Artículos de la Conferencia
VI. R EFERÊNCIAS
At the end of the day, after much hard graft, we have thrown away almost all of the higher order terms except for the seventh order term in x/a in the inverse series for λ . Clearly terms of order O e2(x/a)7 would also be negligible so at last we have justified the use of the spherical approximation in calculating the higher order terms. Note that we could not have assessed the size of the higher order terms without working them out!
One can use the Redfearn series as they stand for they are simple to encode on any computer and they take very little extra computation time. Remember, however, that when these series were first developed it was imperative to simplify the working as much as possible for hand(-machine) calculations. Lee would no doubt have had this in mind when he dropped the sixth order terms in his calculations. To be exact, of the required terms he dropped the term in λ6in the direct series for y, the term in(x/a)6in the inverse series for φ and the term in(x/a)7in the inverse series for λ ; at the same time he included the term in(x/a)5e2in the inverse series for λ although we now see that it is negligible.
Nowadays no ’hard graft’ is required for the series can be readily implemented on a computer algebra program such as Maxima (AppendixH. Such programs are exhibited in SectionH: they can easily be extended to higher orders.
The series developed here are simply extensions of the work of Kr¨uger (1912) and Gauss. However, Gauss developed another form of the TM projection which was clarified and actually used by Schreiber in the survey of Prussia. This Gauss-Screiber projection is derived by the transformation of the ellipsoid to the conformal sphere (Sections 5.9,5.10 followed by the TMS from conformal sphere to the plane. This double projection does not have a uniform scale on the central meridian but Kru¨uger showed that this could be achieved by a further conformal transformation which is based on series linking the conformal and rectifying latitudes. This form of the TME is much more accurate than the Readfearn series.
An implementation is described inKarney(2011) and the computer code code is available atKarney(2010). Such implementations can be used for accurate projections to much wider domains.
Both the Redfearn method and this second method based on Gauss-Screiber involve truncated series. They can be extended to any finite order but there are no general series which permit the attainment of arbitrary precision. This is available by using an exact version of the TME developed by E. H. Thompson and communicated toLee(1976). This solution is exact in the sense that it can be computed to arbitrary precision and it provides a yard stick by which other methods can be assessed. Such comparisons permit one to say that the Gauss-Schreiber method is more accurate than the Redfearn series. One remarkable property of the solution is that it is a finite projection which does not tend to infinity as the longitude separation from the central meridian increases. This method is implemented by Dozier(1980),Stuifbergen(2009) andKarney(2011).
Blank page. A contradiction.
Appendix A
Curvature in 2 and 3 dimensions
A.1 Planar curves
A straight line has zero curvature. The curvature, κ, of a general curve in the plane is defined as the rate of change of the direction of its tangent with respect to the distance travelled along the line:
κ= dθ
ds. (A.1)
If we are given the equation of the curve as y= f (x) with respect to Cartesian axes then it is natural to choose the x-axis as the reference for the direction of the tangent.
The geometry of the small inset in the figure shows that
tan θ = dy
dx = y0(x), cos θ = dx
ds (A.2) Differentiating the first of these statements by s and using the second gives
sec2θdθ
ds = d[y0(x)]
ds = y00(x)dx ds
= y00(x) cos θ . (A.3)
θ
θ
Figure A.1 Now sec θ=±√
1+ tan2θ=±p
1+ y02so we obtain dθ/ds and
κ = ± y00(x)
[1 + y02]3/2. DASH≡ d
dx (A.4)
The choice of sign is a matter for convention in every case. We shall illustrate this point immediately. The unsigned curvature is given by taking the modulus.
The curvature of a circle
For a general circle of radius a at the origin we have x2+ y2= a2so that on the two semi-circles y> 0 and y < 0,
y(x) =±p
a2− x2, y0(x) = ∓x
√a2− x2, y00(x) = ∓a2
(a2− x2)3/2. (A.5) Substituting these values in equation (A.4) we see that the curvature of the upper semicircle is κ=±(−1/a) whilst for the lower semicircle κ = ±(1/a). Now it is conventional to define the curvature of a circle to be positive so we must choose the negative sign in the definition for the case of the upper semicircle and the positive sign for the lower; with these choices of sign we have a constant curvature κ= 1/a. Therefore the curvature of a circle is the inverse of the radius and vice-versa.
The osculating circle and the radius of curvature
The particular circle which touches the curve at P (FigureA.1) and also shares the same curvature at that point is called the ‘osculating circle’ (osculating=kissing) or the ‘circle of curvature’. The radius of this circle defines R, the radius of curvature of the curve at that point. Clearly
R= 1
κ. (A.6)
Curves in parametric form
The previous results related to a curve in two dimensions described by a single function y(x) in Cartesian coordinates. We now consider the situation where these Cartesian coordinates are parameterised by two functions of u; that is the position of a point on the curve is written as r(u) = x(u), y(u). We shall investigate three types of parameterisation: (1) the parameter is assumed to be perfectly general, not necessarily the distance along the path;
(2) the parameter is taken as the arc length s; (3) the parameter is taken as the angle between the tangent and the x-axis.
Case 1: Arbitrary parameterisation: x(u), y(u). SetDOT≡ d du y0(x) = dy
dx =y˙
˙ x, y00(x) = d
du
y˙
˙ x
du
dx =x˙y¨− ˙y¨x
˙ x3 1
R = κ = x˙y¨− ˙y¨x
[ ˙x2+ ˙y2]3/2 DOT≡ d
du . (A.7)
Case 2: Special parameterisation: u→ s. Given x(s), y(s). SetDASH≡ d
Case 3: Special parameterisation: u→ θ. Given x(θ), y(θ). SetDOT≡ d dθ Since θ is the angle between the tangent and the x-axis we have tan θ =dy
dx = ˙y/ ˙x
The Cartesian equation of an ellipse is x2
a2+y2
b2 = 1, (A.10)
where the semi-axes are related to the eccentricity by b= a√
1− e2. Now the ellipse may be obtained by scaling the auxiliary circle by a factor of b/a in the y direction. Since an arbitrary point P0 on the circle is (a cosU, a sinU) the corresponding point on the ellipse is P(a cosU, b sinU).
We call U the ‘parametric’ or ‘reduced’ latitude in cartog-raphy and the ‘eccentric anomaly’ in astronomy.)
We calculate the curvature from equation (A.7) setting:
x= a cosU, y= b sinU,