1. Introduction.
We begin with necessary definitions of the main objects considered here.
Denote by P (w) the oriented hyperplane
with the unit normal vector w in the Euclidean space
Rn+1 , n ³ 2 .
Let Mn Ì Rn+1 be a compact smooth closed hypersurface.
Following [8], [11], the set of the points y Î P (w) such that the straight line containing y and orthogonal to P (w) is tangent to Mn in some point m(y) will be called the apparent contour of the surface Mn in the direction w and will be denoted by C(M, w) . Besides these points m(y) each of these lines can intersect the hypersurface Mn transversally finitely many times, see figures 1 and 2.
It is very easy to see that for a convex smooth surface Mn Ì Rn+1 its apparent contour in any hyperplane P( w) is just the boundary of the orthogonal projection of Mn onto this plane P( w) .
For any hyperplane P Î Rn+1 , containing the zero point, its
intersection with W Ì Sn is non-empty and for all
quadratic form Q over P we have:
Q ( w) = 0 for all w Î P ÇW
implies that Q is degenerate.
F.Pointet calls these sets n -omnidirectional.
Such apparent contours projection data have a natural physical
interpretation: if a thin-walled transparent membrane M
is examined by the laser beams, X-rays or other high-frequency radiation,
the signals spreading along the rays tangent to the inner surface
of the membrane loose much more energy than those which have only
transversal intersections with this membrane, as on the
Figure 2.
Analogous well-known pictures appear naturally
in various domains of pure and applied mathematics.
Theorem 1.
Let M1 and M2 be smooth closed compact surfaces in
R3 such that for any w Î S2 the apparent contours
C(M1 , w) and C( M2 , w) are SO(2) -congruent
in the plane P(w) , and the convex hulls
conv C( M1 , w) and conv C( M2 , w) of these
contours have no rotation symmetries,
then M1 and M2 are either parallel or centrally symmetric
to each other.
To prove the theorem, we determine the map
j: S2 ® S1 as in [5,6]
as follows: given a unit vector w Î S2 , let
j(w) Î S1 be the angle such that the apparent contour
C( M2 , w) is obtained from C( M1 , w) by
rotation through the angle j(w) .
Note that the convex hulls
of these contours are congruent with respect to the same rotations.
As in [5] the asymmetry of the convex hulls
of these contours implies correctness of determination of this map
and its continuity.
The sign of the angle j(w) is determined by the direction
of the normal w, hence we have very important relation
j(- w) = - j(w) .
For the continuous mapping j: S2 ® S1
we shall denote by j-1(0)
and by j-1(p) , respectively, the preimages of the points
0, p Î S1 , i.e. the sets of the vectors w Î S2 such
that the contours C(M1 , w) , C(M2 , w) are
parallel translation equivalent (resp. centrally symmetric to each other).
It is obvious that these sets are closed and centrally symmetric.
Now, suppose that there is a vector w0 Î S2 , such that
p ¹ j(w0) ¹ 0 . Consider all meridians
m(w0, t) on S2 which join the points w0 and
- w0 , where 0 £ t £ 2 p and t parametrizes
the points on the great circle E(w0 ) = P(w0) ÇS2 .
Lemma 1.
For the continuous mapping j, one of the sets
j-1(0) and j-1(p) intersects
all the meridians m(w0 , t) .
The proof repeat the arguments of [5,6], where instead of the apparent
contours we have considered the projections of the convex and visible
bodies.
If all these meridians m (w0, t) intersect the preimage
j-1(p) , we replace the surface M1 by M¢1 which is
obtained by a central symmetry in order to change the roles of the preimages
j-1(0) and j-1(p) .
For the surfaces M¢1 and M2 , all the
meridians m(w0, t) intersect the preimage
j-1(0) , so, we may assume in the sequel that such
a change has been executed if necessary.
The proof of the Theorem 1 is reduced to the consideration of the
following two cases:
a). The set j-1 (0) is not a great circle on S2 .
