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Niels Lubbes

Publications and source records attributed to Niels Lubbes.

At least 19 recordsLinked to original sources
arXiv2026

Bivariate quaternionic factorizations and surfaces that decompose into two circles

We present an algebraic and geometric condition for bivariate quaternionic polynomials of arbitrary bidegree to have a univariate linear left or right factor. We apply this quaternionic factorization theorem to the bidegree (1,1) case and recover a classical theorem of Clifford in elliptic geometry. By applying to the bidegree (2,2) case, we obtain decompositions into two circles of celestial surfaces, namely surfaces in the 3-dimensional sphere that contain two circles through a general point. This results in an alternative proof and refinement for a theorem by Skopenkov and Krasauskas from 2019, which states that a non-quartic celestial surface is Möbius equivalent to either the pointwise product of circles in the unit-quaternions, or an inverse stereographic projection of the pointwise sum of circles in Euclidean space. Our proposed method extends this decomposition result to the quartic case and we show that surfaces are, up to Möbius equivalence and stereographic projections, not both a sum and product of circles.

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arXiv2025

Shapes of surfaces that contain a great and a small circle through each point

We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the pointwise product of circles in the unit quaternions or contain five concurrent circles. We classify the real singular loci of such surfaces and characterize how circles in the surface meet the self-intersection locus.

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arXiv2025

Irreducible components of sets of points in the plane that satisfy distance conditions

For a given graph whose edges are labeled with general real numbers, we consider the set of functions from the vertex set into the Euclidean plane such that the distance between the images of neighbouring vertices is equal to the corresponding edge label. This set of functions can be expressed as the zero set of quadratic polynomials and our main result characterizes the number of complex irreducible components of this zero set in terms of combinatorial properties of the graph. In case the complex components are three-dimensional, then the graph is minimally rigid and the component number is a well-known invariant from rigidity theory. If the components are four-dimensional, then they correspond to one-dimensional coupler curves of flexible planar mechanisms. As an application, we characterize the degree of irreducible components of such coupler curves combinatorially.

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arXiv2024

Calibrating Figures

It is known that a camera can be calibrated using three pictures of either squares, spheres, or surfaces of revolution. We give a new method to calibrate a camera with the picture of a single torus.

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arXiv2024

Translational and great Darboux cyclides

A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2, 3, 4, or 5 circles through a general point. A surface in a unit-sphere of any dimension that contains 2 great circles through a general point contains either 4, 5, 6, or infinitely many circles through a general point. These are some corollaries from our classification of translational and great Darboux cyclides. We use the combinatorics associated to the set of low degree curves on such surfaces modulo numerical equivalence.

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arXiv2023

Calligraphs and sphere realizations

We introduce a recursive procedure for computing the number of realizations of a minimally rigid graph on the sphere up to rotations. We accomplish this by combining two ingredients. The first is a framework that allows us to think of such realizations as of elements of a moduli space of stable rational curves with marked points. The second is the idea of splitting a minimally rigid graph into two subgraphs, called calligraphs, that admit one degree of freedom and that share only a single edge and a further vertex. This idea has been recently employed for realizations of graphs in the plane up to isometries. The key result is that we can associate to a calligraph a triple of natural numbers with a special property: whenever a minimally rigid graph is split into two calligraphs, the number of realizations of the former equals the product of the two triples of the latter, where this product is specified by a fixed quadratic form. These triples and quadratic form codify the fact that we express realizations as intersections of two curves on the blowup of a sphere along two pairs of complex conjugate points.

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arXiv2023

Coupler curves of moving graphs and counting realizations of rigid graphs

A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is called its coupler curve. To each calligraph we uniquely assign a vector consisting of three integers. This vector bounds the degrees and geometric genera of irreducible components of the coupler curve. A graph, that up to rotations and translations admits finitely many, but at least two, realizations into the plane for almost all edge length assignments, is a union of two calligraphs. We show that this number of realizations is equal to a certain inner product of the vectors associated to these two calligraphs. As an application we obtain an improved algorithm for counting numbers of realizations, and by counting realizations we characterize invariants of coupler curves.

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arXiv2021

Projective isomorphisms between rational surfaces

We present a method for computing projective isomorphisms between rational surfaces that are given in terms of their parametrizations. The main idea is to reduce the computation of such projective isomorphisms to five base cases by modifying the parametric maps such that the components of the resulting maps have lower degree. Our method can be used to compute affine, Euclidean and Möbius isomorphisms between surfaces.

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arXiv2021

Reconstruction of rational ruled surfaces from their silhouettes

We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the `apparent contour' of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to $\mathbb{p}^3$ of rational normal scrolls. In the first case, we use the fact that every such surface is the projection of the tangent developable of a rational normal curve, while in the second we start by reconstructing the rational normal scroll. In both instances we then reconstruct the correct projection to $\mathbb{p}^3$ of these surfaces by exploiting the information contained in the singularities of the apparent contour.

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arXiv2021

Reconstruction of surfaces with ordinary singularities from their silhouettes

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.

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arXiv2021

Surfaces that are covered by two pencils of circles

We list up to Möbius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric characteristics are encoded in the Néron-Severi lattices of such surfaces and is of potential interest to geometric modelers and architects. As an application we confirm Blum's conjecture in higher dimensional space and we address the Blaschke-Bol problem by classifying surfaces that are covered by hexagonal webs of circles. In particular, we find new examples of such webs that cannot be embedded in 3-dimensional space.

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arXiv2020

Möbius automorphisms of surfaces with many circles

We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a Möbius automorphism group of dimension at least two. Our theorem generalizes the classical classification of Dupin cyclides.

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arXiv2019

Kinematic generation of Darboux cyclides

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.

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arXiv2018

Minimal degree rational curves on real 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.

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arXiv2018

Computing basepoints of linear series in the plane

We present an algorithm for detecting basepoints of linear series of curves in the plane. Moreover, we give an algorithm for constructing a linear series of curves in the plane for given basepoints. The underlying method of these algorithms is the classical procedure of blowing up points in the plane. We motivate the algorithmic version of this procedure with several applications.

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