Stereographic Projection

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Stereographic Projection ◽  
Spherical Trigonometry ◽  
Arbitrary Triangle ◽  
Cesàro Method

This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave instrument makers a huge advantage and the second provides clear astronomical advantages. The earliest text on stereographic projection is Ptolemy's Planisphere, in which he explains how to use stereographic projection to solve problems involving rising times, suggesting that the astrolabe may have existed already. After providing an overview of the astrolabe, an instrument for solving astronomical problems, the chapter considers how stereographic projection is used in solving triangles. It then describes the Cesàro method, named after Giuseppe Cesàro, that uses stereographic projection to project an arbitrary triangle ABC onto a plane. It also examines B. M. Brown's complaint against Cesàro's approach to spherical trigonometry.

Great Circles and Rhumb Lines on the Complex Plane

2017 ◽  
Vol 70 (3) ◽  
pp. 618-627
Author(s):  
Robin G. Stuart
Keyword(s):  
Complex Plane ◽  
Stereographic Projection ◽  
Riemann Sphere ◽  
Complex Numbers ◽  
Spherical Trigonometry ◽  
Numerical Examples ◽  
Computer Evaluation

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.

The determination of true orientations of fractures in rock cores

1983 ◽  
Vol 20 (2) ◽  
pp. 221-227 ◽  
Author(s):  
J. S. O. Lau
Keyword(s):  
Stereographic Projection ◽  
Management Program ◽  
Reference Line ◽  
Spherical Trigonometry ◽  
Mathematical Methods ◽  
Analytical Geometry ◽  
Deep Boreholes ◽  
Nuclear Fuel Waste ◽  
Apparent Orientation

Determination of the true orientations of fractures in diamond drill cores obtained from deep boreholes in plutonic bodies is an essential requirement of the geoscience component of the Canadian Nuclear Fuel Waste Management Program. A reference line can be painted on the entire length of the rock core, indicating the orientation of the core, and the apparent orientation of the fracture can be measured from this reference line. This paper describes three methods that have been developed to convert the apparent orientation to true orientation, namely, stereographic projection, spherical trigonometry, and analytical geometry. The results obtained from these techniques were compared to assess their relative accuracy. Whereas the graphical method is more readily adaptable for use in the field, the mathematical methods can be computer-programmed and the programs GEOCORE and ORIENTC are available from the Geological Survey of Canada to facilitate the calculation of large volumes of data. Keywords: true orientation, fracture, rock core, stereographic projection, spherical trigonometry, analytical geometry.

Orthogonal Projection

1897 ◽  
Vol 1 (11) ◽  
pp. 97-102
Author(s):  
A. Lodge
Keyword(s):  
Orthogonal Projection ◽  
Stereographic Projection ◽  
Spherical Surface ◽  
The Other ◽  
Spherical Trigonometry

The object of the papers is to consider two modes of representing points and lines on a spherical surface by points and lines on a plane: one method being by orthogonal projection, and the other by stereographic projection. The authors consider that students of spherical trigonometry ought to be able to accurately draw any figure with which they may have to deal, using one or other of the above methods of projection. The following propositions indicate methods of solving the various problems which would arise in connection with such accurate drawing.

Spherical Trigonometry after the Cesaro Method

1946 ◽  
Vol 30 (288) ◽  
pp. 52
Author(s):  
B. M. Brown ◽  
J. D. H. Donnay
Keyword(s):  
Spherical Trigonometry ◽  
Cesàro Method

Spherical Trigonometry after the Cesaro Method.

1946 ◽  
Vol 53 (1) ◽  
pp. 32
Author(s):  
H. V. Craig ◽  
J. D. H. Donnay
Keyword(s):  
Spherical Trigonometry ◽  
Cesàro Method

Heavenly Mathematics

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Stereographic Projection ◽  
Spherical Trigonometry ◽  
Classical Greece ◽  
Religious Rituals ◽  
The Earth ◽  
Celestial Navigation ◽  
History Of ◽  
The Rich ◽  
Ancient Astronomy ◽  
Rich History

This book traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. The book explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. The book conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. It is illustrated throughout with stunning historical images and informative drawings and diagrams. It also features easy-to-use appendices as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.

Aplikasi Teori Geodesi dalam Perhitungan Arah Kiblat: Studi untuk Kota Banjarnegara, Purbalingga, Banyumas, Cilacap, Kebumen

1970 ◽  
Vol 8 (2) ◽  
pp. 329-351
Author(s):  
Marwadi Marwadi
Keyword(s):  
Spherical Trigonometry

Para ulama sepakat bahwa menghadap kiblat menjadi syarat sahnya salat, tetapi mereka tidak sepakat bahwa  orang yang salat wajib menghadap ke bangunan Ka’bah atau ke arah Ka’bah. Untuk kesempurnaan ibadah, diperlukan usaha mencari arah kiblat yang tepat. Teori yang biasa digunakan untuk menghitung arah kiblat adalah teori ilmu ukur segitiga bola (spherical trigonometry). Sekarang, teori geodesi dengan rumus vincenty menjadi teori yang lebih akurat digunakan untuk menghitung arah kiblat daripada teori spherical trigonometry. Tulisan ini berusaha menggambarkan penggunaan teori geodesi dalam perhitungan arah kiblat untuk kota Banjarnegara, Purbalingga, Banyumas, Cilacap dan Kebumen. Teori geodesi menghasilkan arah kiblat untuk kota Banjarnegara 294°40’01.042”, Purbalingga 294°44’11.376”, Banyumas 294°45’25,582”, Cilacap 294°54’21.568”, dan Kebumen 294°44’16.752”. Jika arah kiblat tersebut dibandingkan dengan arah kiblat yang selama ini dipedomani, maka terdapat selisih rata-rata 0°7’32.74”. Dengan adanya hasil perhitungan yang mempunyai tingkat akurasi lebih tinggi, tentu akan menambah keyakinan dalam beribadah, walaupun arah kiblat yang selama ini menjadi pedoman juga masih dalam lingkup menghadap kiblat.

Sign in / Sign up

Export Citation Format

Share Document