Earth section paths
Earth section paths are plane curves defined by the intersection of an earth ellipsoid and a plane. Common examples include the great ellipse and normal sections. Earth section paths are useful as approximate solutions for geodetic problems, the direct and inverse calculation of geographic distances. The rigorous solution of geodetic problems involves skew curves known as geodesics.
Inverse problem
The inverse problem for earth sections is: given two points, and on the surface of the reference ellipsoid, find the length,, of the short arc of a spheroid section from to and also find the departure and arrival azimuths of that curve, and. The figure to the right illustrates the notation used here. Let have geodetic latitude and longitude . This problem is best solved using analytic geometry in earth-centered, earth-fixed Cartesian coordinates.Let and be the ECEF coordinates of the two points, computed using the geodetic to ECEF transformation discussed here.
Section plane
To define the section plane select any third point not on the line from to. Choosing to be on the surface normal at will define the normal section at. If is the origin then the earth section is the great ellipse.. Since there are infinitely many choices for, the above problem is really a class of problems, where is the unit vector in the direction of. The orientation convention used here is that points to the left of the path. If this is not the case then redefine. Finally, the parameter d for the plane may be computed using the dot product of with a vector from the origin to any point on the plane, such as, i.e.. The equation of the plane is thus, where is the position vector of.Azimuth
Examination of the ENU to ECEF transformation reveals that the ECEF coordinates of a unit vector pointing east at any point on the ellipsoid is:, a unit vector pointing north is, and a unit vector pointing up is. A vector tangent to the path is:so the east component of is, and the north component is. Therefore, the azimuth may be obtained from a two-argument arctangent function,. Use this method at both and to get and.
Section ellipse
The intersection of a plane and ellipsoid is an ellipse. Therefore, the arc length,, on the section path from to is an elliptic integral that may be computed to any desired accuracy using a truncated series or numerical integration. Before this can be done the ellipse must be defined and the limits of integration computed.Let the ellipsoid given by, and let.
If then the section is a horizontal circle of radius, which has no solution if.
If then Gilbertson showed that the ECEF coordinates of the center of the ellipse is, where,
the semi-major axis is, in the direction, and the semi-minor axis is, in the direction, which has no solution if.
Arc Length
The above referenced paper provides a derivation for an arc length formula involving the central angle and powers of to compute the arc length to millimeter accuracy, where.That arc length formula may be rearranged and put into the form:
, where
and the coefficients are
To compute the central angle, let be any point on the section ellipse and. Then is a vector from the center of the ellipse to the point. The central angle is the angle from the semi-major axis to. Letting, we have.
In this way we obtain and.
On the other hand it's possible to use Meridian arc formulas in the more general case provided that the section ellipse parameters are used rather than the spheroid parameters. One such rapidly convergent series is given in Series in terms of the parametric latitude. If we use to denote the spheroid eccentricity, i.e., then ≤ ≅. Similarly the third flattening of the section ellipse is bounded by the corresponding value for the spheroid, and for the spheroid we have ≅, and ≅. Therefore it may suffice to ignore terms beyond in the parametric latitude series.
To apply in the current context requires converting the central angle to the parametric angle using, and using the section ellipse third flattening. Whichever method is used, care must be taken when using & or & to ensure that the shorter arc connecting the 2 points is used.
Direct problem
The direct problem is given, the distance, and departure azimuth, find and the arrival azimuth.Section plane
The answer to this problem depends the choice of. i.e. on the type of section. Observe that must not be in span. Having made such a choice, and considering orientation proceed as follows.Construct the tangent vector at,, where and are unit vectors pointing north and east, together with defines the plane. In other words, the tangent takes the place of the chord since the destination is unknown.
