IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
A Note on the Number of Solutions of the
Noncoplanar P4P Problem
Z.Y. Hu and F.C. Wu
AbstractÐIn the literature, the PnP problem is indistinguishably defined as either
to determine the distances of the control points from the camera's optical center or
to determine the transformation matrices from the object-centered frame to the
camera-centered frame. In this paper, we show that these two definitions are
generally not equivalent. In particular, we prove that, if the four control points are
not coplanar, the upper bound of the P4P problem under the distance-based
definition is 5 and also attainable, whereas the upper bound of the P4P problem
under the transformation-based definition is only 4. Finally, we study the conditions
under which at least two, three, four, and five different positive solutions exist in
the distance based noncoplanar P4P problem.
Index TermsÐThe Noncoplanar P4P Problem, rigid transformation, upper bound.
æ
1
INTRODUCTION
THE PnP problem is a classical problem in computer vision,
photogrammetry, and even in mathematics. It was first formally
introduced by Fishler and Bolles in 1981 [1] and later extensively
studied by others, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]
to cite a few. The PnP problem in [1] was defined as:
Given the relative spatial locations of n control points and given
the angle to every pair of control points from the perspective center
(the camera's optical center), find the lengths of the line segments
joining the perspective center to each of the control points.
Since the distances from the perspective center to the control
points should be determined in this definition, we shall in sequel
call this definition a ªDistance-Based Definition.º The distancebased definition was exemplified in Haralick's work about the
P3P problem in [2], which can be summarized as follows:
Let p1 ; p2 ; p 3 be three 3D control points. Assume that the three
distances a ˆ jp2 p 3 j, b ˆ jp1 p3 j, and c ˆ jp1 p2 j, the corresponding three 2D image points q 1 ; q 2 ; q 3 of points p1 ; p 2 ; p3 , and
the camera's intrinsic parameters are all known, determine the
unknown distances s1 ; s2 ; s3 of the points p1 ; p2 ; p 3 from the
camera's optical center.
In summary, the PnP problem under the distance-based
definition is (or equivalently) to determine the distances of the
control points from the camera's optical center.
In the literature, the PnP problem is also defined as determining
the transformation matrix from the object-centered frame to the
camera-centered frame. In [3], the PnP problem is defined as:
Given a set of points with known coordinates in an objectcentered frame and their corresponding projections onto an image
plane and given the intrinsic camera parameters, find the
transformation matrix (three rotation parameters and three
translation parameters) between the object frame and the camera
frame.
. Z.Y. Hu is with the National Laboratory of Pattern Recognition, Institute
of Automation, Chinese Academy of Sciences, P.O. Box, 2728, Beijing
100080, People's Republic of China. E-mail: [email protected].
. F.C. Wu is with the National Laboratory of Pattern Recognition, Institute
of Automation, Chinese Academy of Sciences, P.O. Box 2728, Beijing
100080, People's Republic of China and the Laboratory of Artificial
Intelligence, Anhui University, Hefei, 230039, People's Republic of China.
E-mail: [email protected].
Manuscript received 3 Oct. 2000; revised 9 May 2001; accepted 20 Aug. 2001.
Recommended for acceptance by Z. Zhang.
For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference IEEECS Log Number 112936.
0162-8828/02/$17.00 ß 2002 IEEE
VOL. 24, NO. 3,
MARCH 2002
1
Since the transformation matrices should be determined in this
definition, we shall in sequel call this definition ªTransformation
Based Definition.º In other words, the PnP problem under the
transformation-based definition is (or equivalently) to determine
the transformation matrices from the object-centered frame to the
camera-centered frame.
In the literature, the above two definitions have been commonly
believed to be equivalent and used indistinguishably in practice.
Here, some questions come about: Are the two definitions indeed
equivalent? If not, what are the main differences between them? In
the following sections, these questions will be clarified. Here is a
short summary of our main results:
1.
2.
The two definitions are generally not equivalent.
For the P4P problem, if the four control points are not
coplanar, the maximum number of positive solutions
under the distance-based definition is 5 and this upper
bound is also attainable.
3. For the P4P problem, if the four control points are not
coplanar, the maximum number of admissible solutions
under the transformation based definition is 4, which was
shown in [3], and this upper bound is also attainable,
which will be shown in Section 4 in this work.
