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Great Circle Navigation with Vectorial Methods

Published online by Cambridge University Press:  28 May 2010

Vincenzo Nastro  and
Urbano Tancredi
Vincenzo Nastro
Affiliation:
(University of Naples “Parthenope”, Italy)
Urbano Tancredi*
Affiliation:
(University of Naples “Parthenope”, Italy)
*
(Email: urbano.tancredi@uniparthenope.it)
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Abstract

The present paper is concerned with the solution of a series of practical problems relevant to great circle navigation, including the determination of the true course at any point on the great circle route and the determination of the lateral deviation from a desired great circle route. Intersection between two great circles or between a great circle and a parallel is also analyzed. These problems are approached by means of vector analysis, which yields solutions in a very compact form that can be computed numerically in a very straightforward manner. This approach is thus particularly appealing for performing computer-aided great circle navigation.

Keywords

Great circleVectorsNavigation

Information

Type
Research Article
Information
The Journal of Navigation , Volume 63 , Issue 3 , July 2010 , pp. 557 - 563
Copyright
Copyright © The Royal Institute of Navigation 2010

1. INTRODUCTION

In air navigation it is well known that the Earth can be regarded as a sphere and, as a consequence, the shortest distance between any two points on its surface is an arc of a great circle. Great circles are obtained by the intersection with the surface of the Earth of any plane passing through the Earth's centre. For short distances, the difference between the great circle and the rhumb line is negligible. However, flying on a great circle allows saving considerable distance particularly on a long-range flight in high latitudes. For instance, the distance between London and Tokyo is about 6100 n.m. by rhumb line and 5170 n.m. by great circle, which allows a saving of about 930 n.m.

Nevertheless, unlike the rhumb line that crosses all meridians at the same angle, the angle between a great circle route and the meridians constantly changes as progress is made along the route and is different at every point along the great circle. This implies that a vehicle shall be continuously steered to follow a great circle route. This necessity gives rise to a series of problems to be solved, such as: the determination of the True Course (TC), that is, the angle between the great circle route and the meridians, at any point on the great circle route; the determination of the lateral deviation, or Cross Track Distance (XTK), from a desired great circle route; and the intersection between two great circles or between a great circle and a parallel.

The position of a point P on the Earth's surface of latitude ϕ and longitude λ can be represented on a unit sphere by the unit vector P joining the Earth's centre to the point itself. The P vector has components given by:

(1)
{\bf P} \equiv \lpar \matrix{\! {\cos \phiv \cos \lambda \comma } \tab {\cos \phiv \sin \lambda \comma } \tab {\sin}\,\phiv } \cr}\! \rpar

with reference to an ECEF (Earth Centred Earth Fixed) frame: a right-handed, orthonormal coordinate system whose origin is located at the Earth's centre and has axes fixed to the Earth. Its z-axis points towards the North Pole along the spin axis of the Earth and its x-axis is the intersection of the reference meridian with the equator.

Therefore, great circle navigation can be developed taking advantage of vector analysis, allowing a continuous control of the trajectory and the solution of more complex problems such as the previously mentioned ones. Solutions obtained applying vector analysis to great circle navigation problems have a very compact form, and can be computed numerically in a very straightforward manner. This approach is thus particularly appealing for performing computer-aided great circle navigation.

This problem has been already examined in a previous paper of the first author (Nastro, Reference Nastro2000), and recently discussed in this Journal (Earle, Reference Earle2005; Tseng and Lee, Reference Tseng and Lee2007); the present manuscript reports some of this previous paper's results in a more compact form.

2. EQUATION OF THE GREAT CIRCLE

Figure 1 shows the great circle between the departure point P 1(ϕ1, λ1) and the arrival point P 2(ϕ2, λ2), where θ stands for the shortest distance between these two points. For the sake of simplicity, the meridian passing through P 1 is taken as the reference meridian, implying that the components of the vectors P1 and P2 are:

(2)
\eqalign{\tab {\bf P}_{\bf \setnum{1}} \equiv \left( {\matrix{\! {\cos \phiv _{\setnum{1}} } \tab 0 \tab {\sin \phiv _{\setnum{1}} } \cr}\! } \right) \cr \tab {\bf P}_{\bf \setnum{2}} \equiv \left( {\matrix{\! {\cos \phiv _{\setnum{2}} \cos \rmDelta \lambda } \tab {\cos \phiv _{\setnum{2}} \sin \rmDelta \lambda } \tab {\sin \phiv _{\setnum{2}} } \cr}\! } \right) \cr}

where Δλ=λ2−λ1 is the difference of longitude between such points.

Figure 1.

Representation of the great circle.

