Coverage Area Determination for Conical Fields of View
Considering an Oblate Earth
Marco Nugnes∗and Camilla Colombo†
Politecnico di Milano, 20156 Milan, Italy
University of Sannio, 82100 Benevento, Italy
This paper introduces a new analytical method for the determination of the coverage area
modeling the Earth as an oblate ellipsoid of rotation. Starting from the knowledge of the
satellite’s position vector and the direction of the navigation antenna line of sight, the surface
generated by the intersection of the oblate ellipsoid and the assumed conical ﬁeld of view
is decomposed in many ellipses, obtained by cutting the Earth’s surface with every plane
containing the navigation antenna line of sight. The geometrical parameters of each ellipse
can be derived analytically together with the points of intersection of the conical ﬁeld of view
with the ellipse itself by assuming a proper value of the half-aperture angle or the minimum
elevation angle from which the satellite can be considered visible from the Earth’s surface.
The method can be applied for diﬀerent types of pointing (geocentric, geodetic and generic)
according to the mission requirements. Finally, numerical simulations compare the classical
spherical approach with the new ellipsoidal method in the determination of the coverage area,
and also show the dependence of the coverage errors on some relevant orbital parameters.
Arot = rotation matrix from the geocentric equatorial frame to the local ellipse reference frame
A321 = rotation matrix from the geocentric equatorial frame to the reference frame aligned with the line of sight
a= ellipsoid semi-major axis, km
˜a= semi-major axis of the ellipse originated from the intersection of a generic plane and the oblate ellipsoid, km
b= ellipsoid semi-minor axis, km
b= semi-minor axis of the ellipse originated from the intersection of a generic plane and the oblate ellipsoid, km
d= distance of the intersection plane from the origin, km
E= oblate Earth’s eccentricity
∗Ph.D. Student, Department of Aerospace Science and Technology; email@example.com.
†Associate Professor of Orbital Mechanics, Department of Aerospace Science and Technology; firstname.lastname@example.org.
‡R&D Proposal Manager, Department of Engineering; email@example.com.
arXiv:1906.12318v1 [physics.space-ph] 28 Jun 2019
e= ellipse apse line direction
f= Earth ﬂattening factor
H= satellite’s height, km
i= orbit inclination, rad
m= angular coeﬃcient
n= unit vector normal to a generic plane
o= line-of-sight opposite direction
q= vertical intercept, km
R= rotation matrix
Req = Earth equatorial radius, km
Rpol = Earth polar radius, km
R⊕= mean Earth equatorial radius, km
rP= position vector of a generic point on the local ellipse, km
rs/c= spacecraft position vector, km
rT= position vector of the local ellipse point of tangency with the conical ﬁeld of view, km
s= position vector of the ellipse’s center originated from the intersection of a generic plane and the oblate ellipsoid, km
u= semi-minor axis direction
α= angle between of a generic line and the apse line direction, rad
β= angle between the line of sight and the apse line direction, rad
γ= intermediate angle, rad
∆= discriminant of a second-order equation
ε= elevation angle, rad
η= half-aperture angle, rad
θ= local sideral time, s
Λ= ground range angle, rad
λ= geographic longitude, rad
ν= argument of latitude, rad
ρ= slant range, km
φ= geographic latitude, rad
ψ= roll angle, rad
Ω= right ascension of the ascending node, rad
eq = equatorial
FOV = ﬁeld of view
gc = geocentric
gd = geodetic
hor = horizon
int = intersection
pol = polar
determination of the coverage area associated to a navigation antenna or an optical sensor having a conical ﬁeld
of view is important for the preliminary design of any space mission. This problem has been already analyzed in
the classic literature such as in Vallado [
] and in Wertz and Larson [
]. Both Vallado and Wertz and Larson derive the
formulations to compute the coverage area under the assumption of a perfect spherical Earth. Vallado takes as input the
half-aperture angle of the instrument, assuming a conical ﬁeld of view, for its proof. On the other hand, Wertz and
Larson consider as starting point the elevation angle of the desired location on the Earth. The two approaches achieve
the same ﬁnal result.
The ellipsoidal representation of the Earth has been already analyzed for the missile trajectory computation: knowing
the launch site and target point of a missile the path length, called ground range, is derived above the surface of the Earth.
This problem was explored by Vincenty [
], who oﬀered an iterative algorithm that is nowadays used for the ground
range computation. Also Nguyen and Dixson [
] and Escobal [
] solved this problem in an analytic way deﬁning the
elliptic curve over the surface of an oblate Earth model by the intersection of a plane that passes through the launch site,
the target point, and the center of the oblate spheroid. Clearly in this case, the center of the intersected ellipse is also
the center of the oblate spheroid, leading to a particular case and not a general solution of the problem. A complete
analytical solution has been discussed by Maturi et al. [
], where the center of the ellipse can be diﬀerent from the
center of the Earth. These examples take into account only the ground range computation starting from the launch site
and the target point.
As regards the computation of the coverage area associated to a satellite, Escobal [
] derives in his book a closed-form
solution to the satellite visibility problem. The closed-form solution is a single transcendental equation in the eccentric
anomalies corresponding to the rise and set times for a given orbital pass of a satellite under the assumptions of Keplerian
motion and knowledge of satellite orbital elements, station coordinates and minimum elevation angle. The elevation
angle is the angle between a satellite and the observer’s (ground station’s) horizon plane [
]. Escobal [
] solves also the
transcendental equation by using numerical methods once per satellite period, which is faster than determining the value
of the elevation angle of the satellite with respect to a ground station for each time instant.
