©
19992004 Astronomical Institute of Kharkov University. Department
of Solar, Lunar and Planetary Physics
DSLPP
> Planets > Cartography
Formulas
of the Perspecitive Cartographic Projection for Planets and Asteroids
of Arbitrary Shape
E. V. Shalygin, Yu. I. Velikodsky, and V. V. Korokhin
Kharkov Astronomical Observatory
Sumskaya Ul., 35, Kharkov, 61022, Ukraine. Email:
evgen@astron.kharkov.ua
A short version of this paper was printed in [1]. Here
we presents the full text of the paper. More uptodate version of this
work is avaliable in PDF format
Abstract: Formulas of transformation between
coordinates on image plane, planetocentric coordinates and photometric
conditions of observation for arbitrary planet have been obtained.
An example with ellipsoidal planet has been considered.
Introduction: When processing planetary images
it is necessary to transform image coordinates to planetocentric
coordinates or back. Also at photometric studies it is necessary
to calculate photometric conditions (geometry) of observation for
each point on the planet surface.
Often at realization of such transformations it is supposed
that the planet image is in orthographic projection. But it is true
only if ratio of the planet size to distance to it (i.e. the angular
size of the planet) is negligible for accuracy of the task being
solved. Otherwise it is necessary to suppose that the planet image
is in the perspective projection (this is especially appreciable
at the observations from the board of space vehicles approaching
with planets and asteroids).
Formulas of the perspective projection are often obtained
for a spherical planet or for some special cases of nonspherical
planets. We have tried to make a step to deriving formulas of coordinate
transformation in most general form: let us consider images of planets
of arbitrary shape in the perspective azimuthal projection. Let
us suppose that it is possible to set the form of a planet with
equation F(x,y,z)=0 or a set of such equations for different parts
of the surface. Let us suppose also, that the line of sight is not
directed strongly to the center of the planet, i.e. we deal with
more general Tilted Perspective Projection rather than Vertical
Perspective Projection.
Problem definition: Let us choose a system of
rectangular coordinates (XYZ) in which it is more suitable to set
the shape of the planet surface with a such equation:
F(x,y,z)=0.
If the planet is an ellipsoid, the equation (1) can
be written like this:
,
where A, B, C – ellipsoid semiaxes.
Let us name the coordinate origin (point O) of system
(XYZ) "the center of planet" (Fig.1).
Fig.1
Let us choose a plane of the perspective projection such
that the line of sight crosses it at right angles in a point, placed
on the distance D from the observer, where D  distance between
the observer and the center of the planet. In this plane we shall
consider the image (projection) of a planet, that is equivalent
to the image on a photodetector (as a rule, it is a rectangular
CCDimage). Let us set coordinates on the image (on the projection
plane) x_{P} and y_{P} (let axis X_{P} be
directed to the right, and Y_{P} – upward).
Let us introduce an additional rectangular system of
coordinates (X'Y'Z'), with coordinate origin (point O') in the point
of crossing of the line of sight with the projection plane (here
axis Z' is directed to the observer, and axes X' and Y' coincide
with axes X_{P} and Y_{P} correspondingly). In this
system the observer has coordinates x'=y'=0, z'=D. Let us note that
at D®¥ the perspective projection
approaches to orthographic one.
Let coordinates in all systems be measured in the same
units – for example, in image pixels. Converting to other units
(to kilometers, angular seconds) can be performed, knowing scale
of the image and distance to the planet.
Since we use the Tilted Perspective Projection, let
us consider, that the center of the planet is displaced relative
to the direction of the line of sight on an angle r
(on the coelosphere) in a direction with azimuth y
counted in the image plane from the positive direction of axis Y'
anticlockwise. If Vertical Perspective is enough (it is acceptable
if the distance to the planet much more than its size), in all formulas
it is possible to set r=y=0.
Let us consider the system of planetocentric coordinates:
b – a latitude, l – a longitude. Let us define it
as spherical system of coordinates with center in the center of
the planet, the equator of which places in plane XOZ, axis Y is
directed to a north pole, and axis Z is directed to the point with
coordinates b=l=0.
And now let us derive the formulas of transformation
from coordinates on the image plane (x_{P},y_{P})
to planetocentric coordinates (b,l), and also inverse
transformation and formulas of calculation of the observational
photometric conditions.
Transformation (x_{P},y_{P}) ®
(b,l): Coordinates on the CCDimage (x_{P},y_{P})
are given.
Step A: (x_{P},y_{P})®(x,y,z).