The distance θ between the two given points P 1 and P 2 is given from the dot product between the two corresponding vectors:

(3)
\theta \equals \cos ^{ \minus \setnum{1}} \lpar {\bf P}_{\bf \setnum{1}} \cdot {\bf P}_{\bf \setnum{2}} \rpar

The vector K, representing the great circle pole, can be obtained normalizing to one the vector cross product between P1 and P2, as:

(4)
{\bf K} \equals {{{\bf P}_{\bf \setnum{1}} \times {\bf P}_{\bf \setnum{2}} } \over {\sin \theta }}

The coordinates of the vertex V of the hemisphere of interest (e.g. the northern one in Figure 1), that is, the point on the great circle path that is nearest to the geographic pole, can be obtained using the latitude and the longitude of the great circle pole (λK, ϕK), as follows:

(5)
\matrix{ {\phiv _{V} \equals 90^\circ \minus \phiv_{K} }\semi \tab {\lambda _{V} \equals \lambda _{K} \pm 180^\circ } \cr}

From Figure 2, P2 can be seen as the result of a rotation of the vector P1 around the direction K of an angle θ:

(6)
{\bf P}_{\bf \setnum{2}} \equals \cos \theta \, {\bf P}_{\bf \setnum{1}} \plus \sin \theta \, \left( {{\bf K} \times {\bf P}_{\bf \setnum{1}} } \right)

Analogously, for a generic point P on the great circle at a distance θ1=kθ from P 1 (where k∊[0,1]) the following holds:

(7)
{\bf P} \equals \cos \theta_{\setnum{1}} \, {\bf P}_{\bf \setnum{1}} \plus \sin \theta _{\setnum{1}} \, \lpar {\bf K} \times {\bf P}_{\bf \setnum{1}} \rpar

The vector K×P1 can be expressed in terms of the two vectors P1 and P2, making use of equation (6), yielding:

(8)
{\bf P} \equals \cos \theta_{\setnum{1}} \, {\bf P}_{\bf \setnum{1}} \plus \sin \theta _{\setnum{1}} \, {\bf T}_{\setnum{1}}

where the vector T1≡(T 1x, T 1y, T 1z)=(P2−cos θ P1)/sin θ is orthogonal to P1 and tangent to the great circle at the departure point P1.

Figure 2.

Rotation of the vector P1 of an angle θ.

The relation (8) can be regarded as the equation of the great circle; for instance, if θ1=kθ=0·5θ the vector P is relative to the mid-point of the great circle.

The components of the vector T1 can be expressed in terms of latitude and True Course at the departure point P1, TC1, by applying the sine and the “four part” formulas to spherical triangle P nP 1P 2 of Figure 1, yielding:

(9)
{\bf T}_{\setnum{1}} \equiv \left(\!\! {\matrix{ { \minus \!\sin \phiv _{\setnum{1}} \cos TC_{\setnum{1}} } \tab {\sin TC_{\setnum{1}} } \tab {\cos \phiv _{\setnum{1}} \cos TC_{\setnum{1}} } \cr}\! } \right)

In case the determination of the TC at the departure point P1 is of interest, equation (9) can be exploited to obtain:

(10)
TC \equals \tan ^{ \minus \setnum{1}} \lpar T_{\setnum{1}y} \cos \phiv _{\setnum{1}} \sol T_{\setnum{1}z} \rpar

The above results are easily extended for determining the TC at any point P of the great circle route, by considering the corresponding vector P in place of P1 in obtaining equation (10).

3. DETERMINATION OF THE CROSS TRACK DISTANCE

In air navigation it is necessary to perform a continuous comparison between the present position derived from the airborne navigation system and the desired position on the great circle. The deviation of the current navigation fix P from the great circle is represented in Figure 3 by the length of the arc PP0, where P0 is the closest point of the great circle to P: the distance PP0 is known as XTK (Cross Track Distance). The coordinates of the point P0 are derived from the vector P0 that coincides with the vector T relative to the great circle between K and P:

(11)
{\bf P}_{\bf \setnum{0}} \equals {{{\bf P} \minus \cos \theta _{\setnum{2}} {\bf K}} \over {\sin \theta _{\setnum{2}} }}

where:

(12)
\theta _{\setnum{2}} \equals \cos ^{ \minus \setnum{1}} \lpar {\bf K} \cdot {\bf P}\rpar

At last, noting that the Cross Track Distance is complementary to θ2, it can be computed as XTK=90°−θ2.

Figure 3.

Determination of the Cross Track Distance (XTK).

The availability of the above explicit expression for computing the cross track distance is beneficial for tracking the desired great circle route. In air navigation, for instance, the cross track distance can be coupled to a flight guidance computer that keeps the aircraft on the great circle course by issuing commands based on the current XTK value, computed as previously shown.