] developed a method to solve for satellite-satellite and satellite-ground station visibility periods considering
an oblate Earth deﬁning a new visibility function based on the vertical distance above the plane tangent to a ground
station by using a Fourier series. Exploiting the sinusoidal nature of the visibility curve generated by satellites with
orbital eccentricities less than 0.1, he determines the local periodicity of this curve and then uses a numerical search to
locate rise and set times. This method works well for low eccentricity orbits, but fails for more elliptical orbits because
the visibility waveform becomes aperiodic. Alfano et al. [
] extend the use of the visibility function deﬁned by Lawton
] for all the types of orbit, presenting an algorithm which exploits a parabolic blending, a space curve modeling, to
construct the waveform of the visibility function. The determination of the rise and set times is the starting point for the
coverage analysis since they enclose the region in which the satellite is visible.
In his report, Walker [
] derives circular orbital patterns providing continuous whole Earth coverage modeling
the Earth as a perfect sphere. This assumption can be considered reasonable if the satellites’ altitude is low since the
spherical geometry and the ellipsoidal one are not so diﬀerent. Ma and Hsu [
] proposed a solution for the exact design
of a partial coverage satellite constellation over an oblate Earth, that is, the coverage of certain regions of the Earth with
gap times in coverage no longer than some speciﬁed maximum time, which is based on a visibility function deﬁned in
Chylla and Eagle [
], analogous to the one introduced by Escobal [
]. This method cannot be eﬃcient for a Global
Navigation Satellite System (GNSS) where the coverage is supposed to be global, and there is a signiﬁcant number of
satellites. The problem in using the visibility function is the resolution of the transcendental equation to determine the
region visible from each satellite. Indeed, to the best of our knowledge, no analytical formulations relating the elevation
angle or the half-aperture angle, modeling the navigation signal as a conical ﬁeld of view, exist.
In this paper a new analytical method for the determination of the coverage area, having as input the line of sight of
the satellite, the half-aperture angle of the conical ﬁeld of view, and the satellite position vector considering the Earth as
an oblate ellipsoid of rotation, is presented. Unlike the previous papers where the rise and set times of a satellite are
evaluated from a generic ground station, this approach starts from the position of a generic spacecraft on its orbit and
derives all the locations on the Earth’s surface within its ﬁeld of view. Moreover, it is also numerically shown that,
while for the spherical case the locus of points on the Earth’s surface is a circle, for the oblate ellipsoid case there is no
planar geometric line containing all the points.
This paper is organized as follows: the intersection of a generic plane with an oblate ellipsoid of rotation is
summarized in Sec. II. In Sec. III the derivation of the analytic formulation to compute the coverage area is presented,
and in Sec. IV the method is applied for diﬀerent orbital pointing scenarios. Numerical simulations to validate the
models and the results are carried out in Sec. V together with a direct comparison between the spherical approach and
the new ellipsoidal approach. Finally, Sec. VI concludes the paper.
In this section the state-of-the-art for the determination of the coverage area is described starting from the classical
derivation considering the Earth as a perfect sphere (see Appendix) and moving to the intersection of a generic plane
with the Earth modeled as an oblate ellipsoid of rotation. Apart from the derivation of the intersection of the oblate
ellipsoid of rotation with a generic plane that is taken from [
], the computation of the intersection points of a conical
ﬁeld of view with the Earth’s surface is an original work of this paper (see Sec. III). The main diﬀerence with respect to
the spherical approach is that the Earth is assumed as an oblate ellipsoid of rotation. This can be considered a more
reﬁned model with respect to the previous one since the oblate ellipsoid approximates better the shape of the Earth.
Also in this case, the navigation signal is assumed to be extending as a cone [
] from the center of mass of the
spacecraft with a given half-aperture angle
. Whereas in the spherical approach the intersection of a plane with the
Earth is a circle, this case is diﬀerent as shown in Fig. 1, because the coverage area is a three-dimensional surface and is
not contained in a plane normal to the conical ﬁeld of view.
Fig. 1 Projection of a conical ﬁeld of view on an oblate ellipsoid.
The ﬁrst step is to prove that the trace of the oblate ellipsoid on a generic plane is an ellipse [
], because in this way
it is possible to move from a 3D problem to a planar one as in the spherical Earth. The general equation of a plane is
is a unit vector normal to the plane and
is the distance of the plane from the origin of the
coordinate frame. In this case the vector normal to the plane is coincident with the line of sight of the navigation signal,
while the origin of the coordinate frame is the Earth’s center. The equation of the oblate spheroid can be written as:
is the semi-major axis and
is the semi-minor axis of the oblate ellipsoid. In the speciﬁc case the semi-major
axis is the Earth equatorial radius and the semi-minor axis is the Earth polar radius. The trace of the oblate ellipsoid in
the generic plane is computed deriving the expression of
in Eq. (1) and then substituting in Eq. (2). For
0, it is
possible to write:
is parallel to the equatorial plane and perpendicular to the polar axis. Substituting Eq. (3) in Eq.
Developing the products and rearranging the equation, the result is the following:
which is the equation of a conic. The general equation of a conic is:
Ax2+By2+2Gx +2Fy+2H x y+C=0(6)
This equation represents an ellipse or a circle if its discriminant, ∆, is less than 0. The discriminant of Eq. (5) is:
After some manipulations, the result is obtained as:
which is always less than zero. This means that the intersection of the oblate ellipsoid with a generic plane is an ellipse
or a circle.
