Quartic Surface Research Articles

A Minimal Solution to the Generalised 3-Point Pose Problem

It is a well known classical result that given the image projections of three known world points it is possible to solve for the pose of a calibrated perspective camera to up to four pairs of solutions. We solve the Generalised problem where the camera is allowed to sample rays in some arbitrary but known fashion and is not assumed to perform a central perspective projection. That is, given three back-projected rays that emanate from a camera or multi-camera rig in an arbitrary but known fashion, we seek the possible poses of the camera such that the three rays meet three known world points. We show that the Generalised problem has up to eight solutions that can be found as the intersections between a circle and a ruled quartic surface. A minimal and efficient constructive numerical algorithm is given to find the solutions. The algorithm derives an octic polynomial whose roots correspond to the solutions. In the classical case, when the three rays are concurrent, the ruled quartic surface and the circle possess a reflection symmetry such that their intersections come in symmetric pairs. This manifests itself in that the odd order terms of the octic polynomial vanish. As a result, the up to four pairs of solutions can be found in closed form. The proposed algorithm can be used to solve for the pose of any type of calibrated camera or camera rig. The intended use for the algorithm is in a hypothesise-and-test architecture.

A family of SCFTs hosting all ``very attractive'' relatives of the (2)4Gepner model

This work gives a manual for constructing superconformal field theories associated to a family of smooth K3 surfaces. A direct method is not known, but a combination of orbifold techniques with a non-classical duality turns out to yield such models. A four parameter family of superconformal field theories associated to certain quartic K3 surfaces in CP^3 is obtained, four of whose complex structure parameters give the parameters within superconformal field theory. Standard orbifold techniques are used to construct these models, so on the level of superconformal field theory they are already well understood. All "very attractive" K3 surfaces belong to the family of quartics underlying these theories, that is all quartic hypersurfaces in CP^3 with maximal Picard number whose defining polynomial is given by the sum of two polynomials in two variables. A particular member of the family is the (2)^4 Gepner model, such that these theories can be viewed as complex structure deformations of (2)^4 in its geometric interpretation on the Fermat quartic.

On factorial double solids with simple double points

We study when double covers of P 3 ramified along nodal surfaces are not Q -factorial. In particular, we describe all the Q -factorial double covers of P 3 ramified along quartic surfaces with at most seven simple double points and sextic surfaces with at most 16 simple double points.

Spinning gas clouds with precession: non-degenerate cases

We extend here our earlier treatment (Gaffet B 2003 J. Phys. A: Math. Gen. 36 5211, Gaffet B 2005 Phil. Trans. R. Soc. A accepted) of the evolution of spinning gas clouds with precession, to the non-degenerate cases where the last integral of motion L6 is non-vanishing. The Liouville torus is still found to be transformable into a quartic surface, on which there are 15 conic point singularities. The differential system governing the evolution is integrable by quadratures: there is an integrating factor, and the independent variable u (which is the thermasy of the cloud) admits an exact differential formulation on the Liouville torus as well. This integrable system is conjectured to have the Painleve property with respect to the independent variable u, as in the degenerate cases.

Brauer groups of diagonal quartic surfaces

We describe explicit methods of exhibiting elements of the Brauer groups of diagonal quartic surfaces. Using these methods, we compute the algebraic Brauer–Manin obstruction in two contrasting examples. In the second example, the obstruction is found to be trivial but a computer search reveals no points of small height on the surface.

Nonbirefringence conditions for spacetime

Within the axiomatic premetric approach to classical electrodynamics, we derive under which covariant conditions the quartic Fresnel surface represents a unique light cone without birefringence in vacuum.

Complex curves of genus three, Kummer surfaces and Quillen metrics

Let C be a smooth, complex, projective curve of genus 3. By choosing an unramified double covering of C, the Abel-Prym map yields an embedding of C into a Kummer surface K when C is non-hyperelliptic. We compute the Quillen metric on Open image in new window the determinant of the cohomologies of Open image in new window with respect to the metric on C induced from the flat Kahler metric on K. For the computation of the Quillen metric, we show the exact self-duality of the Heisenberg-invariant Kummer's quartic surfaces.

