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Gnomonic Projection of the Surface of an Ellipsoid

  • Roy Williams

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When a surface is mapped onto a plane so that the image of a geodesic arc is a straight line on the plane then the mapping is known as a geodesic mapping. It is only possible to perform a geodesic mapping of a surface onto a plane when the surface has constant normal curvature. The normal curvature of a sphere of radius r at all points on the surface is I/r hence it is possible to map the surface of a sphere onto a plane using a geodesic mapping. The geodesic mapping of the surface of a sphere onto a plane is achieved by a gnomonic projection which is the projection of the surface of the sphere from its centre onto a tangent plane. There is no geodesic mapping of the ellipsoid of revolution or the spheroid onto a plane because the ellipsoid of revolution or the spheroid are not surfaces whose curvature is constant at all points. We can, however, still construct a projection of the surface of the ellipsoid from the centre of the body onto a tangent plane and we call this projection a gnomonic projection also.

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1Admiralty Manual of Navigation, vol. 3, ch. VI, p. 52. HMSO (1954) or Admiralty Manual of Navigation, vol. 1, Appendix 4, p. 624. HMSO (1987).
2Williams, R., (1996). The great ellipse on the surface of the spheroid. This Journal, 49, 229.

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Gnomonic Projection of the Surface of an Ellipsoid

  • Roy Williams

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