Angle Bisector for an Angle Subtended by a Tangent Line

2008 ◽  
Keyword(s):  
Tangent Line ◽  
Angle Bisector

Femoral Anterior Tangent Line of the Osteoarthritic Knee for Determining Rotational Alignment of the Femoral Component in Total Knee Arthroplasty

2011 ◽  
Vol 26 (2) ◽  
pp. 268-273 ◽  
Author(s):  
Hiroki Watanabe ◽  
Ryuichi Gejo ◽  
Yoshikazu Matsuda ◽  
Ichiro Tatsumi ◽  
Kazuo Hirakawa ◽  
...  
Keyword(s):  
Total Knee Arthroplasty ◽  
Femoral Component ◽  
Knee Arthroplasty ◽  
Rotational Alignment ◽  
Tangent Line ◽  
Osteoarthritic Knee ◽  
Total Knee

scholarly journals Maxillomandibular Plane Angle Bisector (MM) Adjunctive to Occlusal Plane to evaluate Anteroposterior Measurement of Dental Base

2012 ◽  
Vol 13 (4) ◽  
pp. 539-544
Author(s):  
Chanamallappa R Ganiger ◽  
KU Cariappa ◽  
AR Yusuf Ahammed
Keyword(s):  
Clinical Significance ◽  
Orthodontic Treatment ◽  
Clinical Usefulness ◽  
Occlusal Plane ◽  
Reference Plane ◽  
Plane Angle ◽  
B Values ◽  
Angle Bisector ◽  
Lateral Cephalograms ◽  
Relationship Of

ABSTRACT Aim This study was undertaken to analyze the clinical usefulness of the maxillomandibular bisector, its reproducibility, its validity and its relationship to the functional occlusal plane, the bisecting occlusal plane and the nature of its cant. Materials and methods Thirty pretreatment lateral cephalograms, each of adolescents (above 18 years of age) and children (10- 12 years), seeking orthodontic treatment were randomly selected and the Wits technique of anteroposterior measurement was used to compare A-B values measured to the new plane with those measured to the functional occlusal plane (FOP) and to the traditional or bisecting occlusal plane (BOP). Results Present study showed that MM bisector plane is more reproducible and valid reference plane, than the FOP and BOP. Conclusion A new plane, geometrically derived from the dental base planes, has been tested as an occlusal plane substitute for the measurement of anteroposterior jaw relationships. It lies close to but at an angle and inferior to the traditional occlusal planes and is highly reproducible at all times. Clinical significance Maxillomandibular planes angle bisector may be a useful adjunct for the cephalometric assessment of sagittal relationship of the patient. How to cite this article Ganiger CR, Nayak USK, Cariappa KU, Ahammed ARY. Maxillomandibular Plane Angle Bisector (MM) Adjunctive to Occlusal Plane to evaluate Anteroposterior Measurement of Dental Base. J Contemp Dent Pract 2012; 13(4):539-544.

scholarly journals Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

2021 ◽  
Author(s):  
Eswara Venkata Kumar Dhulipala
Keyword(s):  
Euclidean Distance ◽  
Travelling Salesman Problem ◽  
Dynamic Programming Algorithm ◽  
Constant Factor ◽  
Programming Algorithm ◽  
Worst Case ◽  
Angle Bisector ◽  
Set Of Points ◽  
The Given ◽  
Tour Length

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

scholarly journals LEARNING HYPERPLANES THAT CAPTURES THE GEOMETRIC STRUCTURE OF CLASS REGIONS

2013 ◽  
pp. 7-12
Author(s):  
PRAMOD PATIL ◽  
ALKA LONDHE ◽  
PARAG KULKARNI
Keyword(s):  
Decision Tree ◽  
Decision Trees ◽  
Geometric Structure ◽  
Gini Index ◽  
Decision Tree Algorithm ◽  
Top Down ◽  
Angle Bisector ◽  
Eigen Value ◽  
Tree Algorithms ◽  
Left And Right

Most of the decision tree algorithms rely on impurity measures to evaluate the goodness of hyperplanes at each node while learning a decision tree in a top-down fashion. These impurity measures are not differentiable with relation to the hyperplane parameters. Therefore the algorithms for decision tree learning using impurity measures need to use some search techniques for finding the best hyperplane at every node. These impurity measures don’t properly capture the geometric structures of the data. In this paper a Two-Class algorithm for learning oblique decision trees is proposed. Aggravated by this, the algorithm uses a strategy, to evaluate the hyperplanes in such a way that the (linear) geometric structure in the data is taken into consideration. At each node of the decision tree, algorithm finds the clustering hyperplanes for both the classes. The clustering hyperplanes are obtained by solving the generalized Eigen-value problem. Then the data is splitted based on angle bisector and recursively learn the left and right sub-trees of the node. Since, in general, there will be two angle bisectors; one is selected which is better based on an impurity measure gini index. Thus the algorithm combines the ideas of linear tendencies in data and purity of nodes to find better decision trees. This idea leads to small decision trees and better performance.