Here, as it was shown in [5,6], there exist such vectors
a1 , a2 Î j-1 (0) , that
for a dense set of the parameters t* Î E( w0)
there exists a vector
w(t*) Î j-1 (0) Çm (w0, t*) ,
which is noncoplanar with a1 , a2 .
In this case according to the Süss's lemma, [3,5], the surface
M1 (or M1¢ ) can be moved by some parallel translation
so that the obtained surface M1¢¢ and the surface M2
have coinciding apparent contours
for all vectors w Î j-1 (0) .
Hence, the convex hulls of these contours coincide as well.
Note that the convex hull of such a contour is the convex hull
of the projection of a surface on the corresponding plane
P (w) . The lemma 1 implies that for any
w1 Î S2 there exists a vector in j-1 (0)
perpendicular to this w1 Î S2 , hence the
support functions of the convex hulls conv M1¢¢, conv M2
coincide for all directions.
So, the convex hulls of both apparent contours
conv C(M1¢¢, w0) and conv C( M2 ,w0)
coincide but since 0 ¹ j(w0) ¹ p,
one can make the convex hull conv C(M1¢¢,w0)
coincide with conv C(M2 , w0)
by a rotation through the angle j0 , which contradicts
the asymmetry of these convex hulls.
b). The set j-1 (0) is a great circle on S2 .
In this case the Süss's lemma can not be applied, but as in
[5,6] we have
Lemma 2.
If j(w) \not º 0 ,
j(w) \not º p, the function
j(w) is continuous, and j-1 (0)
is a great circle, then the convex hulls of the surfaces
M1 , M2 have a constant width.
The proof of this lemma can be easily deduced from the following facts:
1). In any neighborhood of j-1 (0) there exists
a vector w3 such that the ratio
j( w3)/p is irrational.
2). For an irrational a and for integer numbers k
the angles k pa compose a dense set on the circle.
3). If for two convex bodies V1 , V2 Ì R3 their projections
on any plane have equal perimeters, then the widths of these bodies
coincide in all directions (see, for example [3]).
So, in both cases a) and b) the apparent contours of the surfaces
M1¢¢ and M2 coincide in all directions, thus
the results of F.Pointet [11]
imply the coincidence of the surfaces, and our theorem is proved.
[¯]
We shall call figures SO-similar if they can be superimposed
by a composition of an orientation preserving motion and
a homothety.
Theorem 2.
Let M1 and M2 be smooth compact closed surfaces in R3 .
If for all w Î S2 their apparent contours
C( M1 , w) and C(M2 , w) are SO(2) -similar
and their convex hulls conv C( M1 , w) ,
conv C(M2 , w) have no SO(2) -symmetries
(the ratio of the similitude is not supposed
to be constant, independent of the plane P(w) ),
then these surfaces M1 and M2 in R3 are either parallel
or directly homothetic.
As in [5] the proof of this theorem is carried out in two steps:
1.
we prove that the convex hulls conv C(M1 ,w) and
conv C( M2 , w) are equivalent either with respect to
a homothety H or with respect to a parallel translation T .
2.
since the convex hulls of the projections
of the apparent contours of the surfaces M1 , M2
have no SO(2) -symmetries, then after one of these transformations
H or T the apparent contours C( M1, w) and
C(M2, w) coincide for all w Î S2 .
Now using the theorem of F.Pointet we obtain the coincidence
of the surface M2 and the surface M1¢, obtained from M1
by one of the transformation H or T .
These considerations do not seem to be too artificial from the
practical viewpoint, because the projections
images sometimes have much more complicated structure than a curve
on the display of computer. Many physical phenomena connected with
the wave fronts transformations (caustics, the focus points etc)
usually are studied as the singularities of projections of
hypersurfaces in the 6-dimensional phase space
to the configuration manifold R3 , see for example [1], [2], [4].
In order to describe the projections of smooth
hypersurfaces Mn Ì Rn+1 onto (n+1-k)-dimensional planes
for k ³ 2 , we remind some useful definitions.
Let G(k, n+1-k) be the Grassmann manifold of all k-dimensional
subspaces in Rn+1 .