4. Some sufficient conditions are obtained under which at
least two, three, four, and five positive solutions appear for
the noncoplanar P4P problem under the distance-based
definition.
Prior to any further discussions on the above issues, it is worth
noting at first that the PnP problem is generally different from that
of camera resectioning in order to avoid potential confusion
between the two problems due to their close connection. Camera
resectioning is defined as determining all consistent camera
projection matrices, each with a different perspective center, for a
given set of correspondences between space control points and
image points [12, p. 516]. The main difference is that in the
PnP problem, the camera is the SAME calibrated one, but in
camera resectioning, the camera is unspecified and allowed to
change. Such a difference implies that, in the PnP problem, any
pairs of control points must subtend a fixed angle with different
perspective centers, but in camera resectioning, it only requires
that the pencils of projection rays with different perspective centers
are related by a homography in other words projectively
equivalent, which is much less stringent. In fact, when the number
of point correspondences is less than 6, camera resectioning is
always indeterminate, even if the control points lie in general
position, however, the corresponding PnP problem will in general
have a limited number of solutions and, in most cases, a unique
solution. For example, if all the space points and the perspective
center lie on a special twisted cubic space curve, camera
resectioning will always be indeterminate [12, pp. 520-521] no
matter how many correspondences one has, but the corresponding
PnP problem will generally have a unique solution.
2
The TWO DEFINITIONS ARE IN GENERAL DIFFERENT
The basic difference is that the configurations of control points,
which all satisfy the distance-based definition, sometimes cannot be
related to each other by a rotation and a translation. As shown for the
P4P problem in Fig. 1, both configurations …A; B; C; D† and
…A; B; C; D0 † satisfy the distance-based definition, where O is the
optical center, a; b; c; d are the four image points, line OD is
perpendicular to plane …ABC†, E is the intersecting point, and
jDE j ˆ jD0 E j. In other words, both …jOAj; jOBj; jOC j; jODj† and
…jOAj; jOBj; jOC j; jOD0 j† are positive solutions of the P4P problem
under the distance-based definition. However, it is impossible to
transform configuration …A; B; C; D† into configuration …A; B; C; D0 †
2
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
VOL. 24,
NO. 3,
MARCH 2002
Fig. 1. Both configuration …A; B; C; D† and configuration …A; B; C; D0 † are positive solutions for the P4P problem under the distance-based definition. However, they
cannot both be for the P4P solutions problem under the transformation-based definition.
by a rotation and a translation since configuration …A; B; C; D† is the
reflection of configuration …A; B; C; D0 † with respect to plane …ABC†,
i.e., configuration …A; B; C; D† and configuration …A; B; C; D0 †
cannot both be solutions of the P4P problem under the transformation-based definition.
3
x2 y2 z2
ˆ ˆ ˆ a > 0; a 6ˆ 1;
x1 y1 z1
then we have
jOA2 j ˆ ajOA1 j; jOB2 j ˆ ajOB1 j; jOC2 j ˆ ajOC1 j
THE NONCOPLANAR P4P PROBLEM UNDER THE
DISTANCE-BASED DEFINITION
and
In this section, we show that under the distance-based definition,
the solutions' upper bound of the noncoplanar P4P problem is 5
and this upper bound is also attainable.
3.1
If X 1 and X 2 were linearly dependent, i.e.,
The Upper Bound of the Noncoplanar
P4P Problem is 5
Lemma 1. If the three control points and the camera's optical center are
not coplanar, then the corresponding P3P problem can have at
maximum four different positive solutions.
A proof of Lemma 1 can be found in [5] (Theorem 4, p. 1091).
Lemma 2. As shown in Fig. 2, if
jA2 B2 j ˆ ajA1 B1 j; jA2 C2 j ˆ ajA1 C1 j; jB2 C2 j ˆ ajB1 C1 j:
Equation (2) is contrary to (1), hence the assumption that X 1
and X 2 were linearly independent is untrue.
u
t
Theorem 1. The upper bound of the positive solutions of the noncoplanar
P4P problem is five.