A characterization of the Fermat quartic K3 surface by means of finite symmetries

We characterize the Fermat quartic K3 surface, among all K3 surfaces, by means of its finite group symmetries.

Possible skewon effects on light propagation

We start from a local and linear space-time relation between the electromagnetic excitation and the field strength. Then we study the generally covariant Fresnel surfaces for light rays and light waves. The metric and the connection of space-time are left unspecified. Accordingly, our framework is ideally suited for a search of possible violations of the Lorentz symmetry in the photon sector of the extended standard model. We discuss how the skewon part of the constitutive tensor, if suitably parametrized, influences the Fresnel surfaces and disturbs the light cones of vacuum electrodynamics. Conditions are specified that yield the reduction of the original quartic Fresnel surface to the double light cone structure (birefringence) and to the single light cone. Qualitatively, the effects of the real skewon field can be compared to those in absorbing material media. In contrast, the imaginary skewon field can be interpreted in terms of nonabsorbing media with natural optical activity and Faraday effects. The astrophysical data on gamma-ray bursts are used for deriving an upper limit for the magnitude of the skewon field.

Degeneration of linear systems through fat points on $K3$ surfaces

In this paper we introduce a technique to degenerate K3 surfaces and linear systems through fat points in general position on K3 surfaces. Using this degeneration we show that on generic K3 surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through n = 4 u 9 w fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through n = 4 u 9 w fat points in general position on a general quartic surface in P 3 .

Riemannian light cone from vanishing birefringence in premetric vacuum electrodynamics

We consider premetric electrodynamics with a local and linear constitutive law for the vacuum. Within this framework, we find quartic Fresnel wave surfaces for the propagation of light. If we require vanishing birefringence in vacuum, then a Riemannian light cone is implied. No proper Finslerian structure can occur. This is generalized to dynamical equations of any order.

Refinable bivariate quartic $C^2$-splines for multi-level data representation and surface display

In this paper, a second-order Hermite basis of the space of C 2 -quartic splines on the six-directional mesh is constructed and the refinable mask of the basis functions is derived. In addition, the extra parameters of this basis are modified to extend the Hermite interpolating property at the integer lattices by including Lagrange interpolation at the half integers as well. We also formulate a compactly supported super function in terms of the basis functions to facilitate the construction of quasi-interpolants to achieve the highest (i.e., fifth) order of approximation in an efficient way. Due to the small (minimum) support of the basis functions, the refinable mask immediately yields (up to) four-point matrix-valued coefficient stencils of a vector subdivision scheme for efficient display of C 2 -quartic spline surfaces. Finally, this vector subdivision approach is further modified to reduce the size of the coefficient stencils to two-point templates while maintaining the second-order Hermite interpolating property.

Classification of normal quartic surfaces with irrational singularities

If a normal quartic surface admits a singular point that is not a rational double point, then the surface is determined by the triplet (M,D,E) consisting of the minimal desingularization M, the pullback D of a general hyperplane section, and a nonzero effective anti-canonical divisor E of M. Geometric constructions of all the possible triplets (M,D,E) are given.

Rationality of an Enriques-Fano threefold of genus five

We prove the rationality of a non-Gorenstein Fano threefold of Fano index one and degree eight having terminal cyclic quotient singularities and Picard group . This threefold can be described as the quotient of a double covering of ramified in a smooth quartic surface by an involution fixing eight different points.

Real quartic surfaces containing 16 skew lines

It is well known that there is an open three‐dimensional subvariety Ms of the Grassmannian of lines in ℙ3 which parametrizes smooth irreducible complex surfaces of degree 4 which are Heisenberg invariant, and each quartic contains 32 lines but only 16 skew lines, being determined by its configuration of lines, are called a double 16. We consider here the problem of visualizing in a computer the real Heisenberg invariant quartic surface and the real double 16. We construct a family of points l ∈ Ms parametrized by a two‐dimensional semialgebraic variety such that under a change of coordinates of l into its Plüecker, coordinates transform into the real coordinates for a line L in ℙ3, which is then used to construct a program in Maple 7. The program allows us to draw the quartic surface and the set of transversal lines to L. Additionally, we include a table of a group of examples. For each test example we specify a parameter, the viewing angle of the image, compilation time, and other visual properties of the real surface and its real double 16. We include at the end of the paper an example showing the surface containing the double 16.

