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A set S of points in R n is called a rationally parameterizable hypersurface if there is vector function σ : R n - 1 → R n having as components rational functions defined on some common domain D such that S = { σ (t) : t ∈ D } . A generalized n-dimensional polytope in R n is a union of a finite number of convex n-dimensional polytopes in R n . The Fourier–Laplace transform of such a generalized polytope P in R n is defined by F P (z) = ∫ P e z · x dx . Let γ be a fixed nonzero complex number. We prove that F P 1 (γ σ (t)) = F P 2 (γ σ (t)) for all t ∈ O implies P 1 = P 2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of P . Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres. [ABSTRACT FROM AUTHOR]

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory. [ABSTRACT FROM AUTHOR]

Let S be a set of n points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of S is less than K n 3 for some K = o (n 1 / 7) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size n of an elliptic curve that span less than n 3 / 6 solids containing exactly four points of S . [ABSTRACT FROM AUTHOR]

We consider the X-ray transform in a projective space over a finite field. It is well known (after Bolker) that this transform is injective. We formulate an analog of Gelfand's admissibility problem for the Radon transform, which asks for a classification of all minimal sets of lines for which the restricted Radon transform is injective. The solution involves doubly ruled quadric surfaces. [ABSTRACT FROM AUTHOR]

Let S be a set of n points in real three-dimensional space, no three collinear and not all co-planar. We prove that if the number of planes incident with exactly three points of S is less than Kn2 for some K=o(n1/7) then, for n sufficiently large, all but at most O(K) points of S are contained in the intersection of two quadrics. Furthermore, we prove that there is a constant c such that if the number of planes incident with exactly three points of S is less than 12n2-cn then, for n sufficiently large, S is either a regular prism, a regular anti-prism, a regular prism with a point removed or a regular anti-prism with a point removed. As a corollary to the main result, we deduce the following theorem. Let S be a set of n points in the real plane. If the number of circles incident with exactly three points of S is less than Kn2 for some K=o(n1/7) then, for n sufficiently large, all but at most O(K) points of S are contained in a curve of degree at most four. [ABSTRACT FROM AUTHOR]

Let P be a set of m points and L a set of n lines in $${\mathbb {R}}^4,$$ such that the points of P lie on an algebraic three-dimensional variety of degree $$D$$ that does not contain hyperplane or quadric components (a quadric is an algebraic variety of degree two), and no 2-flat contains more than s lines of L. We show that the number of incidences between P and L is for some absolute constant of proportionality. This significantly improves the bound of the authors (Sharir, Solomon, Incidences between points and lines in $${\mathbb {R}}^4.$$ Discrete Comput Geom 57(3), 702-756, 2017), for arbitrary sets of points and lines in $${\mathbb {R}}^4,$$ when $$D$$ is not too large. Moreover, when $$D$$ and s are constant, we get a linear bound. The same bound holds when the three-dimensional surface is embedded in any higher-dimensional space. The bound extends (with a slight deterioration, when $$D$$ is large) to the complex field too. For a complex three-dimensional variety, of degree $$D,$$ embedded in $${\mathbb {C}}^4$$ (or in any higher-dimensional $${\mathbb {C}}^d$$ ), under the same assumptions as above, we have For the proof of these bounds, we revisit certain parts of [36], combined with the following new incidence bound, for which we present a direct and fairly simple proof. Going back to the real case, let P be a set of m points and L a set of n lines in $${\mathbb {R}}^d,$$ for $$d\ge 3,$$ which lie in a common two-dimensional algebraic surface of degree $$D$$ that does not contain any 2-flat, so that no 2-flat contains more than s lines of L (here we require that the lines of L also be contained in the surface). Then the number of incidences between P and L is When $$d=3,$$ this improves the bound of Guth and Katz (On the Erdős distinct distances problem in the plane. Ann Math 181(1), 155-190, 2015) for this special case, when $$D\ll n^{1/2}.$$ Moreover, the bound does not involve the term O( nD). This term arises in most standard approaches, and its removal is a significant aspect of our result. Again, the bound is linear when $$D= O(1).$$ This bound too extends (with a slight deterioration, when $$D$$ is large) to the complex field. For a complex two-dimensional variety, of degree $$D,$$ when the ambient space is $${\mathbb {C}}^3$$ (or any higher-dimensional $${\mathbb {C}}^d$$ ), under the same assumptions as above, we have These new incidence bounds are among the very few bounds, known so far, that hold over the complex field. The bound for two-dimensional (resp., three-dimensional) varieties coincides with the bound in the real case when $$D= O(m^{1/3})$$ (resp., $$D= O(m^{1/6})$$ ). [ABSTRACT FROM AUTHOR]