Proof. Assume the four control points are …A; B; C; D†, the
camera's optical center is O, and the six distances between the
control points are:
X 1 ˆ … x1
y1
z1 † ˆ … jOA1 j
jOB1 j
jOC1 j †;
X 2 ˆ … x2
y2
z2 † ˆ … jOA2 j
jOB2 j
jOC2 j †
dAB ˆ jABj; dAC ˆ jACj; dBC ˆ jBCj;
dAD ˆ jADj; dBD ˆ jBDj; dCD ˆ jCDj:
and
Since the camera is calibrated, i.e., its intrinsic parameters are
are two different positive solutions of the P3P problem, where O is the
optical center, then X 1 and X 2 must be linearly independent, i.e.,
X 1 6ˆ aX
X2 ; a > 0, and a 6ˆ 1.
Proof. Since X 1 ; X 2 are two different positive solutions of the
P3P problem, the following equalities must hold:
jA2 B2 j ˆ jA1 B1 j; jA2 C2 j ˆ jA1 C1 j; jB2 C2 j ˆ jB1 C1 j:
…1†
known, the following angles can also be considered known,
AB ˆ €AOB; AC ˆ €AOC; BC ˆ €BOC;
AD ˆ €AOD; BD ˆ €BOD; CD ˆ €COD:
As shown in Fig. 3, we can obtain the following six constraints
for the P4P problem via the law of cosines, where …x; y; z; w† ˆ
…jOAj; jOBj; jOC j; jODj† are the four distances to determine
Fig. 2. …A1 ; B1 ; C1 † and …A2 ; B2 ; C2 † are two point configurations of the
P3P problem.
…2†
Fig. 3. The P4P problem has six constraints in total.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,
8 2
x ‡ y2
>
>
>
>
x2 ‡ z 2
>
>
< 2
y ‡ z2
> x2 ‡ w 2
>
>
>
>
y2 ‡ w2
>
: 2
z ‡ w2
2 cos AB xy ˆ d2AB
2 cos AC xz ˆ d2AC
2 cos BC yz ˆ d2BC
2 cos AD xw ˆ d2AD
2 cos BD yw ˆ d2BD
2 cos CD zw ˆ d2CD :
3
2 cos AD xk w ˆ d2AD
2 cos BD yk w ˆ d2BD
2 cos CD zk w ˆ d2CD
x2k
y2k
z2k
…8†
In order for variable w to have two distinct positive solutions in
Suppose system (3) has N 6 different positive solutions,
X j ˆ …xj ; yj ; zj ; wj †, j ˆ 1; 2; 3; ; N 6, since the four control
points …A; B; C; D† are not coplanar, there always exist three
control points which are not coplanar with point O. Without
loss of generality, suppose points …A; B; C† and point O are
not coplanar. According to Lemma 1, the corresponding P3P
problem …O : …A; B; C†† can have at maximum four positive
~ j ˆ …xj ; yj ; zj †, j ˆ 1; 2; 3; ; N 6 must
solutions. Evidently X
all be positive solutions of the P3P problem …O : …A; B; C††.
~ 2; ; X
~ M M 4 are different positive
~ 1; X
Hence, suppose X
solutions of the P3P problem …O : …A; B; C††, based on
~ j ˆ …xj ; yj ; zj †, j ˆ 1; 2; 3; ; N 6 can be classiLemma 1, X
fied into the following two subsets:
~ 1; X
~ 2; ; X
~ M M 4; where X
~ j for i 6ˆ j;
~ i 6ˆ X
1 ˆ X
~ M‡1 ; X
~ M‡2 ; ; X
~N :
2 ˆ X
system (8), the coefficients must satisfy the following constraints:
1
2 cos AD xk
1
2 cos AD xk
ˆ 0; Det
ˆ0
Det
1
2 cos BD yk
1
2 cos CD zk
…9†
or
cos AD yk cos AD zk
ˆ ;
ˆ
cos BD xk cos CD xk
…10†
Similarly from (7), the following constraints must be satisfied:
1
2 cos AD xl
1
2 cos AD xl
ˆ 0; Det
ˆ0
Det
1
2 cos BD yl
1
2 cos CD zl
…11†
or
cos AD yl cos AD zl
ˆ ;
ˆ :
cos BD xl cos CD xl
…12†
From (10) and (12), we have
Clearly, subset 2 must contain at least two elements …N
~ i 2 1 such that
~ M‡j 2 2 , there exists an X
M 2† and, for each X
~ i.
