LXVII. Note on the nodal curve of the developable derived from the quartic equation (a, b, c, d, e)(t, 1)4=0

The perspective 3-point (P3P) problem, also known as pose estimation, has its origins in camera calibration and is of importance in many fields: for example, computer animation, automation, image analysis, and robotics. One possibility is to formulate it mathematically in terms of finding the solution to a quartic equation. However, there is yet no quantitative knowledge as to how control-point spacing affects the solution structure—in particular, the multisolution phenomenon. Here, we consider this problem through an algebraic analysis of the quartic’s coefficients and its discriminant and find that there are significant variations in the likelihood of two or four solutions, depending on how the spacing is chosen. The analysis indicates that although it is never possible to remove the occurrence of the four-solution case completely, it could be possible to choose spacings that would maximize the occurrence of two real solutions. Moreover, control-point spacing is found to impact significantly on the reality conditions for the solution of the quartic equation.

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EXTENSIONS AND RANK-2 VECTOR BUNDLES ON IRREDUCIBLE NODAL CURVES

International Journal of Mathematics ◽

10.1142/s0129167x05003284 ◽

2005 ◽

Vol 16 (10) ◽

pp. 1081-1118

Author(s):

D. ARCARA

Keyword(s):

Moduli Space ◽

Vector Bundles ◽

Rational Map ◽

Nodal Curves ◽

Nodal Curve ◽

Stable Vector Bundles ◽

Rank 2 ◽

Rank 2 Vector Bundles

We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use the natural rational map [Formula: see text] defined by [Formula: see text] to study a compactification [Formula: see text] of the moduli space [Formula: see text] of semi-stable vector bundles of rank 2 and determinant L on C. In particular, we resolve the indeterminancy of ϕL in the case deg L = 3,4 via a sequence of three blow-ups with smooth centers.

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On the Determination of the Properties of the Nodal Curve of a Unicursal Ruled Surface

American Journal of Mathematics ◽

10.2307/2370055 ◽

1906 ◽

Vol 28 (1) ◽

pp. 43

Author(s):

Charles H. Sisam

Keyword(s):

Ruled Surface ◽

Nodal Curve

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Discussions: A Solution of the Quartic Equation

American Mathematical Monthly ◽

10.2307/2298174 ◽

1928 ◽

Vol 35 (10) ◽

pp. 558

Author(s):

Raymond Garver

Keyword(s):

Quartic Equation

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Reflection moveout approximation for a P-SV wave in a moderately anisotropic homogeneous vertical transverse isotropic layer

Geophysics ◽

10.1190/geo2018-0474.1 ◽

2019 ◽

Vol 84 (2) ◽

pp. C75-C83 ◽

Cited By ~ 2

Author(s):

Véronique Farra ◽

Ivan Pšenčík

Keyword(s):

Numerical Solution ◽

Taylor Series Expansion ◽

Pure Mode ◽

Isotropic Layer ◽

Weak Anisotropy ◽

Reflected Waves ◽

Quartic Equation ◽

S Waves ◽

Conversion Point ◽

Quartic Term

A description of the subsurface is incomplete without the use of S-waves. Use of converted waves is one way to involve S-waves. We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry. For its derivation, we use the weak-anisotropy approximation; i.e., we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. Traveltimes are calculated along reference rays of converted reflected waves in a reference isotropic medium. This requires the determination of the point of reflection (the conversion point) of the reference ray, at which the conversion occurs. This can be done either by a numerical solution of a quartic equation or by using a simple approximate solution. Presented tests indicate that the accuracy of the proposed moveout formula is comparable with the accuracy of formulas derived in a weak-anisotropy approximation for pure-mode reflected waves. Specifically, the tests indicate that the maximum relative traveltime errors are well below 1% for models with P- and SV-wave anisotropy of approximately 10% and less than 2% for models with P- and SV-wave anisotropy of 25% and 12%, respectively. For isotropic media, the use of the conversion point obtained by numerical solution of the quartic equation yields exact results. The approximate moveout formula is used for the derivation of approximate expressions for the two-way zero-offset traveltime, the normal moveout velocity and the quartic term of the Taylor series expansion of the squared traveltime.

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Plane-wave propagation in media with electromagnetic anisotropy

European Journal of Physics ◽

10.1088/1361-6404/ac41ad ◽

2021 ◽

Author(s):

David Romero-Abad ◽

Jose Pedro Reyes Portales ◽

Roberto Suárez-Córdova

Keyword(s):

Plane Wave ◽

Electromagnetic Waves ◽

Wave Equations ◽

Refraction Index ◽

Pedagogical Approach ◽

Undergraduate Level ◽

Quartic Equation ◽

Principal Axes ◽

The Subject ◽

Magnetic Tensors

Abstract The propagation of electromagnetic waves in a medium with electrical and magnetic anisotropy is a subject that is not usually handled in conventional optics and electromagnetism books. During this work, we try to give a pedagogical approach to the subject, using tools that are accessible to an average physics student. In this article, we obtain the Fresnel relation in a media with electromagnetic anisotropy, which corresponds to a quartic equation in the refraction index, assuming only that the principal axes of the electric and magnetic tensors coincide. Additionally, we find the geometric location related to the different situations the discriminant of the quartic equation provides. In order to illustrate our findings, we determine the refractive index together with the plane wave equations for certain values of the parameters that meet the established conditions. The target readers of the paper are students pursuing physics at the intermediate undergraduate level.

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