After showing that the intersection of an oblate ellipsoid on a generic plane is represented by an ellipse, it is
necessary to compute the value of the geometric quantities of this ellipse following [
]: the center of the ellipse, its
semi-major axis and its semi-minor axis. Without going into details, the previous quantities are analytically computed
with the following formulas taken from [
]. The center of the local ellipse with respect to the origin of the reference
frame is computed with Eq. (9).
where Eis the oblate Earth’s eccentricity. Naturally, if the generic plane passes through the center of the Earth, then:
; that is, the center of the oblate spheroid will also be the center of the ellipse. If the
conical ﬁeld of view of the navigation signal is divided into diﬀerent planes having in common the satellite’s line of
sight, each plane will generate an ellipse hosting two limiting points for the coverage area, which is the output of the
analysis. The semi-major axis of the local ellipse is deﬁned by:
the magnitude of the semi-major axis of the ellipse and
the semi-major axis of the oblate ellipsoid of rotation
(i.e., the Earth equatorial radius). The direction of the generic unit vector
to be aligned with the apse line of the
intersection ellipse is given by:
The last quantity to be deﬁned is the semi-minor axis of the intersection ellipse. The magnitude of the semi-minor
axis is given by:
representing the semi-minor axis of the ellipse and
the semi-minor axis of the oblate ellipsoid of rotation (i.e.,
Earth polar radius). The direction of the generic unit vector ˆ
uto be aligned with the semi-minor axis is given by:
Figure 2 summarizes the geometric parameters deﬁned in this section putting in evidence the intersection ellipse
with the oblate ellipsoid of rotation.
Fig. 2 Representation of the intersection ellipse with the oblate ellipsoid of rotation.
III. Computation of the Coverage Parameters for an Oblate Earth
In the previous section all the geometric quantities of the intersection ellipse have been determined. This gives the
possibility to move from a 3D view to a planar view and determine the same quantities computed using the spherical
approach. The new problem representation is identiﬁed in Fig. 3.
Fig. 3 Representation of the elliptical ﬁeld-of-view problem.
It is possible now to deﬁne a local reference frame in the plane of the ellipse obtained by the intersection of the
oblate ellipsoid of rotation and a generic plane, where
is the unit direction vector along the semi-major axis and
the unit direction vector along the semi-minor axis. The equation of the ellipse in this planar reference frame is:
is just the ﬁrst coordinate of a generic point in the local reference frame and it has not to be confused with the
magnitude of the unit vector
. The ﬁrst parameter to be determined is the horizon delimited by the points
given by the tangents to the ellipse drawn from the center of mass of the spacecraft. Let
be, respectively, the
generic angular coeﬃcient and the vertical intercept of a line in the local reference frame. The equation of a line in the
local reference frame in the explicit form is:
The two tangents in
, as shown in Fig. 3, have in common the satellite’s center of mass whose coordinates
are known if the satellite’s position vector in the inertial frame is known.
The two points of tangency are obtained as intersection of the equation of the tangent and the equation of the local
Developing the equations, it is possible to get to the following system:
The discriminant, ∆, associated to the second of Eq. (18) is the following:
From the value of the discriminant it is possible to deduce the type of solution. To have the tangency, the value of
needs to be equal to zero. This leads to a second-order equation in the unknown m.
The roots of Eq. (20) are given by these two expressions:
Once the value of the angular coeﬃcient of the two tangents is known, it is possible to compute the points of
intersection with the ellipse, whose coordinates are:
From the equation of the tangents it is possible to derive the other coordinates:
In this way the vectors associated to the points of intersection of the tangents with the local ellipse are deﬁned:
The equivalent of the horizon-ground range angle (Λhor) computed for the spherical case can be identiﬁed also for
the ellipsoidal case considering the angle between these two vectors:
The horizon-boresight angle,
, can be determined in a similar manner knowing the position vector of the two
points of tangency and the satellite’s center of mass, because it is possible to compute the relative position of the
spacecraft with respect to the two points of tangency. The angle between the two relative positions will represent the
cos(ηhor)=(rs/c−rT1) · rs/c
The next step is to derive the ground range angle that is associated to the aperture angle of the instrument on-board
of the spacecraft. Using always the assumption of the conical ﬁeld of view, three diﬀerent situations may be identiﬁed:
geocentric (line of sight coincident with the conjunction with the center of the Earth), geodetic (line of sight aligned
with the local vertical to the Earth’s surface), and generic (moving line of sight) pointing. The diﬀerence between the
three methods from the geometrical point of view is the bisector line of the aperture angle. This subject will be better
explored in the next section. For now, it is suﬃcient to assume that ηis the half-aperture angle with respect to a given
Fig. 4 Highlight of the elliptical geometry.
line that forms an angle
axis as visible from Fig. 4. To determine the angular coeﬃcient needed to compute
the equations of the two secants in
intersecting the local ellipse, it is suﬃcient to draw the perpendicular to
axis from the satellite’s center of mass. In this way, it is possible to make some geometrical considerations using
the right triangles having as vertices
and the satellite’s center of mass and
and satellite’s center of mass. Let
α1and α2be the angles formed by the two secants and the ˆ
eaxis, respectively. It is possible to prove that:
The angular coeﬃcients of the two secants are simply the tangents of the two angles, visible in Fig. 4, deﬁned before.
The next step is to write the generic equation for a line having as angular coeﬃcient
and to solve a
system with the equation of the local ellipse. The result is the same second-order equation (18) with the diﬀerence that
this time the discriminant will be or higher or lower than zero. In the latter case, no intersection with the ellipse are
present. Let the discriminant
be higher than 0, therefore there are two distinct solutions for each angular coeﬃcient
and so for each secant.