~ M‡j ˆ X
X
In the following, we shall show at first the elements in 2 are all
~ M‡j 2 2 , X
~ M‡i 6ˆ X
~ M‡j if
~ M‡i , X
different. In other words, for X
i 6ˆ j.
~ M‡i ˆ X
~ M‡j 2 2 i 6ˆ j, then there
This is because, suppose X
~ k 2 1 such that:
exists X
…4†
Since
xk yk zk
ˆ ˆ ˆa>0
xl
yl
zl
…13†
Based on Lemma 2, (13) is contrary to the assumption that
…xk ; yk ; zk † and …xl ; yl ; zl † are two different positive solutions of the
P3P problem …O : …A; B; C††, which in turn indicates that the
noncoplanar P4P problem cannot have six or more positive
solutions.
3.2
The Upper Bound is also Attainable
An example of the noncoplanar P4P problem, which has five
~ k wM‡i ;
X M‡i ˆ X
~ k wM‡j ;
X M‡j ˆ X
~ k wk ˆ …xk ; yk ; zk ; wk †
Xk ˆ X
positive solutions, is shown in Fig. 7 in Section 5.
4
are three different positive solutions of the P4P problem,
wM‡i ; wM‡j ; wk must be three different solutions of the following
system:
8 2
< xk ‡ w2 2xk w cos AD ˆ d2AD
…5†
y2 ‡ w2 2yk w cos BD ˆ d2BD
: 2k
zk ‡ w2 2zk w cos CD ˆ d2CD :
Since each equation in system (5) is of second order, the system (5)
has at maximum two different solutions. Hence, the assumption that
~ M‡j 2 2 i 6ˆ j is untrue. In other words, all the elements
~ M‡i ˆ X
X
in 2 must be different from each other.
~ M‡i ,
Based on the above discussion, for any two elements X
~ k, X
~ l 2 1 …k 6ˆ l†,
~ M‡j 2 2 …i 6ˆ j†, there exist two elements X
X
such that:
~ k;
~ M‡i ˆ X
X
MARCH 2002
8 2
<w
w2
: 2
w
…3†
u
t
~ M‡j ˆ X
~k
~ M‡i ˆ X
X
VOL. 24, NO. 3,
THE NONCOPLANAR P4P PROBLEM UNDER THE
TRANSFORMATION-BASED DEFINITION
Horaud et al. [3] in 1989 had proven that the upper bound of the
noncoplanar P4P problem is 4 under the transformation-based
definition. The following example shows that this upper bound is
also attainable.
4.1
An Example
Without loss of generality, assume the camera's intrinsic parameter
matrix is the identity matrix, i.e.,
K ˆ I:
The four control points in the object-centered frame are:
M 1 ˆ … 1 0 10 †T ;
T
p
3
1
;
M2 ˆ
10
2
2
T
p
3
1
M3 ˆ
10 ;
2
2
199 T
M 4 ˆ 0 0 20 :
…6†
~ M‡j ˆ X
~ l:
X
…7†
~ k wM‡i and X k ˆ X
~ k wk are two
Since X M‡i ˆ X
different positive solutions of the P4P problem, from (6),
wM‡i ; wk must be two different positive solutions of the following
system:
The four corresponding image points in homogenous
coordinates are:
4
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VOL. 24,
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SOME SUFFICIENT CONDITIONS FOR MULTIPLE
POSITIVE SOLUTIONS OF THE NONCOPLANAR
P4P PROBLEM
In this section, we study some conditions under which at least two,
three, four, and five different positive solutions appear in the
noncoplanar P4P problem under the distance-based definition. We
have the following proposition:
Proposition 1. Given four noncoplanar control points …A; B; C; D†, O is
the system's origin located at the camera's optical center,
a.
b.
c.
Fig. 4. At least two positice solutions exist in this case.
I1 ˆ
I2 ˆ
I3 ˆ
1
10
1
20
1
20
T
0 1 ;
T
p
3
1 ;
20
T
p
3
1 ;
20
d.
