Abstract
In this chapter we show that the apparent contour of a stable embedded closed smooth (not necessarily connected) surface can be modified, using some of the moves illustrated in Chap. 6, to obtain an apparent contour without cusps.
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Notes
- 1.
- 2.
On the surface embedded in \(\mathbb{R}^{3}\) having the graph as its apparent contour (compare with Theorem 5.1.1), these long arcs correspond to a pair of folds on the surface, forming a sort of wrinkle.
- 3.
See Definition 2.2.12.
- 4.
- 5.
The argument is of course valid also in the presence of crossings in between the two cusps.
- 6.
Recall that, when traversing a crossing, either d remains constant, or it jumps by two units, so that its parity remains constant.
- 7.
When G is the apparent contour of an embedded surface \(\Sigma \), the arc a results from a folding of \(\Sigma \).
- 8.
- 9.
It may be helpful to regard this curve as a curve connecting the two corresponding points in (a connected component of) the embedded surface constructed in Chap. 5 Considered at this level, the curve does not self-intersect, and two cusps are connectable if they lie in the same connected component.
- 10.
It is possible to modify the construction so that the whole of im(γ) is \(\mathcal{C}^{\infty }\), requiring that at points of B I the set im(γ) is tangent to the arc. Since this smoothness is not necessary here, we do not add this requirement.
- 11.
When G is the apparent contour of an embedded surface \(\Sigma \), this operation corresponds to the creation of thin long crease (a double fold) that follows that part of γ.
References
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). Elimination of Cusps. 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_8
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