Absolute Planes Research Articles

Plane Fitting in 3D Reconstruction to Preserve Smooth Homogeneous Surfaces

Photogrammetric techniques for weakly-textured surfaces without sufficient information about the R (red), G (green) and B (blue) primary colors of light are challenging. Considering that most urban or indoor object surfaces follow simple geometric shapes, a novel method for reconstructing smooth homogeneous planar surfaces based on MVS (Multi-View Stereo) is proposed. The idea behind it is to extract enough features for the image description, and to refine the dense points generated by the depth values of pixels with plane fitting, to favor the alignment of the surface to the detected planes. The SIFT (Scale Invariant Feature Transform) and AKAZE (Accelerated-KAZE) feature extraction algorithms are combined to ensure robustness and help retrieve connections in small samples. The smoothness of the enclosed watertight Poisson surface can be enhanced by enforcing the 3D points to be projected onto the absolute planes detected by a RANSAC (Random Sample Consensus)-based approach. Experimental evaluations of both cloud-to-mesh comparisons in the per-vertex distances with the ground truth models and visual comparisons with a popular mesh filtering based post-processing method indicate that the proposed method can considerably retain the integrity and smoothness of the reconstruction results. Combined with other primitive fittings, the reconstruction extent of homogeneous surfaces can be further extended, serving as primitive models for 3D building reconstruction, and providing guidance for future works in photogrammetry and 3D surface reconstruction.

The elementary Archimedean axiom in absolute geometry

Absolute planes in which the elementary Archimedean axiom holds satisfy Aristotle’s axiom. Absolute planes satisfying both the elementary Archimedean axiom and Bachmann’s Lotschnittaxiom are Euclidean. The Corollary to Aristotle’s axiom is equivalent to Aristotle’s axiom.

Evaluation of a fast method of EPID-based dosimetry for IMRT and Comparison with 3D EPID-based dosimetry system using conventional two- and three-dimensional detectors for VMAT

Introduction: Electronic portal imaging devices (EPIDs) could potentially be useful for intensity-modulated radiation therapy (IMRT) and VMAT QA. The data density, high resolution, large active area, and efficiency of the MV EPID make it an attractive option. However, EPIDs were designed as imaging devices, not dosimeters, and as a result they do not inherently measure dose in tissue equivalent media. EPIDose is a tool designed for the use of EPIDs in IMRT and VMAT QA that uses raw MV EPID images to estimate absolute dose planes normal to the beam axis in a homogeneous medi. However, because of the inherent challenges of the EPID-based dosimetry, validating and commissioning such a system must be done very carefully, by exploring the range of use cases and using well-proven “standards” for comparison. Converted EPID images were compared to 2D diode array absolute dose. To provide a 3D dosimetric evaluation of a commercial portal dosimetry system using 2D/3D detectors under ideal conditions using VMAT. Material & Method: A 2D ion chamber array, radiochromic film and gel dosimeter were utilized to provide a dosimetric evaluation of transit phantom and pre-treatment fluence’ EPID back-projected dose distributions for a standard IMRT & VMAT plan. In house 2D and 3D gamma methods compared pass statistics relative to each dosimeter and TPS dose distributions. Results: Fluence mode and transit EPID dose distributions back-projected onto phantom geometry produced 2D gamma pass rates in excess of 97% relative to other tested detectors and exported TPS dose planes when a 3%, 3mm global gamma criterion was applied. Use of a gel dosimeter within a glass vial allowed comparison of measured 3D dose distributions versus EPID 3D dose and TPS calculated distributions. 3D gamma comparisons between modalities at 3%, 3mm gave pass rates in excess of 92%. Use of fluence mode was indicative of transit results under ideal conditions with slightly reduced dose definition. Conclusions: 3D EPID back projected dose distributions were validated against detectors in both 2D and 3D. Cross validation of transit dose delivered to a patient is limited due to reasons of practicality and the tests presented are recommended as a guideline for 3D EPID dosimetry commissioning; allowing direct comparison between detector, TPS, fluence and transit modes. The results indicate achievable gamma scores for a complex VMAT plan in a homogenous phantom geometry and contributes to growing experience of 3D EPID dosimetry.

