.42% |
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),
sφ > 1 if φ > φ1
sφ < 1 if φ1> φ > φ2
sφ > 1 if φ < φ2 (5.13)
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.
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.
(Cylindrical Projections )
(Closing Comments )