Then,
Case 1: If only (a) is satisfied, the P4P problem has at least two
positive solutions;
Case 2: If both (a) and (b) are satisfied, the P4P problem has at
least three positive solutions;
Case 3: If (a), (b), and (c) are all satisfied, the P4P problem has at
least four positive solutions;
Case 4: If (a), (b), (c) and (d) are all satisfied, the P4P problem has
five positive solutions.
I 4 ˆ … 0 0 1 †T :
The four consistent transformation matrices are:
1
0
1 0 0
0
C
B
P 1 ˆ … R1 T 1 † ˆ @ 0 1 0
0 A;
0 0
1 20
p
0
P 2 ˆ … R2
B
T2 † ˆ B
@
0
P 3 ˆ … R3
B
T3 † ˆ B
@
0
P 4 ˆ … R4
201
202
p
3
202
10
101
201
202
p
3
202
10
101
99
101
B
T4 † ˆ @ 0
20
101
3
202
199
202
p
10 3
101
p
3
202
0
1
0
10
101
p
10 3
101
199
202
99
101
1990
101
199
202
p
10 3
101
20
101
0
99
101
1
p C
199 3 C;
202 A
10
101
p
10 3
101
Proof. Case 1. Choose a point C 0 on line OC such that
jC 0 Mj ˆ jCMj. As shown in Fig. 4, since DM?CC 0 , we have
jDC 0 j ˆ jDCj. Similarly, we have jBC 0 j ˆ jBCj, jC 0 Aj ˆ jCAj.
Hence, the two configurations …A; B; C; D† and …A; B; C 0 ; D† are
both positive solutions of the P4P problem (Here, we really
mean
1
199
202
p C
199 3 C;
202 A
99
1990
101
101
1
199
101
C
0 A:
1990
101
We can verify that under the above four transformation
matrices P i ; i ˆ 1; 2; 3; 4, the corresponding four projected image
points for each of the four control points are identical and RTi Ri ˆ
I and Det…R
Ri † ˆ 1 i ˆ 1; 2; 3; 4. This indicates that … R i T i †; i ˆ
1; 2; 3; 4 are indeed four consistent solutions of this P4P problem
under the transformation-based definition.
Fig. 5. At least three positive solution exist in this case.
On one of the four projection rays OA, OB, OC, and OD,
say OC, exists a point M such that AM?OC, BM?OC,
DM?OC (See Fig. 4);
OD is perpendicular to plane …ABC†, E is the foot point;
E is also the intersecting point of the bisectors of ABC and
jABj ˆ jACj;
jABj ˆ jACj ˆ jBCj.
… jOAj jOBj
jOC j
jODj †
… jOAj
jOC 0 j
jODj †
and
jOBj
are two positive solutions of the P4P problem. For simplicity,
we simply say the configurations are positive solutions here).
Case 2. From Case 1, we have at least two solutions. In
addition, since OD is perpendicular to plane …ABC†, for the
mirror point D0 of point D with respect to plane …ABC†, we
have jDAj ˆ jD0 Aj, jDBj ˆ jD0 Bj, jDCj ˆ jD0 Cj. Hence, configuration …A; B; C; D0 † is another positive solution of the
P4P problem, as shown in Fig. 5.
Case 3. From Case 2, we have at least three solutions.
Choose a point M 0 on line OB such that jBM 0 j ˆ jCMj. Since
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5
Fig. 6. At least four solutions exist in this case.
Fig. 7. Five positive solutions exist in this case.
the foot point E is also the intersecting point of the bisectors of
ABC and, additionally, jABj ˆ jACj, OAB, and OAC must
be identical. Since AM is the altitude of OAC on side OC, AM 0
must be the altitude of OAB on side OB. Similarly, OBD and
ODC must be identical, hence DM 0 is the altitude of ODB on
side OB. In addition, since OCB is an isosceles, BM is the
altitude on side OC, CM 0 should be the altitude on side OB.
Hence, DM 0 ?OB, AM 0 ?OB, CM 0 ?OB. Choose a point B0 on
line OB such that jB0 M 0 j ˆ jBM 0 j. From Case 1, …A; B0 ; C; D†
must be another positive solution of the P4P problem, hence the
P4P problem must have at least four positive solutions in this
case, as shown in Fig. 6.
