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Charles F. F. Karney

Publications and source records attributed to Charles F. F. Karney.

At least 19 recordsLinked to original sources
arXiv2025

Jacobi's solution for geodesics on a triaxial ellipsoid

On Boxing Day, 1838, Jacobi found a solution to the problem of geodesics on a triaxial ellipsoid, with the course of the geodesic and the distance along it given in terms of one-dimensional integrals. Here, a numerical implementation of this solution is described. This entails accurately evaluating the integrals and solving the resulting coupled system of equations. The inverse problem, finding the shortest path between two points on the ellipsoid, can then be solved using a similar method as for biaxial ellipsoids.

physics.geo-ph↗
arXiv2023

Geodesic intersections

A complete treatment of the intersections of two geodesics on the surface of an ellipsoid of revolution is given. With a suitable metric for the distances between intersections, bounds are placed on their spacing. This leads to fast and reliable algorithms for finding the closest intersection, determining whether and where two geodesic segments intersect, finding the next closest intersection to a given intersection, and listing all nearby intersections. The cases where the two geodesics overlap are also treated.

physics.geo-ph↗
arXiv2023

Geodesics on an arbitrary ellipsoid of revolution

The algorithms given in Karney, J. Geodesy 87, 43-55 (2013), to compute geodesics on terrestrial ellipsoids are extended to apply to ellipsoids of revolution with arbitrary eccentricity. For the direct and inverse geodesic problems, this entails implementing the formulation in terms of elliptic integrals given by Legendre and Cayley. The integral for the area of geodesic polygons is computed in terms of the discrete sine transform of the integrand. In all cases, accuracy close to full machine precision is achieved. An open-source implementation of these algorithms is available.

physics.geo-ph↗
arXiv2023

On auxiliary latitudes

The auxiliary latitudes are essential tools in cartography. This paper summarizes methods for converting between them with an emphasis on providing full double-precision accuracy. This includes series expansions in the third flattening, where the truncation error is precisely measured and where estimates of the radii of convergence are given. Also new formulas are given for computing the rectifying and authalic latitudes with minimal roundoff error.

physics.geo-ph↗
arXiv2023

The area of rhumb polygons

The formula for the area of a rhumb polygon, a polygon whose edges are rhumb lines on an ellipsoid of revolution, is derived and a method is given for computing the area accurately. This paper also points out that standard methods for computing rhumb lines give inaccurate results for nearly east- or west-going lines; this problem is remedied by the systematic use of divided differences.

physics.geo-ph↗
arXiv2022

Sensor trajectory estimation by triangulating lidar returns

The paper describes how to recover the sensor trajectory for an aerial lidar collect using the data for multiple-return lidar pulses. This work extends the work of Gatziolis and McGaughey (2019) by performing a least-squares fit for multiple pulses simultaneously with a spline fit for the sensor trajectory. The method can be naturally extended to incorporate the scan angle of the lidar returns following Hartzell (2020). This allows the pitch and the yaw of the sensor to be estimated in addition to its position.

physics.geo-ph↗
arXiv2014

Sampling exactly from the normal distribution

An algorithm for sampling exactly from the normal distribution is given. The algorithm reads some number of uniformly distributed random digits in a given base and generates an initial portion of the representation of a normal deviate in the same base. Thereafter, uniform random digits are copied directly into the representation of the normal deviate. Thus, in contrast to existing methods, it is possible to generate normal deviates exactly rounded to any precision with a mean cost that scales linearly in the precision. The method performs no extended precision arithmetic, calls no transcendental functions, and, indeed, uses no floating point arithmetic whatsoever; it uses only simple integer operations. It can easily be adapted to sample exactly from the discrete normal distribution whose parameters are rational numbers.

physics.comp-ph↗
arXiv2012

Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen

The solution of the geodesic problem for an oblate ellipsoid is developed in terms of series. Tables are provided to simplify the computation. [This is a transcription of F. W. Bessel, Astronomische Nachrichten 4(86), 241-254 (1825). The text follows the original; however the mathematical notation has been updated to conform to current conventions. Several errors have been corrected and the tables have been recomputed.]

physics.comp-ph↗
arXiv2012

The calculation of longitude and latitude from geodesic measurements

The solution of the geodesic problem for an oblate ellipsoid is developed in terms of series. Tables are provided to simplify the computation. [This is an English translation of F. W. Bessel, Astronomische Nachrichten 4(86), 241-254 (1825). The mathematical notation has been updated to conform to current conventions and, in a few places, the equations have been rearranged for clarity. Several errors have been corrected, a figure has been included, and the tables have been recomputed.]

physics.comp-ph↗
arXiv2012

Algorithms for geodesics

Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of geodesics to be computed.

physics.geo-ph↗
arXiv2011

Geodesics on an ellipsoid of revolution

Algorithms for the computation of the forward and inverse geodesic problems for an ellipsoid of revolution are derived. These are accurate to better than 15 nm when applied to the terrestrial ellipsoids. The solutions of other problems involving geodesics (triangulation, projections, maritime boundaries, and polygonal areas) are investigated.

physics.geo-ph↗
arXiv2011

Transverse Mercator with an accuracy of a few nanometers

Implementations of two algorithms for the transverse Mercator projection are described; these achieve accuracies close to machine precision. One is based on the exact equations of Thompson and Lee and the other uses an extension of Krueger's series for the projection to higher order. The exact method provides an accuracy of 9 nm over the entire ellipsoid, while the errors in the series method are less than 5 nm within 3900 km of the central meridian. In each case, the meridian convergence and scale are also computed with similar accuracy. The speed of the series method is competitive with other less accurate algorithms and the exact method is about 5 times slower.

physics.comp-ph↗
arXiv2006

Quaternions in molecular modeling

Quaternions are an important tool to describe the orientation of a molecule. This paper considers the use of quaternions in matching two conformations of a molecule, in interpolating rotations, in performing statistics on orientational data, in the random sampling of rotations, and in establishing grids in orientation space. These examples show that many of the rotational problems that arise in molecular modeling may be handled simply and efficiently using quaternions.

physics.comp-ph↗
arXiv2005

Using binding free energy to guide ligand design

The molecular distributions obtained from canonical Monte Carlo simulations can be used to find an approximate interaction energy. This serves as the basis of a method for estimating the binding free energy for a ligand to a protein which enables the free energy to be used to direct the design of ligands which bind to a protein with high affinity.

physics.chem-ph↗
arXiv2005

Modeling molecules with constraints

Techniques for simulating molecules whose conformations satisfy constraints are presented. A method for selecting appropriate moves in Monte Carlo simulations is given. The resulting moves not only obey the constraints but also maintain detailed balance so that correct equilibrium averages are computed. In addition, techniques for optimizing the evaluation of implicit solvent terms are given.

physics.comp-ph↗
arXiv2005

Fokker-Planck and Quasilinear Codes

The interaction of radio-frequency waves with a plasma is described by a Fokker-Planck equation with an added quasilinear term. Methods for solving this equation on a computer are discussed.

physics.plasm-ph↗
arXiv2005

Efficiency of Current Drive by Fast Waves

The Rosenbluth form for the collision operator for a weakly relativistic plasma is derived. The formalism adopted by Antonsen and Chu can then be used to calculate the efficiency of current drive by fast waves in a relativistic plasma. Accurate numerical results and analytic asymptotic limits for the efficiencies are given.

physics.plasm-ph↗