Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

In the present paper we study $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the ‘surface of a geodesic triangle’. Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ tetrahedron. Moreover, we generalize the famous Menelaus’s and Ceva’s theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries described by E. Molnár in [6].

Geometric correction for satellite image of Mosul city using 3D methods

The important preprocessing techniques for remote sensing data and geometrical alteration is the geometric correction. In this paper ,it covers two models which are used in to three dimensions the physical model and the projective Transformation for SPOT 2 to determine the geometric correction of study region of Mosul city. The ground control points are located on SPOT 2 satellite image. These models are producing the X residual, Y residual and root mean square error (RMSE) and then comparing the effects of two models. The X residual for SPOT 2 physical model is lower than X residual for projective transformation at 0.1420, while the Y residual for physical model has higher comparing with Y residual for projective transformation at 0.1143and Total RMSE for SPOT 2 physical model is lower than Total RMSE for projective Transformation at 0.1823. The physical model is superior to Projective model, so that, it is highly recommended to be used for very precise application and not to be replaced by these non-physical models. For the physical model, it is clearly that, the altitude of the Ground Control Points (or Check Points) does not effect on its individual RMSE, when using Projective model the RMSE for high altitude was high ,where at low altitude it was low too(extreme relationship).

Arrow categories of monoidal model categories

We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This answers a question posed by Mark Hovey, in the course of his work on Smith ideals. As a corollary, we prove that the projective model structure in cubical homotopy theory is a monoidal model structure. As illustrations we include numerous examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, pro-categories, and topological spaces.

Lattice Coverings by Congruent Translation Balls Using Translation-like Bisector Surfaces in Nil Geometry

In this paper we study the Nil geometry that is one of the eight homogeneous Thurston 3-geometries. We determine the equation of the translation-like bisector surface of any two points. We prove, that the isosceles property of a translation triangle is not equivalent to two angles of the triangle being equal and that the triangle inequalities do not remain valid for translation triangles in general. We develop a method to determine the centre and the radius of the circumscribed translation sphere of a given translation tetrahedron. A further aim of this paper is to study lattice-like coverings with congruent translation balls in Nil space. We introduce the notion of the density of the considered coverings and give upper estimate to it using the radius and the volume of the circumscribed translation sphere of a given translation tetrahedron. The found minimal upper bound density of the translation ball coverings $\Delta \approx 1.42783$. In our work we will use for computations and visualizations the projective model of Nil described by E. Molnár in [6].

Cohn’s Platonic Model and the Regular Irregularities of Recent Popular Multimedia

Richard Cohn’s Platonic model of funky rhythms can be converted into a property that closely matches the asymmetrical temporal successions that most frequently occur in recent popular English-language multimedia. In the case of quintuple and septuple meters, this property also closely matches successions that most frequently occur in other forms of popular music as well. The movies from which the evidence for this claim comes range fromThe Magnificent Sevenof 1960 toKung Fu Panda 3of 2016. Some short but close analyses of filmic scenes demonstrate this property's effectiveness as a hermeneutical tool. A related property of “near realization” recruits Christopher Hasty’s projective model to explain this stylistic bias, and connects these asymmetrical successions with another seemingly dissimilar class of syncopated rhythms favored in popular music.

FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.