Case 4. From Case 3, we have already four positive
solutions. Since additionally jBCj ˆ jACj, similarly to the proof
in Case 3, we can find a point A0 on line OA to make the
configuration …A0 ; B; C; D† be another positive solution of the
P4P problem, thus we obtain five positive solutions, as shown
in Fig. 7.
For example, it can be verified that the four point
correspondences fM i $ I i i ˆ 1; 2; 3; 4g in Section 4 with the
perspective center at the origin satisfy simultaneously (a), (b),
(c), and (d), hence the corresponding P4P problem under the
distance-based definition has five positive solutions.
6
SOME DISCUSSIONS
At this stage, some questions can be asked, for example, how to use
this study in a practical vision system, how to avoid multiple
solutions, and how sensitive are the solutions to noise?
Unfortunately, a clear-cut answer to the above questions is
currently unavailable. In our application, four markers (four light
diodes) fixed on a flying object are used to determine the
instantaneous pose of the object with respect to another flying
object (a base object where a camera is rigidly mounted), which is a
standard noncoplanar P4P problem. It is primordial to have a
unique solution in our application. Concerning how to avoid
multiple solutions, we thought an analytical solution seems very
difficult, even if not impossible, since the problem is in nature
equivalent to determining sufficient conditions of a unique
solution in the noncoplanar P4P problem. To our knowledge, an
analytical solution is unavailable, even for the P3P problem which
is a much simpler one. A Monte Carlo procedure is instead
engaged in this work to experimentally explore the probability of
multiple solutions. The detailed simulation results are shown in
Table 1. The setup of our simulation is as follows: The optical
center is located at the origin and the matrix of camera's intrinsic
parameters is assumed to be the identity matrix. At each trial, four
noncoplanar control points are generated at random within a cube
centered at (0,0,50) and of dimension 60 60 60. Then, the six
distances and six angles from the six different pairs of control
points are computed. Finally, these computed distances and angles
are substituted into system (3) in Section 3 to determine …x; y; z; w†.
^ †, if the following two conditions are both
For a 4-tuple …x^; y^; z^; w
satisfied, then it is considered as a distinct true solution of the
noncoplanar P4P problem.
a.
b.
All the six residuals, each associated with one of the six
equations in system (3), are less than a prefixed threshold
of a very small value ( called ªResidual Thresholdº in
Table 1);
Max…jx^
x^i j; jy^
y^i j; jz^
^
z^i j; jw
^ i j† > 1:5
w
6
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TABLE 1
The Simulation Results Among 100,000 Trials
ªx-solution trialsº in the first column stands for the total number of the trials that give rise to ªxº positive solutions. ªResidual Thresholdº means all the six residuals in
system (3) must be less than this fixed value for each solution.
^ i †i ˆ 1; 2; , where …x^i ; y^i ; z^i ; w
^i† i ˆ
for all …x^i ; y^i ; z^i ; w
1; 2; are distinct solutions established previously.
From Table 1, we can see that the probability of multiple
solutions, and, in particular, that of two solutions, is not negligible
and, consequently, special care must be taken in practice. In
addition, from Table 1, we know that the number of solutions is
relatively stable when the residual threshold does change within a
certain range. The interested reader is referred to references [13],
[14], [15] for more in-depth analysis about the solution's instability.
Last, since the results in Table 1 are numerical solutions, the
maximum number of solutions could be larger than 5, and the
theoretical upper bound proven in Section 3.1.
7
CONCLUSIONS
In this paper, we showed that contrary to what is commonly
believed, the distance-based definition of the PnP problem is, in
general, different from the transformation-based definition. In
particular, we showed that the noncoplanar P4P problem under
the distance-based definition could have as many as five positive
solutions. However, the maximal number of possible solutions is 4
under the transformation-based definition. In addition, we showed
that, under both definitions, their respective upper bound of
solutions is also attainable. Finally, we studied some sufficient
conditions under which two, three, four, and five positive
solutions of the noncoplanar P4P problem exist.
ACKNOWLEDGMENTS
This work was supported by ª973º Program (G1998030502), The
National Science Foundation of China (60033010).
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