This vanishes when h = 1, that is if the projection be stereographic; or for u = 0, that is at the centre of the map. At a distance of 90° from the centre, the greatest alteration is 90° − 2 cot−1 √h. (See Phil. Mag. 1862.)
Clarke’s Projection.—The constants h and k can be determined, so that the total misrepresentation, viz.:
M = ∫β0 { (σ − 1)2 + (σ′ − 1)2 } sin u du,
shall be a minimum, β being the greatest value of u, or the spherical radius of the map. On substituting the expressions for σ and σ′ the integration is effected without difficulty. Put
| λ = (1 − cos β) / (h + cos β); ν = (h − 1) λ, |
| H = ν − (h + 1) loge (λ + 1), H′ = λ (2 − ν + 1⁄3ν2) / (h + 1). |
Then the value of M is
M = 4 sin2 1⁄2β + 2kH + k2H′.
When this is a minimum,
dM / dh = 0; dM / dk = 0
∴ kH′ + H = 0; 2 dH / dh + k dh H′ / dh = 0.
Therefore M = 4 sin2 1⁄2β − H2/H1, and h must be determined so as to make H2 : H′ a maximum. In any particular case this maximum can only be ascertained by trial, that is to say, log H2 − log H′ must be calculated for certain equidistant values of h, and then the particular value of h which corresponds to the required maximum can be obtained by interpolation. Thus we find that if it be required to make the best possible perspective representation of a hemisphere, the values of h and k are h = 1.47 and k = 2.034; so that in this case