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Computation Heading Geodesic from Dresden (Saxony) to Dresden (Ontario) Generation of points on the computed geodesic Points on the geodesic from Dresden (Saxony) to Dresden (Ontario) Error propagation Error propagation during the computation of a geodesic In the library

From three input quantities of a geodesic all other quantities are computed, optionally including an . If two solutions exist, both are computed.

≡ START First steps Deutsch

There are six special curves connecting two given points P and Q on an ellipsoid:

• the geodesic
• the loxodrome
• the great elliptic arc
• the normal section curve at P
• the normal section curve at Q
• the alignement curve

A geodesic is the shortest curve inside a surface, which connects two given points. In our case the surface is always an ellipsoid of revolution approximating the figure of the earth.

In essence, there are six quantities describing the course of a geodesic:

• the differenc of ellipsoidal longitudes Δλ=λ_Q-λ_P from P to Q
• the ellipsoidal latitude φ_P of P
• the ellipsoidal latitude φ_Q of Q
• the arc length s
• the ellipsoidal azimuth α_P from P to Q (forward azimuth)
• the ellipsoidal azimuth α_Q from P to Q (reverse azimuth)

Besides this, the following quantities have some importance:

• the ellipsoidal equatorial azimuth α₀, this is the azimuth, by which the (possibly extended) geodesic traverses the equator, when going from the southern to the northern hemisphere.
• Clairaut's constant C, for which C = sin α₀ holds.
• the bounding ellipsoidal latitude φ_m this is the ellipsoidal latitude of the northermost point of the westerly oriented geodesic (C>0) or of the southernmost point of the easterly oriented geodesic (C<0), which is possibly extended.

Instead of the ellipsoidal latitudes φ for P,Q,m we may also use the reduced latitudes β or the geocentric latitudes ψ .

meridian cross section

P	point on ellipsoid
P′	point on circumscribed sphere
O	geocentre
a	semimajor axis
b	semiminor axis
m	length of meridian arc
ρ	radius of parallel
N	radius of curvature in the prime vertical
φ	ellipsoidal latitude
ψ	geocentric latitude
β	reduced latitude
μ	rectified latitude

≡ START First steps Deutsch Computation

Out of the 15 possible quantities, three must be given, to compute the remaining 12 quantities. The two classical primary problems of the geodesic on the ellipsoid of revolution are:

	given quantities
1st primary problem	φP, αP, s
2nd primary problem	φP, φQ, Δλ

But also almost all other combinations of known quantities are computed. In few cases a computation is practically impossible, either because the problem is in principle theoretically unsolvable or because a solution has not yet been implemented.

In some cases the solution is non-unique. If two solutions exist, both are computed.

If one has computed the ellipsoidal difference of longitudes Δλ , one may use the ellipsoidal longitude λ_P to compute λ_Q or vice versa. This can be computed manually by the function Create coordinate list.

Die computation is partly iterative. The extended setting ''Criterion of convergence'' controls, how many iteration steps are executed.

If both points are close to the equator or one point is near a pole, the computation may be inaccurate, while a warning is issued, or even fail. A solution is pending.

≡ START First steps Deutsch Heading

This symbol shows the rough course of the geodesic PQ:

↗	from southwest to northeast	→	on the equator from west to east
↙	from northeast to southwest	←	on the equator from east to west
↘	from northwest to southeast	↑	on the meridian from south to north
↖	from southeast to northwest	↓	on the meridian from north to south
↷	from southwest to southeast by touching the northern bounding parallel
↶	from southeast to southwest by touching the northern bounding parallel
↺	from northwest to northeast by touching the southern bounding parallel
↻	from northeast to northwest by touching the southern bounding parallel

≡ START First steps Deutsch Geodesic from Dresden (Saxony) to Dresden (Ontario)

We consider the following points in ellipsoidal coordinates referring to :

	ellipsoidal
point	latitude	longitude	height
Dresden (Saxony), centre point of the central building of the	51.037512°	13.735186°	120 m
Dresden (Ontario), St. Andrews Presbyterian Church	42.590278°	-82.181667°	183 m

The second primary problem should be solved on the WGS84 ellipsoid. In the following, the heights are set to zero.

and click ''Compute''

Result: The length of the geodesic is s = 6848049 m. The forward azimuth at the start point is α_P = 303.521454° The reverse azimuth at the end point is α_Q = 45.4313890°.

Problem: Use Loxodrome from Dresden (Saxony) to Dresden (Ontario), to determine the arc length between these points on the loxodrome. (Solution: It is s_Lox = 7373778 m and is therefore almost 8% larger than the arc length on the geodesic.

Now the midpoint M on the geodesic is desired. For this purpose wi start at the start point P (Saxony) with the same forward azimuth α_P = 303.521454°, but we go only half the distance s/2 = 6848049 m/2. This is a first primary problem.

and click ''Compute''

Result: The midpoint M in called Q in this computation. The ellipsoidal latitude of M is φ_M = 57.7794406°The ellipsoidal longitude of M may be computed as follows: λ_M = 13.735186° - 52.931461° = -39.196275° = 320.803725°. The azimuth at point M looking into the direction of Ontario amounts to α_M = 259.3707113°. This corresponds to a heading from southeast to northwest touching the northern bounding parallel. Point M is reached after the geodesic PM reaches the bounding latitude φ_m = 58.3981055° .