4. INTERSECTION OF TWO GREAT CIRCLES

Figure 4 represents two great circles: the first one connecting the points P1 and P2, whose pole is K1, and the second one between P3 and P4 with pole K2. The intersection point I can be derived from the vector I that is orthogonal to the vectors K1 and K2; consequently:

(13)
{\bf I} \cdot \lpar {\bf K}_{\bf \setnum{1}} \minus {\bf K}_{\bf \setnum{2}} \rpar \equals 0

The vector I can be related by equation (8) to the known vectors P1, T1, and to the unknown distance θ1 between P1 and I. Substituting in the above equation, remembering that the dot product of orthogonal vectors is zero, and rearranging, yields the following expression for determining θ1 and, consequently, the intersection point I.

(14)
\theta _{\setnum{1}} \equals \tan ^{ \minus \setnum{1}} \left( { \minus {{{\bf P}_{\bf \setnum{1}} \cdot {\bf K}_{\bf \setnum{2}} } \over {{\bf T}_{\bf \setnum{1}} \cdot {\bf K}_{\bf \setnum{2}} }}} \right)

The above equation can be made specific in particular cases of interest, such as the intersection between a great circle and a meridian of longitude λm, or between a great circle and the equator, by setting:

(15)
\eqalign{\tab \matrix{ {K_{\setnum{2}} \equiv \lpar\! \sin \rmDelta \lambda _{m} \comma \quad \minus\! \cos \rmDelta \lambda _{m} \comma \quad 0\rpar } \tab {{\rm for\ the\ meridian}} \cr} \cr \tab \matrix{ {K_{\setnum{2}} \equiv \lpar 0\comma \quad 0 \comma \quad 1 \rpar } \tab \quad\quad\quad\quad\quad\quad\ \, {{\rm for\ the\ equator}} \cr} \cr}
Figure 4.

Intersection of two great circles.

5. INTERSECTION OF THE GREAT CIRCLE WITH A PARALLEL

The coordinates of the intersection point I1 between a great circle and a parallel can be determined by the knowledge of the distance θ1 between the departure point P1 and I1 (Figure 5).

Figure 5.

Intersection of the great circle with a parallel.

The vector I1 can be obtained by the rotation of the vector P1 around K1 until:

(16)
{\bf I}_{\bf \setnum{1}} \cdot {\bf K}_{\bf \setnum{2}} \equals \sin \phiv _{p}

where K2 is coincident with the unit vector k≡(0, 0, 1) and ϕp stands for the latitude of the parallel. From relation (8) we have:

(17)
\lpar\! \cos \theta _{\setnum{1}} {\bf I}_{\bf \setnum{1}} \plus \sin \theta _{\setnum{1}} {\bf T}_{\setnum{1}} \rpar \cdot {\bf K}_{\bf \setnum{2}} \equals \sin \phiv _{p}

or, equivalently:

(18)
\cos \theta _{\setnum{1}} \sin \phiv _{p} \plus T_{\setnum{1}z} \sin \theta _{\setnum{1}} \equals \sin \phiv _{p}

This relation can be written as:

(19)
a\cos \theta _{\setnum{1}} \plus b\sin \theta _{\setnum{1}} \equals c

Expressing sin θ1 and cos θ1 in terms of the tangent of θ1, the above equation becomes quadratic in tan θ1, whose solution is:

(20)
\theta _{\setnum{1}} \equals \tan ^{ \minus \setnum{1}} \left[ {{{ \minus ab \mp c\sqrt {a^{\setnum{2}} \plus b^{\setnum{2}} \minus c^{\setnum{2}} } } \over {b^{\setnum{2}} \minus c^{\setnum{2}} }}} \right]

In case the discriminant is positive, we have two distinct and real roots (ϕp<ϕv), if it is zero we have a double real root (ϕp=ϕv), whereas when the discriminant is negative there are no real roots (ϕp>ϕv).

CONCLUSION

This paper has presented results for several practical problems relevant to navigation along a great circle route making use of vector analysis. These results are given in a compact form that is suitable for numerical implementation, thus being particularly appealing for computer-aided great circle navigation.

References

REFERENCES

Nastro, V. (2000), Problemi di navigazione ortodromica risolti con notazioni vettoriali, Studi in memoria di Antonino Sposito, 37–47.Google Scholar
Earle, M.A. (2005). Vector solutions for great circle navigation. The Journal of Navigation, 58, 451–457.CrossRefGoogle Scholar
Tseng, W.K. and Lee, H.S. (2007). The vector function for distance travelled in great circle navigation. The Journal of Navigation, 60, 150–164.CrossRefGoogle Scholar
Figure 0

Figure 1. Representation of the great circle.

Figure 1

Figure 2. Rotation of the vector P1 of an angle θ.

Figure 2

Figure 3. Determination of the Cross Track Distance (XTK).

Figure 3

Figure 4. Intersection of two great circles.

Figure 4

Figure 5. Intersection of the great circle with a parallel.