˜a2mP1(mP1es/c−us/c) ± q(˜a4˜
˜a2mP2(mP2es/c−us/c) ± q(˜a4˜
Of course, just two of these solutions are correct and these are the ones on the same side and closer to the center
of mass of the spacecraft. To choose the right solution a distinction according to the quadrant has to be done and the
are involved. For example, in the ﬁrst quadrant the value of
is always less than
2. This means
that once the right solution is identiﬁed, in the speciﬁc case the positive solution, it keeps the same sign. A diﬀerent
reasoning should be done for the other solution because the value of
starts lower than
2and then it becomes higher
than π/2. The complete list of the cases together with the right solution to be picked is reported in Table 1.
Table 1 Decision table to select the right solutions.
First Quadrant Second Quadrant
if α2<= π/2if α2>π/2if α1<= π/2if α1>π/2
eP1= positive solution eP1= positive solution eP1= positive solution eP1= negative solution
eP2= positive solution eP2= negative solution eP2= negative solution eP2= negative solution
Third Quadrant Fourth Quadrant
if α2<= −π/2if α2>−π/2if α1<= −π/2if α1>−π/2
eP1= negative solution eP1= negative solution eP1= negative solution eP1= positive solution
eP2= negative solution eP2= positive solution eP2= positive solution eP2= positive solution
Once the values of nP1and nP2are obtained it is possible to compute the remaining coordinates.
At this stage the position vectors of the two points P1and P2are identiﬁed.
The ground range angle is simply the angle between these two vectors and can be computed as:
The last step is to link the aperture angle,
, with the elevation angle,
. It should be emphasized that, in this
case, there is no close relationship between the aperture angle and the elevation angle, and so the two quantities are
independent. First of all, it is possible to compute the slopes of the tangents to the ellipse in the two points just
determined deriving the equations of the two halves of the ellipse.
It needs to be stressed that the ellipse is not a bijective function, and therefore the derivative cannot be considered.
The ellipse has to be split in two halves and this is the reason why there are two possible solutions for the slope of the
tangents. The equations of the tangents at the points are known because there are one point and the slope available for
each line. Remembering that the elevation angle is the angle between the satellite’s center of mass and the observer’s
(ground station’s) horizon plane [
], it is possible to apply the formulation to compute the angle between two lines
without ambiguity because the elevation angle is always lower than π:
Of course, the two values of the elevation angles, as for the two values of ground range angles, are diﬀerent in the
ellipsoidal approach because the curvature of the Earth is not constant but changes. To be consistent with the previous
methods that introduce a visibility function depending on the minimum elevation angle for a satellite to be visible, it is
useful to express the new approach also in terms of the minimum elevation angle. Basically, there are two solutions that
can be adopted for the resolution of this problem: iterative and analytical.
The iterative procedure starts from the value of the half-aperture angle related to the theoretical horizon, for which
the elevation angle is zero, and computes the elevation angles for both the two sides giving small decrement to the
half-aperture angle. The iteration ends when the diﬀerence between the minimum elevation angle and the actual one is
below a prescribed threshold. The number of iterations is very small because the values of the minimum elevation
angles are in around 5
and the elevation angle is very sensitive to a small variation of the half-aperture angle due to
the curvature of the oblate ellipsoid.
The analytical approach gets the elevation angle as a function of the half-aperture angle in such a way to use directly
the right value to determine the coverage area. Starting from Eq. (34), the expressions
is replaced by Eq. (33)
obtaining the following intermediate result:
where the identity
has been used. The "+" sign is related to the negative half part of the ellipse,
whereas the "-" is associated to the positive one. Finally, by replacing
with Eq. (29) the only unknown in the
equations is the angular coeﬃcient
, which is directly associated to the half-aperture angle
. The advantage of
the last method is to have an analytical relation between the elevation angle and the half-aperture angle. However, the
equation is highly non-linear, is transcendental, and needs to be solved numerically paying attention to the correct initial
guess or interval used.
IV. Determination of the Coverage Region in Relevant Satellite Orbital Scenarios
In this section the ﬁeld of view of a generic satellite is computed in diﬀerent orbital scenarios. Indeed, during the
preliminary mission analysis according to the mission requirements, the attitude of the satellite can be chosen among
three diﬀerent pointings: a geocentric pointing, where the line of sight of the navigation antenna is directed toward
the Earth center; a geodetic pointing allowing the direction of the line of sight to be normal to the local horizon on
the surface of the Earth; and a generic pointing associated to a casual direction to analyze the eﬀects of the attitude
A. Generic Pointing Case
The ﬁrst scenario is most general one involving a moving line of sight. From this analysis, all the simpliﬁed cases
may be retrieved by imposing the right value of the geometric parameters.
To apply this procedure, it is assumed that the position of the satellite’s center of mass is known together with the
direction cosines of the navigation antenna line of sight that can be the output of an attitude and orbit propagator. It is
possible to work both with the spacecraft position vector and direction cosines of the navigation antenna expressed in
the inertial reference frame and relative to a ﬁxed Earth. The output of the procedure will result in position vectors
consistent with the type of representation used. The ﬁrst thing to be done is the determination of the projection of the
line of sight onto the Earth’s surface corresponding to the nadir. Indeed, if the line of sight is not aligned with the
geocentric direction also the footprint changes. It is useful to express the equation of the line of sight in its parametric
form considering one point (the spacecraft center of mass) and the direction cosines:
Because the footprint belong to the Earth’s surface, its coordinates have to satisfy the equation of the oblate ellipsoid:
are the Earth equatorial and polar radii, respectively. After performing some operations, the equation
may be rearranged in the following form:
Two solutions are obtained for the parameter t:
eq ) ± √∆int
where ∆int is the discriminant of the equation and it is equal to:
There are two solutions because this method considers both the entry point and the exit point of the line of sight from
the ellipsoid as shown in Fig. 5.
