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The Gnomonic Net

Published online by Cambridge University Press:  14 March 2018

Harold Hilton
Harold Hilton*
Affiliation:
Magdalen College, Oxford
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Extract

Mr. G. F. Herbert Smith has pointed out the advantages of the gnomonic projection for some special crystallographic problems. It seems to me that much of the labour in drawing the projections could be saved by a device similar to that employed by Fedorow, Wulff, Penfield, &c., in drawing stereographic projections.

Corresponding to the ‘stereographic net’ described by these authors (which is the stereographic projection of lines of latitude and longitude on a plane through a llne of longitude), we have the gnomonic net, which is the projection of the lines of latitude (l) and of longitude (m) on the tangent plane at the point whose latitude and longitude are zero. The lines m are parallel straight lines, the lines l are hyperbolae having double contact with each other at two fixed imaginary points.

Type
Research Article
Information
Mineralogical magazine and journal of the Mineralogical Society , Volume 14 , Issue 63 , October 1904 , pp. 18 - 20
Copyright
Copyright © The Mineralogical Society of Great Britain and Ireland 1904

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References

Page 18 note 1 Min. Mag., 1903, vol. xiii, pp. 309-321.

Page 18 note 2 See, for instance, Zeite. Kryst. Min., 1902, vol. xxxvi, p. 14 ; Phil. Mag., 1903, ser. 6, vol, vi, p. 67.

Page 18 note 3 In the diagram a = 1.56 inches = 39.64 ram.

Page 18 note 4 As in the stereographic net given by Professor G. Wulff in plate II, Zeits. Kryst. Min., vol. xxxvi, which I have found useful.

Page 18 note 5 Unlike the stereographic net, the gnomonic net does not itself serve as a protractor.

Page 19 note 1 The corners of the net correspond to a distance of 71° 45′ from the centre.

Page 19 note 2 Zeits. Kryst. Min., 1902, vol. xxxvi, p. 15; el. Phil. Mag., 1903, ser. 6, vol. vi, p. 68.

Page 19 note 3 If we project figures on a sphere from any point V on to any plane cutting the tangent cone from V to the Sphere in the conic j, then all circles on the sphere project into conics having double contact with j. It follows that the tangents at the intersection of the projections of two orthogonal spherical curves are conjugate with respect to j. If V is on the sphere, j degenerates into a pair of points. In the projection shown in the diagram j is x2 + y2 + a2 = 0.

Page 19 note 4 The method may also be applied to crystals measured with a one-circle goniometer, but the stereogrphic net is more convenient. This will be clear on a careful reading of Wulff, loc. cit., p. 16, problem 3.

Page 20 note 1 For the pencil of lines joining the poles of the faces to the centre of the sphere is cut by any plane in a range whose anharmonic ratio is that of the four faces.

Page 20 note 2 See Min. Mag., 1903, vol. xiii, p. 815.

Page 20 note 3 If we wished the dial to register apparent Greenwich time in longitude d° we should take points whose distances from R are (15 + d)°, (30 + d)° &c.

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Errata