For any x Î G(k, n+1-k) we denote by x^ Î G (n+1-k, k)
its orthogonal complement.
Let G: Mn ® Sn be the Gauss map
which associates to any point of the surface the unit normal vector
in this point. Consider the orthogonal projection
px : Mn ® x^ along the k -dimensional
plane x , and let
S( px ) = G-1 (x^ ÇSn )
be the critical points set of this projection. The set
C(M, x) = px ( S(px)) is called
the apparent
contour of the hypersurface Mn in the direction x .
It is easy to verify that C(M, x) consists of the points
y Î x^ such that the k -dimensional plane containing
y and parallel to x is tangent to Mn in some point.
The following definition and theorem belong to Pointet [11]:
Definition.
Given W Ì G(k, n+1-k) , we say that
W is nk -omnidirectional if for all hyperplanes
p there exists w Î W with w Ì p,
and if for any quadratic form Q over p
Q \midw degenerates for all w Î W with
w Ì p then Q is degenerate.
Theorem.
If the apparent contours C( M1 ,x) ,
C( M2 , x) of smooth hypersurfaces M1 , M2 Ì Rn+1
coincide for
an nk -omnidirectional set
W Ì G(k, n-1)
of k-dimensional planes,
then these hypersurfaces coincide themselves.
First, we consider the apparent contours of the surfaces
in two-dimensional planes.
Theorem 3.
Let M1 and M2 be smooth closed compact hypersurfaces in
Rn+1 , n ³ 2 , such that for any (n-1) -dimensional subspace
x Î G(n-1, 2) the apparent contours C(M1, x^ ) ,
C(M2, x^ ) are SO(2) -congruent in x^, and
the convex hulls
conv C( M1 , x^) and conv C( M2 , x^)
of these contours have no SO(2) -symmetries, then the surfaces
M1 and M2 are either parallel or
centrally symmetric to each other.
Theorem 4.
Let M1 and M2 be smooth closed compact
hypersurfaces in Rn+1 , n ³ 2 , such that for any
(n-1) -dimensional subspace x Î G(n-1, 2) the apparent contours
C(M1, x^ ) and C(M2, x^ ) are SO(2) -similar
and the convex hulls
conv C( M1, x^) and
conv C( M2, x^)
of these contours have no SO(2) -symmetries, then the surfaces
M1 and M2 are either parallel or directly homothetic.
Both these theorems as the
theorem 2 are proved in two steps:
1. we prove that for all x Î G (n-1, 2) the convex hull
conv C(M1, x^ ) and conv C(M2, x^ )
are equivalent with respect to either a homothety H or
a parallel translation T or a central symmetry S ;
2. the condition of asymmetry of these convex hulls imply that
they are congruent with respect to one of these transformations
H , S or T , and the statements of our theorems follow from
[11], theorem 13.
Similar considerations can be reproduced for the apparent contours
C( M, x^) , x Î G(n-2, 3 ) in three-dimensional planes
in Rn+1 .
Theorem 5.
Let M1 and M2 be smooth closed compact hypersurfaces in
Rn+1 , n ³ 3 , for which
(1). the apparent contours C( M1 , x^) and
C( M2 , x^) in every three-dimensional subspace x^
are SO(3) -congruent with respect to some orientation preserving
isometry s(x^) of the plane x^ ;
(2). the convex hulls conv C(M1, x^ ) and
conv C(M2, x^ ) of these contours have no
SO(3) -symmetries, and the width functions of conv M1 ,
conv M2 have finitely many maxima,
then the surfaces M1 and M2 are parallel in Rn+1 .
A similar theorem holds for the isometry s(x^) which
does not preserve the orientation of the planes x^ .
Theorem 6.
Let M1 and M2 be smooth closed compact hypersurfaces in
Rn+1 , n ³ 3 , for which
(1). the apparent contours C( M1 , x^) and
C( M2 , x^) in every three-dimensional subspace x^
are SO(3) -similar;
(2). the convex hulls conv C(M1, x^ ) and
conv C(M2, x^ ) of these contours have no
SO(3) -symmetries and the width functions of conv M1 and
conv M2 have finitely many maxima,
then M1 and M2 are either parallel in
Rn+1 or directly homothetic with a positive coefficient.
