ANALYTIC DESCRIPTION OF A GEOMETRICAL
MODEL OF GPS - ALTIMETRY
Jaroslav Černý1, Milada Kočandrlová1
The satellite altimetry is one of the most important method for the study of the geoid.
A wide list of its application in different branches of geosciences exists nebo is known. The
procedure of the altimetry method is similar to the GPS method. One satellite altimetr
transmits a short signal to the Earth. It reflects back to the satellite or to the CHAMP satellite
(Challenging Minisatellite Payload for Geophysical Research and Application). The problem
is to locate the point of reflection.
Let us fix a cartesian frame of reference in the space such that the origin is in the
satellite S1 (CHAMP) and the satellite S2 (GPS) lies on the y axis. The coordinate plane yz is
determined by the velocity vector of the satellite S1, and the x axis is perpendicular to this
plane. Three quadrics are defined by the situation
All points P that reflect a signal of the constant length d1  d 2  2a , which is
transmitted from the satellite S1 to the satellite S2 lie on an ellipsoid of revolution
(1)
( y  e) 2 x 2  z 2

 1,
a2
b2
where b 2  a 2  e 2 , 2e  d12 (distance of satellites).
A straight circular cone is formed by all points P in which the signal transmitted from
the satellite S2 is reflected to the satellite S1, with the satellite S2 velocity vector and the
reflected ray vector forming the angle . S1 is the vertex of the corresponding conical surface,
the axis of revolution is determined by the satellite S1 velocity vector. The bias slope of the
elements of the conical surface and the axis of revolution is . The equation is derived from
the conical surface equation in the basic position, i.e. with the vertex in the satellite S1 and
with the axis in the z axis
(2)
z 2  t 2 (x2  y 2 ) .
Using the rotation around the x axis under the angle of the x axis and the velocity vector of
the satellite S1 gives us the desired equation. We use t for the cotangent of the angle , s for
sine, and c for cosine. The reflecting points lie on the right circular cone
(3)
(c y  s z ) 2  t 2 x 2  t 2 (s y  c z ) 2 .
The intersection curve of the ellipsoid of revolution (1) and the right circular cone (2) is a
curve of the forth order, which is projected as a hyperbola into the two rotation axes plane.
The implicit equation of this hyperbola is derived by removing the variable x from the
equations (1) and (2) and proceeding accordingly
(4)

1
y 2 (a 2 c 2 (1  t 2 )  t 2 e 2 )  2a 2 c s(1  t 2 ) y z  2b 2 t 2 e y  a 2 s 2 (1  t 2 ) z 2  t 2 b 4  0.
Keywords: Altimetry, surface, curve, quadrics.
Dept. of Mathematics, Faculty of Civil Engineering CTU, Thákurova 7, 16629 Praha 6, cerny@mat.fsv.cvut.cz
If we denote




A  a 2c 2 1  t 2  t 2e2 , B  a 2c s 1  t 2 ,


C  b 2 t 2 e, D  a 2 b 2 s 2 1  t 2 b 2 , E  t 2 b 4 ,
then the equation (4) can be written in the following form
(5)
Ay 2  2Byz  Dz 2  2Cy  E  0.
The matrix of the quadratic form to the conic section (5) is singular. Therefore this conic
section is a pair of intersecting lines. The coordinates of their common point are given by the
following formula
BC   b 2 b 2 c 
  CD
(6)
,
 AD  B 2 AD  B 2    e , es  .


The following equation holds for the intersection points y, z, 0 (homogeneous coordinates)
of both lines (5) and the line of infinity
(7)
Ay 2  2Byz  Dz 2  0.
The solution of this equation
(8)
y  B  B 2  AD
 as (ac  e cos  )


z
A
(ac  e cos  )( ac  e cos  )
defines the vectors of both lines
(9)
u  (as, ac  e cos  ) ,
v  (as, ac  e cos  ).
The following equations give parametric description of the singular conic section (5)
b2
b2c
 a s , z 
 (a c  e cos  )  ,
e
es
where  is a real parameter. The intersection quartic of the ellipsoid and the cone splits into
two ellipses. If we eliminate the parameter  from the equation (10) we obtain the equation of
the planes of these ellipses
(10)
y
(11)
(a c  e cos  ) y  a s z  b 2 cos  .
We describe one part of the cone (3) by means of the parametric equations
(12)
X (u, v)  v cos u, v (c t  s sin u), v (s t  c sin u), u  0,2 , v  R, v  0
We denote  the plane which intersects this surface. Then the equation of  is
(13)
(a c  e cos  ) y  a s z  b 2 cos  .
If we eliminate the parameter v from the parametric equation of the cone (12) we obtain the
parametric equation of the ellipse E
 b 2 sin  cos u
x
,
a  e(c cos   s sin  sin u )
(14)
y
 b 2 (c cos   s sin  sin u )
,
a  e(c cos   s sin  sin u )
z
 b 2 ( s cos   c sin  sin u )
,
a  e(c cos   s sin  sin u )
u  0, 2 .
The plane  intersects the reference ellipsoid. The intersection is an ellipse Er. We know its
equation. The required reflection points are common points of the ellipse E and the ellipse Er.
Some numerical methods are used to found all common points. The right point is uniquely
defined by means of the reflection theorem. Fig. 1 shows the situation.
Fig. 1
The next procedure shows a geometric approach to finding the right reflection point.
There are five possible situations of mutual position of two ellipses in a plane which have at
most four common real points, see Fig. 2.
Fig. 2
A pair of ellipses defines a bundle of conic sections which all pass through the knots of the
bundle (common points of the basic ellipses). The bundle is described by means of the
ellipses E and Er equations. Consider only such example that the ellipses have four real
common points. If we choose a suitable system of coordinates in the plane  the bundle is
described by the equation
F ( x, y )   G ( x, y )  0,   ,
where F and G are equations of ellipses E and Er. The bundle obtains a family of singular
conic sections, pairs of intersecting or parallel lines. They are described by means of a system
of linear equations. Since we are searching the knots of the bundle we can find them as
common points of the singular conic sections, thus solving a system of linear equations. The
practical solution applies the numerical approach.
The illustrative picture Fig. 3 shows the reference sphere, the ellipsoid of revolution
(1) and the conical surface (2). The Fig. 4 shows both ellipses E and Er in the plane .
The research has been supported by the grant MSM 210000010 „Applied Mathematics in
Technical Sciences“.
References:
[1]
Wagner, C., Klokočník, J., Reflection Altimetry for Oceanography and Geodesy,
IAPSO – IABO Symp., Mar del Plata, 2001
[2]
Kočandrlová, M., Geometrický model úlohy GPS – altimetrie, Sborník 27. konference
VŠTEZ, Hejnice 2002, ISBN 80-7015-864-6, pp. 110-114
Fig. 3
Fig. 4