The classical Demoulin transformation is examined in the context of discrete differential geometry. We show that iterative application of the Demoulin transformation to a seed projective minimal surface generates a $${\mathbb {Z}}^2$$ lattice of projective minimal surfaces. Known and novel geometric properties of these Demoulin lattices are discussed and used to motivate the notion of lattice Lie quadrics and associated discrete envelopes and the definition of the class of discrete projective minimal and Q-surfaces (PMQ-surfaces). We demonstrate that the even and odd Demoulin sublattices encode a two-parameter family of pairs of discrete PMQ-surfaces with the property that one discrete PMQ-surface constitute an envelope of the lattice Lie quadrics associated with the other. [ABSTRACT FROM AUTHOR]

We consider the Betti numbers of an intersection of k random quadrics in $$\mathbb {R}\text {P}^n$$ . Sampling the quadrics independently from the Kostlan ensemble, as $$n \rightarrow \infty $$ we show that for each $$i\ge 0$$ the expected ith Betti number satisfies In other words, each fixed Betti number of X is asymptotically expected to be one; in fact as long as $$i=i(n)$$ is sufficiently bounded away from n / 2 the above rate of convergence is uniform (and in this range Betti numbers concentrate to their expected value). For the special case $$k=2$$ we study the expectation of the sum of all Betti numbers of X. It was recently shown (Lerario, in Proc. Am. Math. Soc. 143:3239-3251, ) that this expected sum equals $$n+o(n)$$ ; here we sharpen this asymptotic, showing that (the term $$\tfrac{2}{\sqrt{\pi }}n^{1/2}$$ comes from contributions of middle Betti numbers). The proofs are based on a combination of techniques from random matrix theory and spectral sequences. In particular (1) is based on a reduction that requires an average count of the number of singular quadrics in a random pencil; this count turns out to be related to the derivative at zero of the gap probability $$f_{\beta , n}$$ in finite Gaussian $$\beta $$ -ensembles ( $$\beta =1,2,4$$ ). We provide also new computations for this quantity and as n goes to infinity: [ABSTRACT FROM AUTHOR]

We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric. [ABSTRACT FROM AUTHOR]

We give a complete description of the Voronoi diagram, in ℝ3, of three lines in general position, that is, that are pairwise skew and not all parallel to a common plane. In particular, we show that the topology of the Voronoi diagram is invariant for three such lines. The trisector consists of four unbounded branches of either a nonsingular quartic or of a nonsingular cubic and a line that do not intersect in real space. Each cell of dimension two consists of two connected components on a hyperbolic paraboloid that are bounded, respectively, by three and one of the branches of the trisector. We introduce a proof technique which relies heavily upon modern tools of computer algebra and is of interest in its own right. This characterization yields some fundamental properties of the Voronoi diagram of three lines. In particular, we present linear semi-algebraic tests for separating the two connected components of each two-dimensional Voronoi cell and for separating the four connected components of the trisector. This enables us to answer queries of the form, given a point, determine in which connected component of which cell it lies. We also show that the arcs of the trisector are monotonic in some direction. These properties imply that points on the trisector of three lines can be sorted along each branch using only linear semi-algebraic tests. [ABSTRACT FROM AUTHOR]

Let P be a (non-necessarily convex) embedded polyhedron in R 3, with its vertices on the boundary of an ellipsoid. Suppose that the interior of $P$ can be decomposed into convex polytopes without adding any vertex. Then P is infinitesimally rigid. More generally, let P be a polyhedron bounding a domain which is the union of polytopes C 1, . . ., C n with disjoint interiors, whose vertices are the vertices of P. Suppose that there exists an ellipsoid which contains no vertex of P but intersects all the edges of the C i. Then P is infinitesimally rigid. The proof is based on some geometric properties of hyperideal hyperbolic polyhedra. [ABSTRACT FROM AUTHOR]

We solve the following geometric problem, which arises in several {\mbox three-dimensional} applications in computational geometry: For which arrangements of two lines and two spheres in ${\Bbb R}^3$ are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics. Replacing the spheres in ${\bf R}^3$ by quadrics in projective space ${\Bbb P}^3$, and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the two lines and two quadrics have infinitely many transversals and tangents: in the nine-dimensional projective space ${\Bbb P}^9$ of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety. [ABSTRACT FROM AUTHOR]