given by:
$, -arctan[(1-e?)tan] (10)
2 CENTRAL PROJECTION
2.1 Vertical Projection onto a Tangent Plane T
(Minimal Distance)
From Fig.(4), we have for the point P:
y,=LP tanA =H tan(A-A,)
So we get:
AXsin®, -AZsin®,
AZsinó, * AXcosó,
Yı (11)
Also
NY y, Y»
PK LPsinA,
X;
where LP=AX/sinA. So we get
DH TITI
AZsino, *AXcosó,
(12)
To derive the mapping equation in terms of the
geocentric coordinates (¢,A,h) of the point P, P, and L
we have:
For the point P:
X,=PJ= (N+h)cosdpcosdA
Y "PS -(N * h)cosásindA (13)
Z -JO-[N( -e?) -h]sinó
For the point P, :
X, -MO-(N,*h,) cos,
y 0 (14)
Z, -P,M-IN, -e^) *h]sinó,
X, -RO-RM*MO"[H-*h,*N ]cosó,-Rcoso,
Y,-0 (45)
Z,=LR=[H+h,+N (1-e?]sind,=R cos,
Substituting from the above geocentric coordinates of the
points P, P,, and L into equations (11) and (12), we
obtain after some trignometrical and algebraic reductions
the mapping equations:
334
CsindA
G -Ssino, - Ccos$,cosdA
(16)
F+Scos®, -Csin®,cosdÀ
G -Ssin, -Ccosd,cosdÀ
y,=H
where
c= 0s
a
d
s= N(1-e Los id
a
$
(17)
x ?N sin,cos®,
a
p Ht * N, 1 - e?sin?$)
a
After expanding the forms of sine and cosine of (A-A)
and substitiuting into equations (16), we get the general
mapping formulas of the vertical perspective projection of
the rotational ellipsoid onto the tangent plane of minimal
distance:
CcosA sinA -CsinA cosA
G-Ssind, -Ccos,cosÀ, cosÀ -Ccos®, sin, sind
x, =H
(18)
F+Scos®, -Csin®,cosÀ,cosÀ -Csin®,sinÀ,sinÀ
G-Ssinh, -Ccos, cos, cosÀ -Ccos,sinA sinA
» (19)
2.2 The mapping equations using geocentric latitude
of the perspective center
The mapping equations (18) and (19) can also be
modified if the geocentric latitude ¢, of the perspective
center L is given as in the case of some satellite
problems. So we have:
cos,
R,=(H+h,+N,
cos,
e R cos($, - 6 ie)
a
(20)
FS R,cos(®, -®,)
a
e?N sind cos
0, 7d, arcein | onte
0
The value of ¢, may be found by a rapidly converging
iteration, with an initial value of ¢,=¢, and using
equation (20) to obtain Ro. The new value R. is used to
obtain the next approximation of d, . The values of R
and 6, are iterated until the change in d) is considered
negleigible.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996