scholarly journals Suns are convex in tangent directions

2019 ◽  
Vol 484 (2) ◽  
pp. 131-133
Author(s):  
A. R. Alimov ◽  
E. V. Shchepin
Keyword(s):  
Linear Space ◽  
Unit Sphere ◽  
Normed Linear Space ◽  
Tangent Line ◽  
Chebyshev Set ◽  
Tangent Direction

A direction d is called a tangent direction to the unit sphere S of a normed linear space s  S and lin(s + d) is a tangent line to the sphere S at s imply that lin(s + d) is a one-sided tangent to the sphere S, i. e., it is the limit of secant lines at s. A set M is called convex with respect to a direction d if [x, y]  M whenever x, y in M, (y - x) || d. We show that in a normed linear space an arbitrary sun (in particular, a boundedly compact Chebyshev set) is convex with respect to any tangent direction of the unit sphere.

On the Definition of a Tangent-Line

1964 ◽  
Vol 71 (10) ◽  
pp. 1099-1103 ◽  
Author(s):  
H. A. Thurston
Keyword(s):  
Tangent Line ◽  
Definition Of

scholarly journals Osculating primes to curves of intersection in 4-space, and to certain curves in n-space

1973 ◽  
Vol 18 (4) ◽  
pp. 325-338 ◽  
Author(s):  
R. H. Dye
Keyword(s):  
Linear Combination ◽  
Complete Intersection ◽  
Tangent Line ◽  
Simple Point ◽  
Irreducible Curve

An irreducible curve in S4, projective 4-space, may arise as the complete intersection of three given irreducible threefolds. At a simple point P on such a curve there is an osculating solid, and we would like to have its equation. This solid, necessarily containing the tangent line to the curve at P, belongs to the net spanned by the tangent solids at P to the threefolds. We seek the appropriate linear combination of the known equations for these tangent solids.

A Tangent-Line Approach for Effective Density Used in the Ideal Mixing Rule: Part II—Evaluation of Mixing Characteristics of Oil/Gas Systems and Application Criteria

SPE Journal ◽  
2020 ◽  
Vol 25 (06) ◽  
pp. 3160-3185
Author(s):  
Zehua Chen ◽  
Daoyong (Tony) Yang
Keyword(s):  
Tangent Line ◽  
Solid Particles ◽  
Effective Density ◽  
Mixing Rule ◽  
Ideal Mixing ◽  
Mixing Characteristics ◽  
Oil Gas ◽  
Solvent Fraction ◽  
The Ideal ◽  
Real Density

Summary Although in Part I of this study (Chen and Yang 2020) we developed a tangent-line approach for effective density that is more general, robust, and flexible than the methods proposed by Saryazdi (2012) and Saryazdi et al. (2013), its application is only limited to heavy-oil/bitumen-associated mixtures [i.e., specifically, it has only been applied to bitumen-rich liquid phase (denoted as L2)]. As indicated in Part I, the density of nitrogen (N2)/hydrocarbon mixtures cannot be accurately predicted by using the ideal mixing rule (IM) with either real density or effective density. Not only do we need to explain and evaluate the observed deviations and patterns, but also the density prediction of solvent/Fraction 1 systems [i.e., Fraction 1 of the Athabasca bitumen, which has a molecular weight (MW) of 268.8 g/mol, as reported in Azinfar et al. (2018a, 2018b, 2018c)] needs to be improved for practical use. In this study, we evaluate the mixing characteristics of different molecules in a mixture using the tangent-line approach. By evaluating and comparing performances of the IM with effective density (IM-E) and the IM with real density (IM-R), the observed patterns and deviations together with those calculated from the Westman equation indicate that the oil/gas molecules somewhat behave like solid particles in mixing. Accordingly, we further modify the effective density used in the IM to bridge the gap between the IM-E and the IM-R. The database has been extended to light-oil/gas systems such as black oils, volatile oils, gas condensates, carbon dioxide (CO2) miscible fluids, sour gases, and wet/dry gases. The IM with modified effective density (IM-ME) has also been applied to solvent/Fraction 1 systems and the C2 or C3 or n-C4-extraction L1 phase (bitumen-related mixtures) with better accuracy. Also, we develop new criteria for the uses of the IM-E, IM-ME, and IM-R that can cover the density predictions for almost all types of oil/gas systems in the petroleum industry with high accuracy. The performances of the IM are thoroughly evaluated and compared with the volume-translated (VT) Peng-Robinson equation of state (EOS) (VT PR EOS), from which the deviations provide new insights for accurately quantifying the mixture density in a more robust and reliable manner.

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