Because in the parametric equation of the line of sight the coordinates of the spacecraft are ﬁxed at a given time
instant and the direction cosines are related to the pointing toward the Earth’s center, it happens that the greater the
magnitude of the parameter
, the greater the distance of the computed point from the satellite’s center of mass is. So,
the exact solution is the one associated to the minimum value of the parameter t.
The next step is to transform the 3D representation of the problem into a planar one in order to use the techniques
developed in the previous section. For this reason, more than one rotation matrix should be introduced in order to
express the coordinates of the spacecraft into the local reference frame deﬁned before. The ﬁrst series of rotations allows
to align the ˆ
iaxis of the inertial frame with the direction of the line of sight as shown in Fig. 6:
Fig. 5 Intersection points of a generic line of sight with the oblate ellipsoid of rotation.
1) A ﬁrst positive rotation around the ˆ
k-axis of an angle equal to the longitude associated to the line of sight.
2) A second negative rotation around the ˆ
j0-axis of an angle equal to the latitude associated to line of sight.
It is easy to compute the longitude and the latitude of the line of sight because the direction cosines are known. This
operation has to be done considering the same direction but the opposite pointing. Indeed, the line of sight is directed
from the spacecraft toward the Earth and it is necessary to introduce the opposite direction ˆ
By applying the conversion from Cartesian coordinates to spherical coordinates, the result is the following:
the longitude (in-plane) and latitude (elevation) angles associated to the line of sight. Once the two
angles have been derived, the two rotation matrices can be written:
cos(λint )sin(λi nt )0
−sin(λint )cos(λi nt )0
0 0 1
cos(φint )0 sin(φi nt )
0 1 0
−sin(φint )0 cos(φi nt )
that aligns the ˆ
iaxis of the geocentric inertial frame with the line of sight of the navigation antenna.
(a) Longitude rotation (b) Latitude rotation
Fig. 6 First series of rotations.
The projection of the conical ﬁeld of view onto the Earth’s surface cannot be expressed using a planar function if the
ellipsoidal model is adopted. Because it is diﬃcult to derive an analytical expression for such three-dimensional surface,
it is more convenient to deﬁne it in terms of couples of points lying on this surface computed with Eqs. (29) and (30).
For this reason, all the planes generated by the rotation of the
plane around the line of sight, corresponding to the
axis of the new reference frame, have to be considered and for each of them a couple of points are derived. This means
that a new rotation matrix of an angle ψ, which is variable in the range [0, π], needs to be introduced:
1 0 0
The ﬁnal rotation matrix will be a combination "321" with the following form:
cos(φint )cos(λi nt )cos(φint )sin(λi nt )sin(φint )
−cos(ψ)sin(λint ) − sin(ψ)sin(φi nt )cos(λint )cos(ψ)cos(λi nt ) − sin(ψ)sin(φint )sin(λi nt )sin(ψ)cos(φint )
sin(ψ)sin(λint ) − sin(φi nt )cos(λint )cos(ψ) − sin(ψ)cos(λint ) − sin(φi nt )sin(λin t )cos(ψ)cos(ψ)cos(φint )
A plane in the space is unequivocally identiﬁed if the direction cosines of a line normal to the plane itself and a
point lying on the plane are known because it is necessary to compute the distance
of the plane from the origin of the
reference system. After the rotation of the reference system, the normal to each plane is simply identiﬁed by the unit
, expressed in the rotated reference frame, as shown in Fig. 7. This normal must be projected in the
inertial reference frame and for this reason the inverse rotation matrix needs to be deﬁned. One of the properties of the
rotation matrices is that the inverse operation is equivalent to the transposition of the matrix, and so it is possible to
The other information needed to write the analytical expression of the plane is also known since the center of mass
of the spacecraft lies on all the planes being part of the axis of rotation around which each plane is obtained. For this
reason the distance dof each plane from the origin of the inertial reference system is:
represent the coordinates of the spacecraft position vector in the inertial frame. If a Keplerian
motion is assumed for the spacecraft, the spacecraft position vector in the inertial frame can be derived from the
knowledge of its orbital parameters. Following Bate et al. :
cos(Ω)cos(ν) − sin(Ω)cos(i)sin(ν)
Fig. 7 Transformation from the geocentric inertial frame to the generic frame aligned with the line of sight.
The last step is to align the
reference system with the local reference system of the ellipse
in the general case, the line of sight do not pass through the center of the ellipse but it is coplanar with the apse line
direction. Therefore, once the apse line direction in the inertial frame is obtained with Eq. (11), the angle between the
line of sight and the apse line axis is computed:
is the vector connecting the nadir and the satellite’s center of mass as depicted in Fig. 8. This vector can be
expressed in the inertial frame thanks to the relative law:
rline =rs/c−rNadir (48)
Fig. 8 Representation of the local reference frame and the local ellipse.
It is convenient to express the previous quantities in the
reference frame aligned with the line of sight.
Indeed, in this case it is possible to identify if the angle between the two directions is associated to a clockwise or
counterclockwise rotation. Another rotation matrix around the
-axis which is coincident with the normal to the plane
of the intersected ellipse is computed:
0 0 1
matrix previously computed in Eq. (43) is left multiplied for the
matrix, the ﬁnal rotation matrix from
the inertial reference frame to the local reference frame of the ellipse is obtained as shown in Fig. 9.
