CIPA HERITAGE DOCUMENTATION - CIPA - Icomos

X
Z
Ο T
x*
The Collinearity Equations or the
Corrected Equations of the Horizontal
Direction and of the Vertical Angle
The point coordinates
Y
Let’s considerer three orthogonal reference systems (figure 2):
a) a local terrain system [OT;X,Y,Z] with vertical Z axis (“terrain
system”);
b) an auxiliary system [O;X´,Y´,Z´] parallel to the terrain system
with vertical Z axis, with origin in the centre
O(X0,Y0,Z0) of the sphere, (“sphere system”);
c) a system [O;X*,Y*,Z*] centred in the centre O of the sphere
and oriented parallel to the panorama borders, with Z axis
not vertical, from now on called “panorama system”.
Given an arbitrary object point P(X,Y,Z), it is:
x´
z´
O
X´ = X – X0 Y´ = Y – Y0 [4]
Z´ = Z – Z0 The spherical coordinates of the point P in the panorama
system, are (figure 2):
The angular corrections
X*=d · sin φ · sin θ
Y*= d · sin φ · cos θ [5]
Z*= d · cos φ
The corrected spherical coordinates (Xʹ,Yʹ,Zʹ) can be derived
from Xʹ,Yʹ,Zʹ of the sphere system, with a rotation matrix where
the angular corrections, da x, da y around the two horizontal axes,
are small enough:
R
X*
V R
1 0 -da V R
y X-X V R
V
d sinusin f
S W S
W S 0 W S
W
SY*
W = S 0 1 da .
xW
S Y-Y0W= S d cos usin fW
S
W S
S S
Z* W
W W
W
Sday
-dax
1 W S Z-Z0WS d cos f W
T X T
X T X T
X
Y´
Z´ X´
Figure 2. Relationship between object coordinates, spherical coordinates,
before and after the correction of verticality.
z*
y*
P0
Z*
P
Y´
X*
y*
[6]
By dividing the first by the second one we get:
x X*
r1( X- X0) + r2( Y- Y0) + r3( Z-Z0) u = u0+ = atg = atg =
r Y*
r ( X- X ) + r ( Y- Y ) + r ( Z-Z) 4 0 5 0 6 0
Xl-daZ y l
= atg
Yl+ da .
[7]
Zl
x
Where θ 0 is the zero bearing of the origine, R radius, x and
y image coordinates of the panorama. As it is known, the zero
bearing is the clokwise angle between the northern direction
and the direction of the origine of the angles, the left border of
the panorama in this case. We derive:
. X*
Xl-daZ x atg .
y l
= R b- u0+ l = R f-
u atg
Y*
0+
Yl+ da . p [8]
Zl
From the third:
Zl
r3( X- X0) + r6( Y- Y0) + r9( Z-Z0) f = a cos = a cos
=
d
d
=
O´
z´
a cos
P´
K´
- dayXl+ daY x l+ Zl
[9]
d
2 2 2 2 2 2
where d = Xl + Yl + Zl= X* + Y* + Z*
is the distance of the sphere center O from point P, invariant
in the two reference systems.
The preceding equations are the equations of collinearity
for the spherical panoramas or the correct equations of the
horizontal direction and the vertical angle corrected to take
into account the missed verticality of the axis of the sphere.
They must be linearized near approximate values of the parameters
and coordinates and then adjusted in block, according
to a surveying technique already set-up (2, Fangi,
2004). The restitution takes places by means of the eqns.
[7] and [9]. The approximated values are supplied by a classical
procedure where the initial values of the correction angles
are set to zero, or by means of the relative orientation
described in 4.
tHe multI-Image sPHerIcal Panoramas as a tool For arcHItectural survey
P
y´
x´
b
K´´
P´´
O´´
Figure 3. The coplanarity of two spherical panoramas.
z´´
x
y´´
x´´
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