Non-Cartesian Coordinates and Quadric Surfaces

2003 ◽  
pp. 39-59
Keyword(s):  
Cartesian Coordinates ◽  
Quadric Surfaces

Vector geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Vector Space ◽  
Velocity Vector ◽  
Cartesian Coordinates

This chapter defines the mathematical spaces to which the geometrical quantities discussed in the previous chapter—scalars, vectors, and the metric—belong. Its goal is to go from the concept of a vector as an object whose components transform as Tⁱ → 𝓡ⱼ ⁱTj under a change of frame to the ‘intrinsic’ concept of a vector, T. These concepts are also generalized to ‘tensors’. The chapter also briefly remarks on how to deal with non-Cartesian coordinates. The velocity vector v is defined as a ‘free’ vector belonging to the vector space ε‎3 which subtends ε‎3. As such, it is not bound to the point P at which it is evaluated. It is, however, possible to attach it to that point and to interpret it as the tangent to the trajectory at P.

scholarly journals Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
G. Panou ◽  
R. Korakitis
Keyword(s):  
Initial Value Problem ◽  
Numerical Solutions ◽  
Oblate Spheroid ◽  
Initial Value ◽  
Data Set ◽  
Cartesian Coordinates ◽  
Geodesic Equations ◽  
Research Article ◽  
Fast Solution ◽  
Geodesic Problem

AbstractThe direct geodesic problem on an oblate spheroid is described as an initial value problem and is solved numerically using both geodetic and Cartesian coordinates. The geodesic equations are formulated by means of the theory of differential geometry. The initial value problem under consideration is reduced to a system of first-order ordinary differential equations, which is solved using a numerical method. The solution provides the coordinates and the azimuths at any point along the geodesic. The Clairaut constant is not used for the solution but it is computed, allowing to check the precision of the method. An extensive data set of geodesics is used, in order to evaluate the performance of the method in each coordinate system. The results for the direct geodesic problem are validated by comparison to Karney’s method. We conclude that a complete, stable, precise, accurate and fast solution of the problem in Cartesian coordinates is accomplished.

Transformations depending on sets of associated points

1952 ◽  
Vol 48 (3) ◽  
pp. 383-391
Author(s):  
T. G. Room
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Three Dimensional ◽  
Cubic Curves ◽  
Birational Transformations ◽  
Quadric Surfaces ◽  
Base Points ◽  
Dimensional Projective Space ◽  
Quartic Curves

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

Envelopes of moving quadric surfaces

1992 ◽  
Vol 9 (4) ◽  
pp. 299-312 ◽  
Author(s):  
J. Flaquer ◽  
G. Gárate ◽  
M. Pargada
Keyword(s):  
Quadric Surfaces

scholarly journals The Pairing Computation on Edwards Curves

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Hongfeng Wu ◽  
Liangze Li ◽  
Fan Zhang
Keyword(s):  
Geometric Interpretation ◽  
Tate Pairing ◽  
Quadric Surfaces ◽  
Edwards Curves ◽  
Explicit Formulae ◽  
Pairing Computation ◽  
Twisted Edwards Curves ◽  
Group Law

We propose an elaborate geometry approach to explain the group law on twisted Edwards curves which are seen as the intersection of quadric surfaces in place. Using the geometric interpretation of the group law, we obtain the Miller function for Tate pairing computation on twisted Edwards curves. Then we present the explicit formulae for pairing computation on twisted Edwards curves. Our formulae for the doubling step are a little faster than that proposed by Arène et al. Finally, to improve the efficiency of pairing computation, we present twists of degrees 4 and 6 on twisted Edwards curves.

Some Comments on Sherratt's Principle

1977 ◽  
Vol 9 (12) ◽  
pp. 1417-1419
Author(s):  
R Vaughan
Keyword(s):  
Probability Density ◽  
Polar Coordinates ◽  
Cartesian Coordinates

The term ‘density of dwellings' used by geographers is conceptually different to the term ‘probability density of dwellings' used by statisticians. In Cartesian coordinates the numerical values only differ by a scaling constant, but in polar coordinates this is not the case. To help clear up the resulting confusion, this paper attempts to show the relation between the two concepts.

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