Another way to get the ﬁnal rotation matrix from the geocentric reference system to the local reference system
is to remember the deﬁnition of rotation matrix as the projections of the unit vectors of the rotated reference
frame onto the initial reference frame. This is easily done once the position vectors of the rotated reference frame are
expressed in the inertial reference frame. By using Eqs. (11), (13), and (44) to derive, respectively, the unit vectors
n0in the inertial reference frame, the matrix Arot is:
Fig. 9 Transformation from the geocentric inertial frame to the local ellipse reference frame.
The last passage is to convert the coordinates of the satellite’s center of mass from the inertial to the local frame:
with Arot =RβA321 (51)
Everything is ready to apply the technique analyzed in Sec. III and to get the intersection points
intersection points are deﬁned in the local reference system. Therefore, with the inverse rotation matrix the points are
expressed in the inertial reference frame.
The procedure is not completed because the center of the local reference frame is not coincident with the center of
the inertial reference frame. This means that an additional operation must be performed to obtain the right points. By
applying the laws of the relative reference systems, we obtain:
B. Geodetic Pointing
While for the spherical approach there is no diﬀerence between geocentric and geodetic pointing, for the ellipsoidal
approach this diﬀerence is present due to the geometry considered. Indeed, in an oblate ellipsoid of rotation the
conjunction of a point with the center is diﬀerent from the local vertical of that point to the surface as shown in Fig. 10.
However, some some simpliﬁcations can be performed to the generic pointing to derive the same results.
Fig. 10 Geodetic latitude representation.
The ﬁrst thing to be emphasized is that, also in this case, the local reference frame is not centered in the origin of the
inertial reference system. The nadir formulation can be reformulated in a lighter form considering the properties of the
ellipsoid. Following Curtis :
q1− (2f−f2)sin2(φgd )
is the local sideral time,
is the height of the satellite and
represents the ﬂattening of the Earth equal to
0.00335 according to the WGS84 model representation of the Earth [
]. The relative position vector of the spacecraft
with respect to the nadir may be evaluated in the inertial reference frame. The method proceeds in a similar way like the
generic pointing with the only diﬀerence that in the determination of the A321 matrix the geodetic latitude is used.
C. Geocentric Pointing Case
The geocentric pointing case can be retrieved from the generic pointing approach imposing the value of the direction
cosines equal to the direction cosines of the satellite’s position vector.
V. Numerical Simulations
In Sec. III the formulas to compute the intersection of a conical ﬁeld of view with the Earth modeled as an oblate
ellipsoid of rotation have been derived. The same formulas have been applied to compute the coverage area for diﬀerent
types of pointing. The ﬁrst step is to verify that the points computed with the new approach are actually on the Earth’s
surface modeled as oblate ellipsoid of rotation. Each point that lies on the surface of the ellipsoid shall fulﬁll the
where Req and Rpol are the Earth equatorial and polar radii and their numerical values are 6378.137 km and 6356.752
km, respectively, according to the World Geodetic System (WGS) 1984 [
]. It is possible to compute the value obtained
substituting the coordinates of each point obtained from the formulas and compare with respect to the right-hand
side equal to 1. This operation can be performed considering the three types of pointing. After several simulations
considering also diﬀerent values of the half-aperture angle,
, for a geocentric pointing the maximum error experienced
is equal to 9
km, whereas for a geodetic pointing the error decreases to 9
km. The maximum error is still
the same for a generic pointing where the line of sight, and so the conical ﬁeld of view, is not ﬁxed but it is moving.
After verifying the formulations derived, another step is to show that the intersection of the conical ﬁeld of view
with the oblate ellipsoid is not a planar line, but a three-dimensional surface. This operation can be done considering a
simple linear algebra operation. Indeed, considering four generic points belonging to the coverage area, it is possible to
prove if they are coplanar computing the following determinant:
If the value of the determinant is equal to zero, it means that the four points are coplanar. Also in this case, after
several simulations the value of the determinant is always ﬁnite and diﬀerent from zero. Therefore, it is not possible to
express the intersection of a conical ﬁeld of view with an oblate ellipsoid as a function of a single variable.
Table 2 Procedure for the computation of the coverage area
Input Satellite Position rs/c, line of sight direction ˆ
Step 1 Line-of-sight intersection with the equatorial plane Pint [Eqs. (36) and (38)]
Step 2 Determination of the geographic longitude λint and latitude φint [Eq. (40)]
Step 3 Alignment of the ˆ
Iaxis of the geocentric frame with the line-of-sight direction ˆ
Step 4 Intersection of the plane normal to ˆ
n0with the oblate ellipsoid of rotation
Step 5 Computation of the geometric parameters of the resulting ellipse [Eqs. (9-13)]
Step 6 Alignment of the ˆ
Iaxis of the geocentric frame with apse line direction ˆ
Step 7 Computation of the two ellipse points belonging to the coverage area [Eqs. (29), (30), (50), and (51)]
Step 8 Rotation of the plane normal to ˆ
n0around the line-of-sight direction ˆ
Step 9 New intersection with the oblate ellipsoid of rotation
Step 10 Derivation of group of points belonging to the coverage area
Step 11 Interpolation of the points to get the analytical formulation of the surface
Step 12 Identiﬁcation of the points inside and outside the coverage area
This is the ﬁrst main diﬀerence with respect to modeling the Earth shape as a sphere. The only solution to get
an analytical formulation of the previous three-dimensional surface is to rely on an interpolation that introduces an
approximation. The interpolation procedure is more reﬁned as the number of points used to perform the interpolation
increases. However, it is possible to compute as many points as needed using the procedure described in Sec. III.
