Reconstruction of surfaces from given contours and silhouettes

Reconstruction of surfaces from given contours and silhouettes

We study a problem of the reconstruction of an algebraic surface from a finite collection of curves. Naturally, we cannot expect meaningful answer, unless we restrict to special class of curves and/or surface. Recall that a contour curve with respect to a certain viewpoint is a set of points of a surface whose tangent plane contains the viewpoint. The silhouette is then a projection of a contour from this point into some plane. Hence we can imagine a silhouette as a boundary of shadow of the surface with light-source at the viewpoint and the reconstruction of a surface from given silhouettes is a typical problem from descriptive geometry. In first part of the talk we will provide a method for reconstruction of a smooth surfaces in P^3 and we will partially answer the question how many contours/silhouettes are needed to determine a surface uniquely. While the second part will be devoted to a rational ruled surfaces - which form a special class of surfaces with rational contour curves.