In geometry, the '''apparent contour''' of 3D object with surface <math>S</math>, with respect to a point light source, <math>\mathbf{p}</math> and a screen, is the image of those points on the surface where the rays from <math>p</math> are tangent to the surface. If the surface is well-behaved the apparent contour includes the boundary of the silhouette of the surface as it can include segments inside the boundary. It is theoretically possible to reconstruct the surface from a series of apparent contours, an important problem in computer vision.<ref name="an outline">{{cite journal|first1=Peter|last1=Giblin|title=Apparent contours: an outline|url=https://royalsocietypublishing.org/doi/10.1098/rsta.1998.0212|journal=Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences|date=15 May 1998|pages=1087–1102|volume=356|issue=1740|doi=10.1098/rsta.1998.0212|bibcode=1998RSPTA.356.1087G |url-access=subscription}}</ref><ref name="GiblinWeiss"> {{cite conference|last1=Giblin|first1=P. J.|last2=Weiss|first2=R. S.|year=1987|title=Reconstruction of surfaces from profiles.|conference=First Int. Conf. on Computer Vision|place=London|publisher=IEEE|pages=136–144|archive-url=https://web.archive.org/web/20240707070135/https://www.liverpool.ac.uk/~pjgiblin/papers/Reconstruction-from-profiles1987.pdf|archive-date=7 July 2024|url=https://www.liverpool.ac.uk/~pjgiblin/papers/Reconstruction-from-profiles1987.pdf}}</ref><ref>{{cite journal|first1=Giovanni|last1=Bellettini|first2=Valentina|last2=Beorchia|first3=Maurizio|last3=Paolini|first4=Franco|last4=Pasquarelli|title=Shape Reconstruction from Apparent Contours|url=https://link.springer.com/book/10.1007/978-3-662-45191-5|journal=Computational Imaging and Vision|date= 2015|volume=44 |publisher=Springer|issn=1381-6446|doi=10.1007/978-3-662-45191-5|isbn=978-3-662-45190-8 |url-access=subscription}}</ref>

thumb|A simple apparent contour

==Definition==

thumb|A cusp on an apparent contour, generate when the ray passes along an asymptotic direction

Let <math>\mathbf{c}</math> be a point light source, <math>S</math> a surface with unit normals <math>\mathbf{n}</math>, and <math>P</math> a unit sphere centred on <math>\mathbf{c}</math>. Define two sets:

<math display="block">\begin{align} \Gamma &= \{ \mathbf{r} \in S : (\mathbf{r}-\mathbf{c}) \cdot \mathbf{n} = 0 \} \\ \gamma &= \{ \mathbf{p} : p = (\mathbf{r} -\mathbf{c}) / \| \mathbf{r} -\mathbf{c} \|, \mathbf{r} \in \Gamma \} \end{align} </math>

<math>\Gamma</math> is called the ''contour generator'', and <math>\gamma</math> the ''apparent contour''. The definition can be generalised to project onto a plane, or have parallel light rays.

The apparent contour can be considered as the envelope of projections of a family of curves on the surface.

The apparent contour can have cusps when the ray has higher order contact with the surface. This occurs when the ray is in asymptotic direction in a hyperbolic region of the surface. Higher singularities occur when the ray is tangent to a parabolic line or flat umbilic on the surface. Only a limited <ref>{{cite conference|first1=R.|last1=Cipolla|first2=G.|last2=Fletcher|first3=P.|last3=Giblin|title=Surface geometry from cusps of apparent contours|date=June 1995|pages=858–863|doi=10.1109/ICCV.1995.466847|conference=International Conference on Computer Vision (ICCV) }}</ref>

thumb|A more complex apparent contour, showing a rhamphoid cusp when the ray is tangent to a parabolic line, and a beak singularity when the ray is tangent to a flat umbilic

==References== {{Reflist}}

Category:Differential geometry of surfaces