Angular eccentricity and linear eccentricity

Angular eccentricity is one of many parameters which arise in the study of the

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...

. It is denoted here by α (alpha). It may be defined in terms of the

eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-center, in geometry * Eccentricity (graph theory) of a v ...

, ''e'', or the aspect ratio, ''b/a'' (the ratio of the

semi-minor axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both focus (geometry), foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major wikt: ...

and the

semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the long ...

): :

\alpha=\sin^\!e=\cos^\left(\frac\right).
 \,\!

Angular eccentricity is not currently used in English language publications on mathematics, geodesy or map projections but it does appear in older literature. Any non-dimensional parameter of the ellipse may be expressed in terms of the angular eccentricity. Such expressions are listed in the following table after the conventional definitions.Rapp, Richard H. (1991). ''Geometric Geodesy, Part I'', Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus, Ohi

/ref> in terms of the semi-axes. The notation for these parameters varies. Here we follow Rapp: :: The alternative expressions for the flattenings would guard against large cancellations in numerical work.

References

{{Reflist

External links

Toby Garfield's APPENDIX A: The ellipse[Archived_copy
/nowiki>..html" ;"title="rchived copy">[Archived copy
/nowiki>.">rchived copy">[Archived copy
/nowiki>.br>Map Projections for Europe (pg.116)
Geodesy Conic sections