The problem is solved using a function which takes in input a series of points of the same generic dimension, and the
coeﬃcients of the independent variables desired as output of the function to express the interpolation. This function is
named Polyﬁtn [
] and is available on the MathWorks website. For the determination of the analytical expression of
the three-dimensional surface only the odd coeﬃcients of the independent variables (i.e., ”
”) are considered
in such a way to distinguish in a unique way the right octant. Table 2 summarizes the complete procedure for the
computation of the coverage area starting from the position of the satellite and the half-aperture angle
of the navigation
A. Sphere Versus Oblate Ellipsoid
It is important to compare the results obtained considering the Earth as a sphere with the ones obtained considering
the oblate Earth. The aim of the analysis is to prove that the spherical model is a ﬁrst-order model that can be used
whenever the accuracy required is not very tight. The error between two identical quantities evaluated with the two
diﬀerent models is computed and the variation of this error versus relevant parameters such as the eccentricity, the
inclination of the orbit, and the half-aperture angle is analyzed starting from a Galileo-like satellite orbit as reference
whose parameters are given in Table 3.
Table 3 Nominal parameters used for the numerical simulations
Height, km Eccentricity Inclination, deg RAAN, deg Pericenter anomaly, deg η, deg
23,229.32 0 56 0 0 10
0 2 4 6 8 10 12 14
Fig. 11 Ground range diﬀerence evolution in one orbit for geocentric pointing.
In Fig. 11 the ground range error computed as the diﬀerence between the spherical and the ellipsoidal ground range
is plotted for a single revolution of the satellite whose orbital parameters are reported in Table 3. The ground range is the
distance measured between two points belonging to the coverage area that are symmetrical with respect to the projection
of the satellite on the Earth’s surface (nadir). Repeating the simulation by changing the orbit shows that the behavior of
this variable is the same. This plot is somehow expected and it is an additional proof for the validity of the ellipsoidal
model. Indeed, the minimum diﬀerence is experienced whenever the spacecraft is nearby the equator, whereas the
maximum diﬀerence is obtained as the spacecraft is moving toward the poles. This happens since the spherical and the
ellipsoidal geometry are more similar at the equator and reach the maximum deviation at the poles.
It is important to emphasize the magnitude of the ground range diﬀerence. The maximum deviation is 20 km and
this number can be considered signiﬁcantly high for many applications, such as navigation mission and GNSS services,
where the precision required is on the order of meters. Therefore, the spherical model is not completely wrong but it
cannot be used in such accurate applications.
Next to the ground range diﬀerence, another important variable is the elevation angle because it is used inside the
requirements for a navigation satellite to determine the limit of visibility. For a spherical model the elevation angle
is strictly related to the half-aperture angle and it is a constant due to the spherical symmetry. This is not valid for
the ellipsoidal approach because the geometry is diﬀerent and the curvature changes according to the position of the
spacecraft. In Fig. 12 the behavior of the elevation angle error is presented. The error is computed as the diﬀerence
0 2 4 6 8 10 12 14
Fig. 12 Elevation angle diﬀerence evolution in one orbit for geocentric pointing.
between the minimum elevation angles associated to a particular conical ﬁeld of view modeling the Earth’s surface as a
sphere and as an oblate ellipsoid. In a sphere there is no diﬀerence between the two sides of the conical ﬁeld of view
with respect to the line of sight. For an ellipsoid the curvature from one side is diﬀerent from the other one. The points
of intersection of the two lines occurs when the spacecraft is at the equator because it is the only case in which the
spacecraft sees a symmetrical geometry with respect to the line of sight. In all the other cases the curvature of the
ellipsoid changes continuously. It is important to stress that an error of 0
on the elevation angle is relatively high and
is associated to a wide area on the Earth’s surface. For this reason there is the possibility that some regions on the
Earth’s surface are inside the ﬁeld of view of the satellite or the other way around.
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Fig. 13 Elevation angle diﬀerence vs orbit eccentricity for geocentric pointing.
It is also interesting to understand how the elevation angle changes according to the type of orbit. In Fig. 13 the
two lines are overlapped because the maximum value of the elevation angle error is computed for each orbit and these
maximum values are the same for the two elevation angles. The elevation angle error does not increase linearly with
the eccentricity of the orbit. This is right since the higher the eccentricity, the higher the distance that the spacecraft
reaches. If the distance from the Earth increases, the spacecraft can look at a larger region characterized by a greater
curvature with respect to the spherical case where there is a change in the elevation angle only linked to the distance of
the spacecraft. In Fig. 14 the elevation angle error is showed as a function of the inclination. This behavior is related to
0 20 40 60 80 100 120 140 160 180
Fig. 14 Elevation angle diﬀerence vs orbit inclination for geocentric pointing.
the geometry of the ellipsoid. Indeed, as the inclination of the orbit increases it is easier to look at points belonging to
diﬀerent curvatures of the ellipsoid and to localize the maximum deviations. Note that this is not the elevation angle
diﬀerence during the orbit but its maximum value for diﬀerent inclinations. This is the reason why the elevation angle
error keeps constant in a particular range of inclination angles.
