ation is reduced to a
five rows according
| * U2U1Z»4 * (1 5)
153 22 ou" su :
22 1 1
11 Z12 Z20 Z21 Z22)-
S
RI
€
oordinate systems
CRITICAL
INS
to projective image
/ vanishing of the
2.1)
Ag (Tab. 1) will be
caused by all point
dition. In order to
, the components of
d originally by plane
formed to the object
4). Using (1.1), the
ical dyadic product
ER (2.2)
nponents of Ag by
| according to (1.4),
Tab.1: Coefficients of the four functional matrices A; (i=5,6, 7,8)
A TOW | Zoo 201 Z02 Z41 212 Z20 221 222 a
Ag 0 1 0 0 0 0 0 0 0 0
1 1 1 0 1 0 0 0 0 -1
2 1 0 1 0 0 1 0 1 0
3
lr ow w uw u$ w uw ww |v
7
A; 1 — 1 0 1 0 0 0 0 -1
2 € 0 1 0 0 1 0 1 0
3
: — wu Uu uj uy uj uj us us uy us us -ui
7
Ag 2 omia cum 1 0 0 1 0 1 0
3
:|— — w (u-Du uwuj uu wu uu [uu
7
As 3
: —MÀu— 50M r1)uf. wu. wu. uut. (Ua 1)U2 u? -u
y
det(Ag)=det(A7)=
1 0 1 0 0 0 0
I cL gd
"|n; na mG MANS Mh Nang nung =O
t & ddr dd qd Xd
Because of the fact that det(Ac) results from det(A7)
by subtracting the first column from the third one, and
det(A;) results from det(Ac) by subtracting its first
column from its fourth and from its sixth one, all de-
terminants are equivalent or in other words, the
condition (2.1) indicates singularity also for A; and Ag
(Tab. 1) and vice versa. Therefore the relation
det(As)=
ard its EATON ne S d pde: Jyies-Quv. vies. ye(Qg-QUY. YeQUy. Ye(Q3- Qv. |=g
| vias YrQB6Yk VEQHYk VOL = Y«QUOYK
ponents equal to 1 “ri 0
2.4)
det(A7)= (2.
Df, nA No NAME NAN is equivalent to the previous condition (2.1). As the
=a qd & qy da | identic denominators of one row may be cancelled
NK Ji : FA ii out, (2.4) converts to
| nik Mick Mange NME MAMA) > 7 À f 7
a did dy dat dj | y.Quyk. Y, Qi2Yk Y. QooYk Yi Q2¥k V4 Q22Yk =0
_| Ne MM DRDA Dx — DAD (2.5)
d; dd dd d d in which the indices of the Q, refer simply to the
né nn no, —nAnk nn) respective unknowns z, of Tab. 1. It is obvious that
"^d dd d dd “da; |” (2.3) each component of this final formulation of the initial
condition det(Ag)=0 represents a quadratic form. This
Nj Ne MN NAME Nan Y i i
"pue TES ax od fact will be of particular interest in the following
K 3 K UC ic C MRUK je discussion.
Do BA DADA Di, DM
di di didi d, dk
The subdeterminants (symbolized by one row k,
k=2....7) are composed exclusively by the spatial
coordinates of the respective object points from which
the geometry of the critical configuration may be
derived.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
3. DISCUSSION OF CRITICAL SITUATIONS
A first inspection of the subdeterminants of (2.3)
shows that in the case of complete coplanarity, that is
by means of (1.4)
yya70. dye 1*(u-1)y *(2-1)yko. ni Hyg
and hence
