Abstracts

Abstracts will be reproduced and distributed in printed form to all participants of Conference at the beginning of the Conference.  Abstracts should be submitted electronically to mml@math.nsc.ru . Submission is also possible by fax or by ordinary mail to
Dr. Olga Klimenko
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630090, Russia
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Abstracts are due by May 31, 2002.
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{\bf ON RECONSTRUCTION OF THE TRANSPARENT SURFACES
FROM THEIR APPARENT CONTOURS}\\[3mm]
{\bf V.\,P.\,Golubyatnikov}\\[2mm]
{\it Novosibirsk, Sobolev Institute of Mathematics,\\
E-mail: glbtn@math.nsc.ru}
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\noindent

The problems of reconstruction of multidimensional objects from information on their plane projections are considered in various disciplines of pure and applied mathematics. Here we study the uniqueness questions in the problem of reconstruction of the shape of a smooth hypersurface from the shapes of its apparent contours. As it was shown by F.\,Pointet [2] if the apparent contours $C(M_1,\omega)$, $C(M_2,\omega)$ of smooth  hypersurfaces $M_1$,
$M_2 \subset \mathbb{R}^{n+1}$ coincide for a sufficiently large set $W$ of directions $\omega \in S^n$ then these hypersurfaces coincide themselves. Using this theorem and the methods of [1] we obtain the following results:
\noindent {\bf Theorem 1.} {\sl Let $M_1$, $M_2 \subset \mathbb{R}^3$ be smooth closed compact surfaces such that for any $\omega \in S^2$ the apparent contours $C(M_1,\omega)$ , $C(M_2,\omega)$ are equivalent with respect to some orientation-preserving motion of the plane $P(\omega)$ and the convex hulls $conv (C(M_1,\omega))$, $conv (C(M_2,\omega))$ of these contours have no rotation symmetries, then

M_{1}=F(M_{2})

\label{e:1}

where $F:\mathbb{R}^3\longrightarrow \mathbb{R}^3$ is either
parallel translation or central symmetry.}
\noindent {\bf Definition.} {\
sl The figures are called SO-similar if they are superimposed by a composition of an orientation preserving motion and a homothety.}
\noindent {\bf Theorem 2.}  {\sl Let $M_1$, $M_2 \subset \mathbb{R}^3$ be smooth compact closed surfaces and for all $\omega \in S^2$ their apparent contours $C(M_1,\omega)$, $C(M_2,\omega)$ are SO-similar and their convex hulls $conv(C(M_1,\omega))$, $conv(C(M_2,\omega))$ have no rotation symmetries (the ratio of the similitude is not supposed to be constant, independent of the plane $P(\omega)$), then the formula (\ref{e:1})holds for some $F$ which is either parallel translation or homothety.}
The work was supported by NATO grant OUTR.CLG 970357.
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\noindent
1. Golubyatnikov~V.P., On unique recoverability of visible
compacta from their projections. {\it Math. USSR Sbornik} (1992) {\bf 73}, No.\,5, 1--10.
\noindent 2. Pointet~F., Separation of hypersurfaces. {\it J. Geom.} (1997)
{\bf 59}, No.\,2, 114--124.
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