Abstract
In this chapter we illustrate the results and report the figures from the paper [3]. More specifically, we shall prove that there exists a finite set of simple, or elementary, moves (also called rules) on labelled apparent contours, such that the following property holds: the images \(\Sigma _{1}\) and \(\Sigma _{2}\) of two stable embeddings of a closed smooth (not necessarily connected) surface M in \(\mathbb{R}^{3}\) are isotopic if and only if their apparent contours can be connected using finitely many isotopies of \(\mathbb{R}^{2}\), and a finite sequence of elementary moves or of their inverses (sometimes called “reverses”).
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Notes
- 1.
- 2.
Namely, the fact that there are no other moves, besides those in the list of Sect. 6.1, necessary to connect two apparent contours of isotopic surfaces.
- 3.
- 4.
A realization in space of these moves involves two folds of the surface which can be “far one from the other”.
- 5.
The move L can be realized in space by considering the surface in Fig. 1.4, by gradually reducing the “hill”. The inverse of a move B can be realized by straightening the central part of a depression in a long “wave” with two parallels arcs corresponding to the crease and the valley of the wave.
- 6.
Compare also with Definition 8.1.1.
- 7.
See Chap. 5
- 8.
See Theorem 2.1.14.
- 9.
- 10.
Hence, the critical points of \(u_{\vert Y _{j}}^{}\) are nondegenerate.
- 11.
By definition, points of Y 3 are considered as critical points of u.
- 12.
The converse statement also holds true, as a consequence of the Isotopy Extension Theorem (see, for instance, [11, Theorem 1.3, p. 180], see also [19, pp. 157–201]). Namely, suppose that γ, e1 and e2 are as in (6.5). Then, γ induces an isotopy from e1(M) to \(\mathbb{R}^{3}\), which extends to an \(\mathbb{R}^{3}\)-ambient isotopy with compact support.
- 13.
Notice that the map \(\alpha \in \mathcal{C}^{\infty }(\mathcal{S}, \mathbb{R}^{3}) \rightarrow F_{\alpha } \in \mathcal{C}^{\infty }(\mathcal{S},\mathcal{T} )\) is continuous.
- 14.
Note also that, defining \(f_{\varphi _{t}}\) as in (2.2), we have \(f_{\varphi _{t}}(x) = \#\{m \in M: F_{\gamma }(m,t) = (x,t)\}\) for any \((x,t) \in \mathcal{T}\).
- 15.
For t ∈ [0, 1], we use the notation \(\tilde{\gamma }_{t}(\cdot ) =\tilde{\gamma } (\cdot,t)\) .
- 16.
- 17.
See [18, Proposition 5.4] for related problems.
- 18.
To be more specific, [4, Fig. 9(row 1, column 1)] corresponds to move L; [4, Fig. 9(2,1)] corresponds to move B; [4, Fig. 9(3,1)] corresponds to move S; [4, Fig. 10(1,1)] corresponds to move K; [4, Fig. 10(1,2)] corresponds to move C; [4, Fig. 10(2,1)] corresponds to move T. Theorem [4, 3.2.3] is a simpler version of [4, Theorem 3.5.5], in which the height function is not considered, and hence is more close to Corollary 6.6.2. It follows by combining the classifications given by [10, 16, 21].
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). Completeness of Reidemeister-Type Moves on Labelled Apparent Contours. In: Shape Reconstruction from Apparent Contours. Computational Imaging and Vision, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45191-5_6
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