Abstract. Laman graphs model planar frameworks that are rigid for a general choice of distances between the vertices. There are finitely many ways, up to isometries, to realize a Laman graph in the plane. Such realizations can be seen as solutions of systems of quadratic equations prescribing the distances between pairs of points. Using ideas from algebraic and tropical geometry, we provide a recursive formula for the number of complex solutions of such systems.
First we solve the problem of finding minimal degree families on toric
surfaces by reducing it to lattice geometry. Then we describe how to find
minimal degree families on, more generally, rational complex projective
surfaces
We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere, then the set of minimal degree rational curves that cover the surface is either empty or of dimension at most two. Moreover, if these curves are of minimal degree over the real numbers, but not over the complex numbers, then almost all the curves are smooth. Our methods lead to an algorithm that takes as input a real surface parametrization and outputs all real families of rational curves of lowest possible degree that cover the image surface.
We define a cone curve to be a reduced sextic space curve which lies on a quadric cone and does not go through the vertex. We classify families of bitangent planes of cone curves. The methods we apply can be used for any space curve with ADE singularities, though in this paper we concentrate on cone curves.An embedded complex projective surface which is adjoint to a degree one weak Del Pezzo surface contains families of minimal degree rational curves, which cannot be defined by the fibers of a map. Such families are called minimal non-fibration families. Families of bitangent planes of cone curves correspond to minimal nonfibration families. The main motivation of this paper is to classify minimal nonfibration families.We present algorithms wich compute all bitangent families of a given cone curve and their geometric genera. We consider cone curves to be equivalent if they have the same singularity configuration. For each equivalence class of cone curves we determine the possible number of bitangent families and the number of rational bitangent families. Finally we compute an example of a minimal non-fibration family on an embedded weak degree one Del Pezzo surface.
We state a relation between two families of lines that cover a quadric surface in the Study quadric and two families of circles that cover a Darboux cyclide.
In this paper, we consider the classification of singularities [P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction. I, II, III, Proc. Camb. Philos. Soc. 30 (1934) 453-491] and real structures [C. T. C. Wall, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math. 1987(375/376) (1987) 47-66, ISSN 0075-4102] of weak Del Pezzo surfaces from an algorithmic point of view. It is wellknown that the singularities of weak Del Pezzo surfaces correspond to root subsystems.We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.
We present algorithms for reconstructing, up to unavoidable projective automorphisms, surfaces with ordinary singularities in three dimensional space starting from their silhouette, or "apparent contour" -namely the branching locus of a projection on the plane -and the projection of their singular locus.
Laman graphs model planar frameworks which are rigid for a general choice of
distances between the vertices. There are finitely many ways, up to isometries,
to realize a Laman graph in the plane. In a recent paper we provide a recursion
formula for this number of realizations using ideas from algebraic and tropical
geometry. Here, we present a concise summary of this result focusing on the
main ideas and the combinatorial point of view.Comment: Extended abstract; the long version is arxiv:1701.0550
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