This theorem is reduced to the previous one by means of the homothety
of M1 with coefficient equal to the ratio of the diameters of
the hypersurfaces M1 and M2 .
Analogous statement holds for the similarities which do not
preserve the orientation of x^ , in this case the initial
hypersurfaces are equivalent with respect to some homothety with
a negative coefficient.
As it was shown in [7], the simplicity of the Riemannian metric g
can be verified from the boundary observations, i.e., if the condition
of uniqueness of the geodesic line gp,q holds for every pair
p, q Î ¶B , and this line depends smoothly
on its endpoints, then the metric g is simple.
The Riemannian metric g on the manifold Bn+1 induces
the canonical isomorphism
g* : T Bn+1 ® T* Bn+1 of the tangent
and cotangent bundles of Bn+1 and all its submanifolds.
Consider a smooth oriented manifold Mn immersed into the interior
of the ball Bn+1 endowed by a simple Riemannian metric. For any
point x Î Mn , denote by v(x) the unit vector of the exterior
normal to Mn at the point x . Let g(x) be
the geodesic line starting from this point in the direction v(x)
and let d(x) be the distance between x and the boundary
¶Bn+1 . It is easy to see that the positive function
d: Mn ® R1 is continuous.
Any geodesic line of a simple metric in Bn+1 arrives to its
boundary ¶Bn+1 in both its directions, see, for example,
[12]. Denote by y(x) = expx (d(x) ·v(x)) Î ¶Bn+1
the intersection of the line g(x) with
¶Bn+1 = Sn ,
here expx ( ) is the exponential map of the tangent space
Tx Bn+1 to Bn+1 .
Let m(x) Î Ty(x) be the unit tangent vector to g(x)
at the point y(x) . Consider the covector
k* (x) = g* ( m(x)) Î T*y(x) Sn defined by
k* (x) (w) = áw, m(x) ñ for all
w Î Ty(x) Sn , here á, ñ denotes
the scalar product generated by the metric g .
Such a covector plays the role of the gradient of the length l(x)
of the geodesic segment of g(x) between x and y(x) .
We define the map Y: Mn ® T* Sn by
the formula
Y(x) = ( y(x), k* (x) ) .
Lemma 7.
The image Y(Mn ) is a Lagrangian manifold in T* Sn .
The proof follows from the determination of the optical length of
the geodesic lines as the generating function of a germ of a Lagrangian
manifold, see [2].
Note, that this image Y(Mn ) can be interpreted as a trace
of the wave front generated by Mn Ì Bn+1 and registered
on the boundary ¶Bn+1 .
Theorem 8.
Let Mn be a smooth oriented compact closed connected manifold
immersed into the interior of the ball Bn+1 . Assume that the image
Y(Mn) has only transversal self-intersections
in T* Sn . Then this image and the location of
the point x0 Î Mn of the
minimum of the function d determine this hypersurface
Mn uniquely.
To prove this theorem, note that in some neighborhood of the point x0
the hypersurface Mn is uniquely determined because the gradient
of a function determines this function up to a constant summand.
The uniqueness of reconstruction of Mn
``in the large'' follows from its connectedness and compactness.
This theorem can be extended to the cases of manifolds of higher
codimensions. Consider a smooth oriented compact closed connected
manifold Mk immersed into the interior of Bn+1 as above,
1 £ k < n + 1 .
Let N Mk be the normal bundle of Mk in Rn+1 ,
and let S N Mk be its spherical subbundle. Denote by S Nx Mk
the (n-k) -dimensional unit sphere in Nx Mk which is the
orthogonal complement of the tangent space Tx Mk of the manifold
Mk at the point x .
Given a unit vector a Î S Nx Mk , consider the geodesic line
g(x, a) starting from the point x Î Mk in the
direction a^Tx Mk . Let
Theorem 9.