B. Visibility Period
The last application involving the proposed approach is the determination of the intervals of viewing for a generic
satellite considering prescribed ground stations. Sentinel-2A satellite has been chosen together with three ground
stations to simulate a real scenario:
1) ASI Matera Laser Ranging Observatory (MLRO) 
2) ESTRACK Maspalomas radio antenna ground station 
3) KSAT Svalbard satellite station 
Table 4 shows Sentinel-2A osculating orbital elements obtained from the two-line Elements (TLE) set at the epoch
25/02/2019 8:40:17 .
Table 4 Sentinel-2A osculating orbital elements at epoch 25/02/2019 08:40:17
axis, km Eccentricity Inclination, deg RAAN, deg Pericenter
7167.129 0.000132 98.5657 132.4338 76.3371 238.7960
Sentinel-2A position has been propagated for one solar day under the assumption of Keplerian motion, and the
rise/set times with respect to the ground stations have been computed considering a minimum elevation angle of 5
This means that the interval of viewing is deﬁned as the time as long as the elevation angle of the ground station with
respect to the satellite is higher than 5
. Table 5 summarizes the results of the numerical simulations presenting the
geographic coordinates of the three ground stations according to the WGS84 [
] ellipsoidal model of Earth and the
rise/set times associated to each passage of the satellite for each ground station.
Table 5 Rise/set times for Sentinel-2A
Ground station Longitude, deg Latitude, deg Height, m Rise times, s Set times, s
Matera 16.7046 40.6486 536.9 18798;24489;30561;
Maspalomas -15.6338 27.7629 205.1 30426;36321;74811;
Svalbard 11.8883 78.9067 474.0
This paper has shown a new analytical method for the determination of the coverage area associated to a spacecraft
navigation antenna or an optical sensor having a conical ﬁeld of view when the Earth is modeled as an oblate ellipsoid
of rotation. In particular, the satellite’s position vector, the direction of the navigation line of sight, and the half-aperture
angle of the conical ﬁeld of view assumed for the propagation of the signal are required as input data.
The major achievement has been the conversion of the set of equations used to derive the most relevant parameters
of the coverage problem (i.e., ground-range angle, elevation angle, horizon-boresight angle) when modeling the Earth’s
surface as a sphere into a new set where the shape of the Earth’s surface is an oblate ellipsoid of rotation. Another
important aspect has been the deﬁnition of a proper analytical formulation to derive an algorithm with low computational
time. The results of the new analytical method have been compared to the ones obtained considering the Earth’s surface
modeled as a sphere. Such results show a signiﬁcant reﬁnement in the determination of the coverage parameters. Finally,
the new geometry of the problem allows dealing with diﬀerent types of pointing scenarios such as the geodetic/nadir
Appendix: Spherical Model
The spherical approach is the simplest way to solve the problem of computing the Earth coverage [
]. Indeed, this
model assumes the Earth as a perfect sphere, which is a ﬁrst order approximation. This approximation can be considered
good if the level of accuracy and precision required from the problem is in the order of kilometers. Figure A1 represents
the geometry used for the ﬁeld-of-view (FOV) from a 3D view.
Fig. A1 Geometry for the ﬁeld of view in the Earth-centered Earth-ﬁxed frame.
The general problem associated with the calculation of the ground range angle (
), which is the convex angle
measured between the two lines connecting the center of the Earth with the extreme intersections of the signal conical
ﬁeld of view with the Earth’s surface (Fig. A2), is to determine the amount of surface that can be seen given the
satellite’s position vector and the half-aperture angle at the satellite. If the navigation signal is modeled as a cone
with a given half-aperture angle extending from the center of mass of the satellite, the spherical symmetry gives the
possibility to study the problem in a planar way as shown in Fig. A2. The point directly below the satellite is called
nadir point. The half-aperture angle,
, deﬁnes the angular displacement from the nadir direction. From simple
geometrical considerations, the maximum coverage area can be determined tracing the two tangents to the sphere from
the spacecraft’s center of mass. The horizon-boresight angle,
, can be computed considering the right triangle OHS
with the following formula:
is the Earth mean equatorial radius and
is the distance of the satellite from the Earth center. The sine
expression is suﬃcient because this angle will never exceed
. Using the same right triangle, also the horizon
ground range angle, Λhor, can be determined using planar trigonometry:
For the slant range to the horizon, ρhor, planar trigonometry is also used:
It is possible to get a general expression for the slant range by examining the oblique triangle SPO in Fig. A2. First,
the intermediate angle, γ, is calculated using the sine law for oblique triangles:
Fig. A2 Planar representation of the ﬁeld-of-view problem.
In particular, this intermediate angle should be always be higher than 90
. The generic slant range to any point on
the Earth is computed by using the following equation:
As a veriﬁcation, the slant range value is always between the height of the satellite and the slant range to the horizon.
Finally, it is possible to determine the ground range angle using the sine law:
Also in this case, the ground range angle needs to be less than
and less than the horizon ground range angle. It
is also possible to convert the ground range from an angle to a length by simply multiplying by the Earth radius:
Λkm =R⊕ΛFOV (A7)
where the ground range angle needs to be expressed in radians. The last quantity of interest is the elevation angle,
which is the angle between the tangent to the sphere in a point and the conjunction of the point of tangency with the
satellite’s center of mass. Looking at Fig. A2, the elevation angle, the ground range angle, and the half-aperture angle
are related to each other by the following equation:
The geographic coordinates of the two points associated to the aperture angle can be obtained and in this way all the
important quantities for the coverage analysis are determined.
The research leading to these results has received funding from the European Research Council (ERC) under the
European Union Horizon 2020 research and innnovation program as part of the project COMPASS (Grant agreement
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