Zur Kontrolle gehen wir von Q (Ontario) mit dem Rückwärtsazimut To check this, we go from Q (Ontario) with reverse azimuth α_Q again half the distance into the direction of P.

and click ''Compute''

Result: The midpoint M is called P in this computation. The ellipsoid latitude and the azimuth at point M agree with high accuracy. The ellipsoid longitude of M is computed as follows: λ_M = -82.181667° + 42.985389° = -39.196278° = 320.803722°. Also this value agrees very well with the previous computation.

The point M lies southeast of Cape Farewell (southernmost piont of Greenland) near the borderline between the Labrador basin and the Irminger sea.

≡ START First steps Deutsch Generation of points on the computed geodesic

The problem is to generate equidistant points on the computed geodesic and to collect them in a coordinate list . The first point A of this list must not necessarily be the point P and the last point E of this list must not necessarily be the point Q. Instead of this, these points are specified by there arc lengths s_A, s_E along the geodesic referring to s_P = 0 . For example, the point A=P would be addressed by s_A = 0 and the point E=Q would be addressed by s_E = s. These are also the default values, if no arc lengths are specified. For example, the midpoint of the geodesic PQ would by addressed by s/2 . A or E may well be outside PQ, but not extremely far away of P and Q.

Since up to now we only worked with ellipsoidal longitude differences Δλ , we need an absolute ellipsoidal longitude λof one point out of P or Q or A or E.

The equidistancy of the points and the belonging to the desired geodesic can be checked by means of . Side lengths and equatorial azimuths must be identical for all polyline sides. Equatorial azimuths must also be identical with the specification α₀ . The equidistancy has a high, but not the highest accuracy.

Example: If you want only the points P and Q in the coordinate list, chose n = 2 as point number and leave the areas for the arc lengths blank.

The coordinate list consists of ellipsoidal latitudes and longitudes. All output of ellipsoidal heights is zero. These coordinates can be converted to other system types by .

The coordinate list may have one of the following column formats :

Coordinates

Only the coordinates are saved.

Pointname coordinates

The sequential arc length are used as pointnames.

Pointname code coordinates

The sequential arc lengths are used as code.

≡ START First steps Deutsch Points on the geodesic from Dresden (Saxony) to Dresden (Ontario)

On the geodesic from Dresden (Saxony) to Dresden (Ontario) we compute nine equidistant points, the first point is Dresden (Saxony) and the last point is Dresden (Ontario):

and click ''Create''

The fifth point is the midpoint and is obtained by φ = 57.77944707385° λ = -39.1962115331° = 320.8037884669° . The last digits differ slightly from the midpoint previously obtained, because, as has been stated, the equidistancy is not always realised by the highest accuracy. If these points are computed by , a correct and constant equatorial azimuth of α₀ = 328.3121600° is obtained, but the polyline side lengths differ by at most 7 m.

Now, we want to generate additional points between the two northernmost points, and the distance should be 1000 m. For this purpose we chose the arc lengths s_A = 2568018.4 m; s_E = 3424024.5 m and the point spacing Δ = 1000 m.

and click ''Create''

We obtain 856 points with a true distance between Δ_min = 1001.1757 m and Δ_max = 1001.1768 m. The difference to the desired value Δ = 1000 m is due to the fact that the arc length is not an integer multiple of the selected point distance, such that the point distance has been adapted slightly. The equatorial azimuths all amount to α₀ = 328.3121600. The northernmost point has an ellipsoidal latitude of φ_m = 58.39810555 at an arc length of φ_m = 2684154.9. These results agree exactly with the values for the bounding parallel computed above.

≡ START First steps Deutsch Error propagation

If the computational task makes it likely that internal rounding errors falsify the output, a warning is issued.

If given quantities are not known errorfree, you can compute, how much these errors propagate to the computed quantities. See .

≡ START First steps Deutsch Error propagation during the computation of a geodesic

Using the computation tool also meridian arc lengths can be computed by setting Δλ = 0 . In this way for the meridian arc of the Bessel-Ellipsoid between the parallels φ_P = 59° and φ_Q = 60° you obtain the arc length s = 111390.561 m. Using the computation tool you obtain the same value, but with more digits: s = 111390.5612847 m

If reversely you want to compute from φ_P = 59°, φ_Q = 60°, s = 111390.561 m back to Δλ = 0 , you do not obtain a result. The reason is that s has been rounded down. A tiny extension to s = 111390.562 m is working and yields two solutions Δλ = ±0.00025° ≈ ±0.9''. Geometrically we intersect a parallel by an ellipsoidal circular arc, which yields two intersection points. They are about 27 m distant from the desired meridian. The reason for this big true deviation is the unfavourable error propagation of this inappropriate section.

One could have grasped this situation without knowing the true solution by propagation the error in s . In the following example we assume a maximum error of Δs = 0.001 .

and click ''Compute''

This yields a deviation in Δλ of 0.00011° ≈ 0.4''.

Problem: If you also specify error values for φ_P and φ_Q , you do not obtain a result. Please find out, why.

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