Locate arrival point
This is a 2-d problem in span, which will be solved with the help of the arc length formula above. If the arc length, is given then the problem is to find the corresponding change in the central angle , so that and the position can be calculated. Assuming that we have a series that gives then what we seek now is. The inverse of the central angle arc length series above may be found on page 8a of Rapp, Vol. 1, who credits Ganshin. An alternative to using the inverse series is using Newton's method of successive approximations to. The inverse meridian problem for the ellipsoid provides the inverse to Bessel's arc length series in terms of the parametric angle. Before the inverse series can be used, the parametric angle series must be used to compute the arc length from the semi-major axis to,. Once is known apply the inverse formula to obtain, where. Rectangular coordinates in the section plane are. So an ECEF vector may be computed using. Finally calculate geographic coordinates via using Bowring's 1985 algorithm, or the algorithm here.Azimuth
Azimuth may be obtained by the same method as the indirect problem: and.Examples
The great ellipse
The great ellipse is the curve formed by intersecting the ellipsoid with a plane through its center. Therefore, to use the method above, just let be the origin, so that . This method avoids the esoteric and sometimes ambiguous formulas of spherical trigonometry, and provides an alternative to the formulas of Bowring. The shortest path between two points on a spheroid is known as a geodesic. Such paths are developed using differential geometry. The equator and meridians are great ellipses that are also geodesics. The maximum difference in length between a great ellipse and the corresponding geodesic of length 5,000 nautical miles is about 10.5 meters. The lateral deviation between them may be as large as 3.7 nautical miles. A normal section connecting the two points will be closer to the geodesic than the great ellipse, unless the path touches the equator.On the WGS84 ellipsoid, the results for the great elliptic arc from New York, = 40.64130°, = -73.77810°
to Paris, = 49.00970°, = 2.54800° are:
= 53.596810°, = 111.537138° and = 5849159.753 = 3158.293603. The corresponding numbers for the geodesic are:
= 53.511007°, = 111.626714° and = 5849157.543 = 3158.292410.
To illustrate the dependence on section type for the direct problem, let the departure azimuth and trip distance be those of the geodesic above, and use the great ellipse to define the direct problem. In this case the arrival point is
= 49.073057°, = 2.586154°, which is about 4.1 nm from the arrival point in Paris defined above. Of course using the departure azimuth and distance from the great ellipse indirect problem will properly locate the destination, = 49.00970°, = 2.54800°, and the arrival azimuth = 111.537138°.
Normal sections
A normal section at is determined by letting . Another normal section, known as the reciprocal normal section, results from using the surface normal at. Unless the two points are both on the same parallel or the same meridian, the reciprocal normal section will be a different path than the normal section. The above approach provides an alternative to that of others, such as Bowring. The importance of normal sections in surveying as well as a discussion of the meaning of the term line in such a context is given in the paper by Deakin, Sheppard and Ross.On the WGS84 ellipsoid, the results for the normal section from New York, = 40.64130°, = -73.77810°
to Paris, = 49.00970°, = 2.54800° are:
= 53.521396°, = 111.612516° and = 5849157.595 = 3158.292438.
The results for the reciprocal normal section from Paris to New York is:
= 53.509422°, = 111.624483° and = 5849157.545 = 3158.292411.
The maximum difference in length between a normal section and the corresponding geodesic of length 5,000 nautical miles is about 6.0 meters. The lateral deviation between them may be as large as 2.8 nautical miles.
To illustrate the dependence on section type for the direct problem, let the departure azimuth and trip distance be those of the geodesic above, and use the surface normal at NY to define the direct problem. In this case the arrival point is
= 49.017378°, = 2.552626°, which is about 1/2 nm from the arrival point defined above. Of course, using the departure azimuth and distance from the normal section indirect problem will properly locate the destination in Paris.
Presumably the direct problem is used when the arrival point is unknown, yet it is possible to use whatever vector one pleases. For example, using the surface normal at Paris,, results in an arrival point of = 49.007778°, = 2.546842°, which is about 1/8 nm from the arrival point defined above. Using the surface normal at Reykjavik will have you arriving about 347 nm from Paris, while the normal at Zürich brings you to within 5.5 nm.
The search for a section that's closer to the geodesic led to the next two examples.
The mean normal section
The mean normal section from to is determined by letting. This is a good approximation to the geodesic from to for aviation or sailing. The maximum difference in length between the mean normal section and the corresponding geodesic of length 5,000 nautical miles is about 0.5 meters. The lateral deviation between them is no more than about 0.8 nautical miles. For paths of length 1000 nautical miles the length error is less than a millimeter, and the worst case lateral deviation is about 4.4 meters.Continuing the example from New York to Paris on WGS84 gives the following results for the mean normal section:
= 53.515409°, = 111.618500° and = 5849157.560 = 3158.292419.