Infrared intensities of furan, pyrrole and thiophene: beyond the double harmonic approximation

Infrared intensities of the fundamental, overtone and combination transitions in furan, pyrrole and thiophene have been calculated using the variational normal coordinate code MULTIMODE. We use pure vibrational wavefunctions, and quartic force fields and cubic dipole moment vector surfaces, generated by density functional theory. The results are compared graphically with second-order perturbation calculations and with relative intensities from experiment for furan and pyrrole.

The internal coordinate path Hamiltonian; application to methanol and malonaldehyde

The internal coordinate path Hamiltonian is introduced for the study of the vibrations of molecules which have one large amplitude motion. The Hamiltonian is represented in terms of a one path coordinate and 3N—7 normal coordinates. The variational method is used to solve the Schrödinger equation. The molecules studied are methanol and malonaldehyde. For methanol the internal coordinate is a dihedral angle, for malonaldehyde it is the difference in the distances between the migrating hydrogen and the neighbouring oxygen atoms. For methanol there is little coupling between the path and the normal coordinates and so no complications were encountered in the calculations which used harmonic surfaces generated by density functional and Møller—Plesset theory. Fundamental frequencies were predicted. Malonaldehyde is a different story. There is substantial coupling between the path coordinate and several of the normal coordinates. This introduces many complications: an anharmonic surface is essential and large variational configuration interaction calculations are essential for convergence. Furthermore, because the Coriolis terms require the evaluation of derivatives of both the nuclear coordinates and the normal coordinate eigenvectors along the path, great care must be taken with these numerical procedures. B3LYP predicts too low a transition state which overemphasizes the large Coriolis terms near the transition state. This may be one of the reasons why our fundamental vibrations are in poor agreement with observation. It is most encouraging that the tunnelling splitting is 58 cm−1 (obs. 21.56 cm−1), obtained with our quartic density functional surface.

The multivariate quartic NURBS surfaces

In this paper, we construct a kind of multivariate quartic nonuniform rational B-spline (NURBS) surfaces by using bivariate quartic B-spline bases in the multivariate spline space S 4 2(△ mn (2)), and discuss some properties of this kind of NURBS surfaces with multiple knots on the type-2 triangulation. Compared with the bicubic (rational) Bézier surfaces, the new multivariate NURBS surfaces on the knot vectors of the form U={0,0,0,0,1,1,1,1} and V={0,0,0,0,1,1,1,1} have similar properties at the four edges of the surfaces. Several examples show that our multivariate B-spline surfaces are better than the corresponding bicubic Bézier surfaces.

Calculation of spectroscopic parameters and vibrational overtones of methanol

A high-level quartic ab initio potential energy surface of methanol has been used to calculate spectroscopic constants of the 12CH3OH molecule. These include coefficients of quartic anharmonic resonance terms, Darling-Dennison constants, for stretching states. A model expressed in terms of dimensionless normal coordinates has been employed in the calculation of O—H and C—H stretching vibrational states in high-overtone regions. Both cubic Fermi and quartic Darling-Dennison anharmonic coupling terms have been included in the model in order to take into account strong resonances between different states. The nonlinear least-squares method has been used to optimize some of the model parameters employing experimental term values of 12CH3OH as data. Vibrational assignments are suggested for the first C—H stretching overtone region.

The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

Let X be a smooth projective curve of genus g⩾2 defined over an algebraically closed field k of characteristic p>0. Let M X ( r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius map F : X→X 1 induces by pull-back a rational map V : M X 1 (r)→ M X(r) . We determine the equations of V in the following two cases (1) ( g, r, p)=(2,2,2) and X nonordinary with Hasse–Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) ( g, r, p)=(2,2,3). We also show the existence of base points of V, i.e., semi-stable bundles E such that F ∗E is not semi-stable, for any triple ( g, r, p).