History of the stereographic projection and its early use in geology

Abstract

The origin of the polar and equatorial stereographic projections can be traced back to Hipparchos (d. 125 bc). The equatorial projection became an essential tool in astronomy and from the 2nd to mid‐17th Century it enabled the development of the astrolabe as a navigational tool and later became important in mapmaking. Examples of equatorial stereographic nets by the astronomer al‐Zarquǎlǎ (ad, 10701 and the cartographer Blaeu (1624) are illustrated. By the early 19th Century, mathematics was beginning to be applied outside the fields of astronomy, physics and mechanics. The stereographic net was first applied to mineralogy by Neumann (1823) and its use was subsequently popularized by Federov (1892), inventor of the universal microscope stage, and by the introduction of the Wulff net (1902). It also proved of considerable use in the correct interpretation by W.L. Bragg of Laue X‐ray diffraction patterns of crystals which led to the development of X‐ray crystallography. The Lambert (1772) equal‐area projection, developed for cartographic use and subsequently termed the Schmidt net by Sander in 1925, was essential to the development of petrofabric analysis in the late 1920s. The transition to macroscopic structural analysis, which largely took place over the period 1930–55 is briefly touched upon.

Stereographic coordinates for simpler far-field radiation analysis

“…Wulff net) and geology. A detailed historical account of stereographic projection development and application has been complied by Howarth [7]. Key mathematical properties of the projection have been summarized by Rosenfeld and Sergeeva [8].…”

“…Key mathematical properties of the projection have been summarized by Rosenfeld and Sergeeva [8]. Despite the wide-spread use of stereographic projection in geo-sciences [9], [10], as well as engineering and material science [8], [7], [11], [12], [13], it remains relatively unknown in the field of applied electromagnetism. In this section we provide two definitions of the stereographic projection: geometric and algebraic.…”

“…For purposes of this section it is easier to derive main results in Cartesian coordinates xyz and then to connect them to stereographic coordinates using Eq. (7,8,9).…”

“…Using Eq. (7,8) the expressions for the great circles on a hemi-sphere in stereographic coordinates become: u (ψ; Φ, θ 0 , φ 0 ) = cos Φ cos θ 0 cos φ 0 sin ψ− − sin Φ sin φ 0 sin ψ + sin θ 0 cos φ 0 cos ψ × × 1 − cos Φ sin θ 0 sin ψ + cos θ 0 cos ψ −1 (16) v (ψ; Φ, θ 0 , φ 0 ) = cos Φ cos θ 0 sin φ 0 sin ψ+ + sin Φ cos φ 0 sin ψ + sin θ 0 sin φ 0 cos ψ × × 1 − cos Φ sin θ 0 sin ψ + cos θ 0 cos ψ (a) The actual shape of the two great circles in the 3d space. The great circles, orange and blue, are plotted as sets of disconnected dots distributed uniformly (fixed steps in ψ) along the great circle.…”

SP2: a computer program for plotting stereographic projection and exploring crystallographic orientation relationships

“…Stereographic projection has long been one of the most powerful research tools for crystallographers in astronomy, physics, mechanical engineering, geography and materials science since its use was popularized by Federov, inventor of the universal microscope stage, and by the introduction of the Wulff net (Howarth, 1996). Stereographic projection enables crystallographers to explore the threedimensional crystal planes and orientations of a crystal with an arbitrary crystal structure on a two-dimensional plane.…”