Uniform Twister Plane Generator

Random plane generators may use various types of the random number algorithms to create multidimensional planes. At the same time, the discrete Descartes random planes have to be uniform. The matter is that using the concept of the uncontrolled random generation may lead to a result of weak quality due to initial sequences having either insufficient uniformity or skipping of the random numbers. This article offers a new approach for creating the absolute twisting uniform two-dimensional Descartes planes based on a model of complete twisting sequences of uniform random variables without repetitions or skipping. The simulation analyses confirm that the resulted random planes have an absolute uniformity. Moreover, combining the parameters of the original complete uniform sequences allows a significant increase in the number of created planes without using additional random access memory.

Classification of general absolute planes by quasi-ends

General (i.e. including non-continuous and non-Archimedean) absolute planes have been classified in different ways, e.g. by using Lambert–Saccheri quadrangles (cf. Greenberg, J Geom 12/1:45-64, 1979; Hartshorne, Geometry; Euclid and beyond, Springer, Berlin, 2000; Karzel and Marchi, Le Matematiche LXI:27–36, 2006; Rostamzadeh and Taherian, Results Math 63:171–182, 2013) or coordinate systems (cf. Pejas, Math Ann 143:212–235, 1961 and, for planes over Euclidean fields, Greenberg, J Geom 12/1:45-64, 1979). Here we consider the notion of quasi-end, a pencil determined by two lines which neither intersect nor have a common perpendicular (an ideal point of Greenberg, J Geom 12/1:45-64, 1979). The cardinality ω of the quasi-ends which are incident with a line is the same for all lines hence it is an invariant \({\omega_\mathcal{A}}\) of the plane \({\mathcal{A}}\) and can be used to classify absolute planes. We consider the case \({\omega_\mathcal{A}=0}\) and, for \({\omega_\mathcal{A} \geq 2}\) (it cannot be 1) we prove that in the singular case \({\omega_\mathcal{A}}\) must be infinite. Finally we prove that for hyperbolic planes, ends and quasi-ends are the same, so \({\omega_\mathcal{A}=2}\) .

On Characterization of Absolute Geometries

Starting from a general absolute plane A = (P, L, α, ≡) in the sense of Karzel et al. (Einfuhrung in die Geometrie, p. 96, 1973), Karzel and Marchi introduced the notion of a Lambert–Saccheri quadrangle (L-S quadrangle) in Karzel and Marchi (Le Matematiche LXI:27–36, 2006): A quadruple (a, b, c, d) of points of P, no three collinear, is a L-S quadrangle, if \({\overline{a,d}\bot\overline{a,b}\bot\overline{b,c}\bot\overline{c,d}}\). Denoting the foot of a on the line \({\overline{c, d}}\) with \({a^{\prime}=\{a\bot\overline{c,d}\}\cap \overline{c,d}}\), the L-S quadrangle (a, b, c, d) is called rectangle, hyperbolic or elliptic quadrangle if \({a^{\prime}=d,\; a^{\prime}\,{\in}\, ]c,d[}\) or\({a^{\prime}\,{\notin}\, ]c,d[\cup \{d\}}\) respectively. Let LS be the set of all L-S quadrangles and LSr, LSh or LSe the subset of all rectangles, hyperbolic or elliptic L-S quadrangles respectively. In Karzel and Marchi (Le Matematiche LXI:27–36, 2006) it was claimed that either LS = LSr or LS = LSh or LS = LSe. To this classification we add five further classifications of general absolute planes by using “distance” [defined in Karzel and Marchi (Discrete Math 308:220–230, 2008)] or the notions of “interior” and “exterior” angle, introduced in Karzel et al. (Resultate Math 51:61–71, 2007) and considering besides Lambert–Saccheri quadrangles, also triangles in particular right-angled triangles. For Lambert–Saccheri quadrangles (a, b, c, d) the relations between distances of the diagonal points (a, c) and (b, d) or between the “midpoint” \({o:=\overline{a,c}\cap\overline{b,d}}\), and the corner points a, b, c, d give us possibilities for complete characterizations. Using triangles (a, b, c) and denoting by m and n the midpoints of (a, b) and (a, c) we classify the absolute planes by the relations between the distances |b, c| and 2|m, n|. All our main results are summarized at the end of the introduction.