Let us write the equation of the right line corresponding to the
line of sight (in system (XYZ)):
,
where (x_{N}, y_{N}, z_{N}) 
coordinates of the observer, (x_{A}, y_{A}, z_{A})
 coordinates of a point on the projection plane corresponding to
(x_{P}, y_{P}). Coordinates of the observer can
be obtained like this:
where b_{0}, l_{0} – planetocentric
coordinates of the point under the observer. Coordinates (x_{A},y_{A},z_{A})
can be obtained from (x_{P},y_{P}) by transforming
from (X'Y'Z') to (XYZ) with several consecutive rotations and translations
of axis. Let (x',y',z')=(x_{P},y_{P},0), and make
this transformation (if r=y=0
then step 1 can be omitted):
1) transformation to system of coordinates (x_{1},y_{1},z_{1})
with coordinate origin in the center of planet:
a) rotation on angle y
around axis Z' to place the center of planet in plane Y_{1a}O'Z_{1}_{a}:
b) translation of coordinate origin to the observer
point N:
c) rotation on angle r
around axis X_{1b} to place the center of planet on axis
Z_{1c}:
d) translation of coordinate origin to the center
of planet:
2) rotation on position angle to place a central
meridian of the planet along axis Y_{2}:
where P_{0} – the position angle of the planet
on the image, counted from position "the north is above"
anticlockwise (let us note that if not r=y=0,
planes X_{1}OY_{1} and X'O'Y'are not parallel, therefore
a such "position angle" should be separated in two parts:
y (in X_{1}OY_{1}) and
P_{0}y (in X'O'Y'), where P_{0}y
is position angle, counted from direction from the center of planet
to the center of Tilted Perspective Projection);
3) rotation on angles b_{0} and
l_{0} to transform to system (XYZ):
This is a general form of transformation (X'Y'Z') to
(XYZ). In our case (x_{A},y_{A},z_{A})=(x,y,z).
Or in the matrix presentation this transformation looks
like this:
To calculate coordinates of a point on the surface of
the planet, we should solve system (3) together with equation (1),
finding unknown x, y, z. We should (if it is necessary) find all
solutions – i.e. all points of crossing of the right line with the
surface and to choose a point with smallest distance to the observer,
i.e. with smallest value of (x_{N}x)^{2}+(y_{N}y)^{2}+(z_{N}z)^{2}.
In the case of ellipsoid (2) problem is reduced to the
solving a quadratic equation. This solution can be written like
this:
where
One of two solutions (6) with smaller (x_{N}x)^{2}+(y_{N}y)^{2}+(z_{N}z)^{2}
is corresponded to nearside of the planet.
Step B: (x,y,z)®(b,l).
Then under the definition (b,l) we have:
Here the term with sign functions removes ambiguity
of arctan function.
Transformation (b,l) ®
(x_{P},y_{P}): Planetocentric coordinates (b,
l) are given.
Step A: (b,l)®(x,y,z).
Coordinates of a point on the planet surface (x,y,z) can be
found with solving system of the equations:
where the first two equations describe a right line corresponding
to direction from the center of planet with spherical coordinates
(b,l); and the last equation describe surface of the
planet; and we find the cross point.
In the case of ellipsoid the solution looks like this:
where .
Step B: (x,y,z)®(x_{P},y_{P}).
Let us convert coordinates (x,y,z) to system (X'Y'Z') with transformation
that is inverse to (5):
This point (x',y',z'), observer point (0,0,D) and point
(x_{p},y_{p},0) (in system (X'Y'Z')) are placed
on one right line. Thus following relation must be valid:
.
So, , .
Transformation (x,y,z) ®
(a,i,e): To
find the phase angle a, the angles of
incidence i and emergence e, let
us use coordinates (x,y,z), obtained in the Step A in any of described
above transformations.
Let us obtain a direction of a normal n to the
surface, differentiating function (1) in point (x,y,z):
,
and then obtain a vector of direction to observer N:
,
where (x_{N}, y_{N}, z_{N}) 
coordinates of the observer (4). The angle of emergence e
we find as an angle between vectors n and N:
,
For obtaining the angle of incidence i and the
phase angle a we should calculate coordinates
of the Sun. We know planetocentric coordinates of the point on the
surface under the Sun (b_{S},l_{S}) and distance
from the Sun to the planet center r_{S} (as a rule, it is
possible to set r_{S} equal to infinity).
Let us obtain Sun coordinates in system (XYZ):
Thus the vector of a direction to the Sun is
.
Then the angle of incidence i and phase angle
a can be calculated like this:
, .
Often instead of values (a,i,e)
it is convenient to use values (a,j,l),
where j  photometric latitude, l
 photometric longitude. By definition:
cosi=cosj×cos(al);
cose=cosj×cosl.
Conclusion: Thus, using above mentioned transformations
of coordinates, it is possible to process images of any objects,
shape of which can be presented like (1). The observer and a light
source can be placed on arbitrary distance from the planet. We should
use the following constants, determining observational conditions:
b_{0}, l_{0}, P_{0}, D, b_{S}, l_{S},
r_{S}, r, y.
If it is impossible to set the shape of a planet analytically
it is necessary to use its discrete representation and then the
solution of systems (3)+(1) and (7) should be found numerically.
References:
1. E.V. Shalygin, Yu.I.Velikodsky, and V.V.Korokhin.
Formulas of the Perspecitive Cartographic Projection for Planets
and Asteroids of Arbitrary Shape. // Lunar and Planet. Sci. 34rd.
Abstract #1946. 2003. LPI. Houston.
