Affine Transformation Of Coordinates Research Articles

Diffraction of pulse shear waves in a tunnel cavity of curvilinear section in an anisotropic massif

The problem of diffraction of longitudinal shear waves in the form of periodic triangular pulses by a cylindrical tunnel cavity in a rectilinearly orthotropic unbounded massif is reduced to a number of problems on the diffraction of harmonic shear waves of different relative length. Each of the problems on stationary diffraction is solved in series in the basis solutions of shear vibrations equations by using an affine coordinate transformation of the original domain. It is proposed to use a point method of least squares to seek the coefficients of the series. The influence of the degree of anisotropy of the elastic properties of the massif on the shear stress concentration at the boundary of a circular section is analyzed for a different position of the incident triangular pulse front relative to center of the cavity.

Regular gravitational lagrangians

The Einstein action with vanishing cosmological constant is for appropriate field content the unique local action which is regular at the fixed point of affine coordinate transformations. Imposing this regularity requirement one excludes also Wess-Zumino counterterms which trade gravitational anomalies for Lorentz anomalies. One has to expect dilatational and SL ( D) anomalies. If these anomalies are absent and if the regularity of the quantum vertex functional can be controlled then Einstein gravity is renormalizable.

An Extended New General Relativity as a Reduction of \overlinePoincare Gauge Theory of Gravity: Generators of Internal and Coordinate Transformations

An extended new general relativity (E. N. G. R.) is formulated as a reduction of Poincare gauge theory (P. G. T.) of gravity whose gauge group is the covering group of the Poincare group. Fundamental fields in the theory are a Higgs-type field <p', and the translational gauge potential A'p and a matter field ,pa. The components e'p of the duals of the vierbein fields are given by e'p =f7t<p'=<p',p+A'p. Internal translation and the field <p',. which the conventional new general relativity lacks, play fundamental roles in E. N. G. R. as well as in P. G. T. The restrictions on the Lagrangian density, identities and differential conservation laws following from the (local internal translation ® global SL(2, e)) ® general coordinate transformations are given. For the specific gravitational Lagrangian LT = - (1/3K)t klm tklm +(1/3K)v'v. + a3a'a., the generators of internal Poincare and general affine coordinate transformations for an isolated system are examined. The results are quite parallel to those in P. G. T., but the former is not trivial reductions of the latter. The following is worth noticing also in E. N. G. R.: For the case in which {<p',A'p, ,pa} with <p''''''e(O)'pY P + <p(0)' is employed as the set of il).dependent variables, (a) the generators of (internal translation ® SL(2, e)) give the energy-momentum and the total angular momentum for an isolated system, and (b) the generators of ~eneral affine coordinate transformations vanish and are trivial.

An Extended New General Relativity as a Reduction of $\overline{\mbox{Poincar\'{e}}}$ Gauge Theory of Gravity

An extended new general relativity (E. N. G. R.) is formulated as a reduction of Poincaré gauge theory (P, G. T.) of gravity whose gauge group is the covering group of the Poincaré group. Fundamental fields in the theory are a Higgs-type field ψ k , and the translational gauge potential Akµ and a matter field φ a . The components ekµ of the duals of the vierbein fields are given by ekµ = ∇ *µ ψ k = ψ k,µ + Akµ . Internal translation and the field ψ k , which the conventional new general relativity lacks, play fundamental roles in E. N. G. R. as well as in P, G. T. The restrictions on the Lagrangian density, identities and differential conservation laws following from the ( local internal translation ⊗ global SL (2, C ) ) ⊗ general coordinate transformations are given. For the specific gravitational Lagrangian LT = -(1/3κ) tklmtklm + (1/3κ) vkvk + a3akak , the generators of internal \overlinePoincaré and general affine coordinate transformations for an isolated system are examined. The results are quite parallel to those in P, G. T., but the former is not trivial reductions of the latter. The following is worth noticing also in E. N. G. R.: For the case in which {ψ k , Akµ , φ a } with ψ k ≃ e (0) kµyµ + ψ (0) k is employed as the set of independent variables, (a) the generators of ( internal translation ⊗SL (2, C )) give the energy-momentum and the total angular momentum for an isolated system, and (b) the generators of general affine coordinate transformations vanish and are trivial.

Coordinate Conversion from Nad to Nad

A method for converting North American Datum (NAD) 27 coordinates to NAD 83 coordinates is presented. The method is tested on over 1,000 control points, known in both NAD 27 and NAD 83, obtained from the National Geodetic Survey. Input data are maintained as geodetic latitude and longitude to allow for nationwide application. In order to use the transformation equations of plane geometry, latitudes and longitudes are projected to transverse Mercator X, Y with a unique projection for each point considered to be unknown in NAD 83. The methodology uses three or four (depending on the type of transformation) nearby points, known both in NAD 27 and NAD 83, to determine transformation coefficients which are then applied to a point considered to be known only in NAD 27. Results from the use of first‐degree conformal, second‐degree conformal, and affine coordinate transformation equations are presented. The final results indicate that greatest accuracy is obtained with the use of first‐ and second‐order control points only. The use of the method in practical applications is discussed.