Eine geometrische Kennzeichnung der reellen absoluten Ebenen

A new geometric characterization of the real absolute planes is presented, which is based upon few and simple axioms concerning properties of a congruence relation. The ordering properties are developed from two axioms concerning triangles and circles. We use essential results of reflection geometry in order to prove that the structures under consideration have the well-known representations.

On Algebraic Structures Related to Beltrami–Klein Model of Hyperbolic Geometry

A new approach to the algebraic structures related to hyperbolic geometry comes from Einstein’s special theory of relativity in 1988 (cf. Ungar, in Found Phys Lett 1:57–89, 1988). Ungar employed the binary operation of Einsteins velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry in full analogy with Euclidean geometry (cf. Ungar, in Math Appl 49:187–221, 2005). Another approach is from Karzel for algebraization of absolute planes in the sense of Karzel et al. (Einfuhrung in die Geometrie, 1973). In this paper we are going to develop a formulary for the Beltrami–Klein model of hyperbolic plane inside the unit circle \({\mathbb D}\) of the complex numbers \({\mathbb C}\) with geometric approach of Karzel.

On M. T. Calapso’s Characterization of the Metric of an Absolute Plane

We show that the equivalence of the rectangle existence axiom to an axiom used by Maria Teresa Calapso to characterize the Euclidean metric can be proved inside rather weak, purely metric axiom systems for absolute planes.

Legendre-like Theorems in a General Absolute Geometry

In this paper the axiomatic basis will be a general absolute plane A = (\({\mathcal{P}}, {\mathcal{L}}, \alpha, \equiv\)) in the sense of [6], where \({\mathcal{P}}\) and \({\mathcal{L}}\) denote respectively the set of points and the set of lines, α the order structure and ≡ the congruence, and where furthermore the word “general” means that no claim is made on any kind of continuity assumptions. Starting from the classification of general absolute geometries introduced in [5] by means of the notion of congruence, singular or hyperbolic or elliptic, we get now a complete characterization of the different possibilities which can occur in a general absolute plane studying the value of the angle δ defined in any Lambert–Saccheri quadrangle or, equivalently, the sum of the angles of any triangle. This yelds, in particular, a Archimedes-free proof of a statement generalizing the classical “first Legendre theorem” for absolute planes.

Loops with reflection germ: a characterization of absolute planes

We give a characterization of non-Euclidean absolute planes using the concept of fibered right loop with reflection germ.

Eine Bemerkung �ber absolute Ebenen

By separating the compatibility conditions for incidence, order and congruence the axioms for absolute planes in [3] are put in a somewhat more lucid form.

Zur Existenz gleichseitiger Dreiecke in H-Ebenen

It is shown, by indicating how to construct one with ruler and gauge, that there are equilateral triangles in absolutes planes which need not satisfy the circle axiom. However, it is not possible to construct an equilateral triangle with given base in absolute planes, even if they satisfy bachmann'sLotschnittaxiom or the Archimedean axiom.

Towards the absolute planes: a new calibration of the bolometric corrections and temperature scales for Population II giants

We present new determinations of bolometric corrections and effective temperature scales as a function of infrared and optical colors, using a large database of photometric observations of about 6500 Population II giants in Galactic Globular Clusters (GGCs), covering a wide range in metallicity (-2.0<[Fe/H]<0.0). New relations for BC_K vs (V-K), (J-K) and BC_V vs (B-V), (V-I), (V-J), and new calibrations for T_eff, using both an empirical relation and model atmospheres, are provided. Moreover, an empirical relation to derive the R parameter of the Infrared Flux Method as a function of the stellar temperature is also presented.

Reflection groups and K-loops

The notion “reflection group” (Γ,\(\mathcal{D}\)) was introduced in order to give group theoretical characterizations of absolute planes. Here we consider “reflection groups with midpoints” and associate to each of them an incidence structure\(\mathfrak{G}\). Then\((\mathcal{D},\mathfrak{G})\) is an incidence space which dimension can assume any value. The motion group Γ together with the set\(\mathcal{D}\) of all reflections in points of a Euclidean or hyperbolic geometry are examples of reflection groups with midpoints. Furthermore the set\(\mathcal{D}\) can be turned into a K-loop. The precise results are summarized in the theorems at the end of the paper.

A characterization of isometries of absolute planes

A characterization of isometries of absolute planes