Let Mk be a smooth oriented compact closed connected manifold
immersed into the interior of the ball Bn+1 . Assume that the image
Y(S N Mk) has only transversal self-intersections
in T* Sn . Then this image and the location
of the point x0 Î Mk of the minimum of the
function d uniquely determine this manifold Mk .
Theorem 10.
Let Mk be as above, then the image Z( S N Mk ) determines
this manifold in Bn .
The proofs of these theorems are similar to that of theorem 8, i.e.,
are based on the theory of the wave fronts transformation, see [2].
The results on the uniqueness of reconstruction of surfaces from the
wave fronts observations were obtained in the situation of general
position. Simple examples show that in the case of non-transversal
self-intersections of the traces of the wave fronts the statements
of the theorems 8 and 9 are not true.
As it was shown by C.M.Petty and J.R.McKinney [9],
there exist affinely nonequivalent central symmetric
coaxial convex bodies of revolution in R3 ,
whose projections on any plane are similar. Hence, the asymmetry
of these convex hulls in the
theorem 2, 4, 6
is essential, and the question of its necessity in the
theorems 1, 3, 5 remains open.
Here, the main difficulty is connected with the symmetries of the
odd orders, which can depend on the plane of the projection.
The results on the uniqueness of reconstruction of surfaces from the
wave fronts observations were obtained in the situation of general
position. It is easy to see, that in the case of non-transversal
self-intersections of the traces of the wave fronts, the statements
of the theorems 7 and 8 are not true.
Figure 1: An apparent contour of Saint-Exupérie's surface.
F.Pointet has shown that if the apparent contours
C( M1 ,w) and C( M2 , w)
of smooth hypersurfaces M1 , M2 Ì Rn+1
coincide for a sufficiently large set W of directions
w Î Sn , then these hypersurfaces coincide themselves.
The condition on such a set W was formulated in [11] as follows:
Figure 2: Transversal and tangent intersections of
membrane with straight lines.
Another very useful case of the apparent contours observations is connected
with the phase shift of the signals spreading along the rays tangent
to the caustics. This effect is well-known in the theory of the seismic
waves propagation and is explained by the stationary phase method.
2. Reconstruction of 2-dimensional surfaces.
Using results of F.Pointet and the methods elaborated in [5,6],
we obtain the basic result of this section:
3. Multidimensional generalizations.
Theorems 1 and 2 have analogues in the
higher dimensional spaces.
4. Reconstruction of the surfaces from the exponential maps
of their normal bundles.
The apparent contours projection data are connected with the
natural map
G: STMn ® TSn
G^: SNMn ® Sn
A Riemannian metric g on a compact manifold B with the boundary
¶B is called simple if every pair of points
p,q Î B can be joined by the unique geodesic line gp,q
of this metric whose all points with the possible exception of its
endpoints belong to the interior of B , and such gp,q
depends smoothly on p and q . We shall suppose that such a simple
metric g is defined in the ball Bn+1 Ì Rn+1 .
The Riemannian metrics of this type are often considered in the inverse
kinematic and dynamic problems of seismology [1], [4],
in tomography [12], and in other wave fronts investigations [2].
y(x, a) = expx (l(x, a) ·a)
Y: S N Mk ® T* Sn ; Z: S N Mk ® ¶Bn+1 ×R1
Y(x, a) =
æ
è
y(x, a); k* (x, a)
ö
ø
; Z(x, a) =
æ
è
y(x, a); l(x, a)
ö
ø
.
5. Concluding Remarks
We have obtained a collection of results on uniqueness of reconstruction
of the shapes of surfaces from the shapes of their apparent contours.
The conditions of asymmetry of the convex hulls of these
contours, which we require here, is connected with our topological
approach, and does not seem to be natural from the viewpoint of the
initial statement of the problem: if two surfaces in R3 have
congruent projections on any plane, how different can they be?
The authors still have no idea whether this restriction in
the theorem 1 can be omitted. A very particular case was mentioned
in [3]:
If projections of two compact convex centrally symmetric bodies
in R3 have equal perimeters (or areas) for any plane,
then these bodies are parallel translation equivalent.