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.

Wavefronts, light rays and caustic associated with the refraction of a plane wavefront by a conospherical lens

2021 ◽  
Author(s):  
José Israel Galindo-Rodríguez ◽  
Gilberto Silva-Ortigoza
Keyword(s):  
Optical Axis ◽  
Spherical Surface ◽  
Bessel Beam ◽  
Optical Element ◽  
The Other ◽  
Two Dimensional ◽  
Light Rays ◽  
Plane Wavefront ◽  
The Relationship ◽  
Dimensional Surface

Abstract The aim of the present work is to introduce a lens whose faces are a conical surface and a spherical surface. We illuminate this lens by a plane wavefront and its associated refracted wavefronts, light rays and caustic are computed. We find that the caustic region has two branches and can be virtual, real or one part virtual and the other real, depending on the values of the parameters characterizing the lens. Furthermore, we present a particular example where one of the branches of the caustic region is constituted by two segments of a line, one part is real and the other one virtual. The second branch is a two-dimensional surface with a singularity of the cusp ridge type such that its Gaussian curvature is different from zero. It is important to remark that for this example, the two branches of the caustic are disconnected. Because of this property and the result obtained by Berry and Balazs on the relationship between the acceleration of an Airy beam and the curvature of its corresponding caustic, we believe that using this optical element one could generate a scalar optical accelerating beam in the region where the caustic is a two-dimensional surface of revolution, and at the same time a scalar optical beam with similar properties to the Bessel beam of zero order in the region were the real caustic is a segment of a line along the optical axis.

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.

A pseudohedron

2004 ◽  
Vol 88 (512) ◽  
pp. 226-229
Author(s):  
H. Martyn Cundy
Keyword(s):  
The Other ◽  
Regular Polygons ◽  
Spherical Trigonometry ◽  
Equilateral Triangles ◽  
Infinite Set

The other day I received a long tube from a Canadian stranger containing a large poster featuring over a hundred polyhedra, including ‘all 92 Johnson polyhedra’. This term, though probably unfamiliar this side of the pond, was not completely unknown to me; it means convex polyhedra, excluding the regular and Archimedean ones, all of whose faces are regular polygons. Of course, as usual, we have to exclude those naughty polyhedra whose faces go around in pairs collecting squares (prisms) or equilateral triangles (antiprisms) and don’t know when to stop. The word convex is vital, otherwise there would be another infinite set. A lot of them are rather trivial, like sticking pyramids on the faces of a dodecahedron, but they include the deltahedra and many other interesting members. But they have at least one imitator who didn't quite make the grade. Trying to discover why, and how to coach him so that he would, I found that my spherical trigonometry was getting rather rusty so I set out to make one and see what was happening. I thought perhaps other readers would like to share this piece of antiresearch.

LXXIV. Spherical trigonometry reduced to plane

1752 ◽  
Vol 47 ◽  
pp. 441-444 ◽  
Keyword(s):  
The Other ◽  
Spherical Trigonometry ◽  
Single Word

It is observable, that the analogies of spherical trigonometry, exclusive of the terms co-fine and co-tangent, are applicable to plane, by only changing the expression, sine or tangent of side, into the single word, side: so that the business of plane trigonometry, like a corollary to the other, is thence to be inferr’d.

ARISTOTLE ON GEOMETRIC PERFECTION IN THE PHYSICAL WORLD

Mnemosyne ◽  
2003 ◽  
Vol 56 (4) ◽  
pp. 463-479
Author(s):  
Theokritos Kouremenos
Keyword(s):  
Spherical Surface ◽  
Physical World ◽  
The Other ◽  
The Sun ◽  
The Moon ◽  
Geometric Properties ◽  
Geometric Objects ◽  
Physical Objects

Although Aristotle is usually thought to deny the existence of physical objects with perfect geometric properties, J. Lear has argued that for Aristotle geometric properties can be perfectly instantiated in the physical world. In support of this thesis Lear has pointed mainly to de An. 403a10- 6, where Aristotle seems to admit the existence of physical objects with so perfect geometric properties that the edge of one touches the spherical surface of the other at a point. In this paper I argue that de An. 403a10- 6 does not commit Aristotle to the perfect instantiation of geometric properties in the physical world because the two objects assumed to touch each other at a point in this passage are not physical, as Lear takes it, but geometric: consequently, de An. 403a10-6 cannot be taken as evidence that geometric properties are perfectly instantiated in physical objects, from which geometric objects are abstracted. In Cael. 287b14-21, however, Aristotle notes that unlike the heaven a sphere made by a craftsman cannot be perfectly spherical and, in general, that no human artifact of whatever shape can be as geometrically perfect as the spherical heaven. This passage leaves no doubt that Aristotle denies the perfect instantiation of geometric properties in the sublunary region of his universe: some geometric properties are perfectly instantiated only in the superlunary region where the aether , the material of the heaven as well as of the celestial spheres that produce the apparent motions of each planet (the sun and the moon included), forms geometrically perfect spheres.

Concerning Cross Track Distance at Mid-Longitude (3)

2005 ◽  
Vol 58 (1) ◽  
pp. 152-153
Author(s):  
Paul Hickley
Keyword(s):  
The Other ◽  
Spherical Trigonometry ◽  
Map Projection ◽  
Solid Geometry ◽  
Cross Track

I am grateful to both Dr Ponsonby and Sqn Ldr Hoare for their responses to my original article and would like to thank them for replying. It is interesting that they have come up with such different approaches, one based on solid geometry (but not spherical trigonometry) and the other based on a map projection, which both give exact or near-exact answers.

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.

Maximum disorientation angles between crystals of any point groups and their corresponding rotation axes

2008 ◽  
Vol 41 (4) ◽  
pp. 803-807 ◽  
Author(s):  
Youliang He ◽  
John J. Jonas
Keyword(s):  
Computer Program ◽  
Input Data ◽  
Quaternion Algebra ◽  
Stereographic Projection ◽  
The Other ◽  
Vertex Enumeration ◽  
Rotation Axes ◽  
Symmetry Operators ◽  
Point Groups ◽  
Disorientation Angle

The symmetry-reduced misorientation,i.e.disorientation, between two crystals is represented in the angle–axis format, and the maximum disorientation angle between any two lattices of the 32 point groups is obtained by constructing the fundamental zone of the associated misorientation space (i.e.Rodrigues–Frank space) using quaternion algebra. A computer program based on vertex enumeration was designed to automatically calculate the vertices of these fundamental zones and to seek the maximum disorientation angles and respective rotation axes. Of the C_{32}^2 = 528 possible combinations of any two crystals, 129 pairs give rise to incompletely bounded fundamental zones (i.e.zones having at least one unbounded direction inR3); these correspond to a maximum disorientation angle of 180° (the trivial value). The other 399 pairs produce fully bounded fundamental zones that lead to nine different nontrivial maximum disorientation angles; these are 56.60, 61.86, 62.80, 90, 90.98, 93.84, 98.42, 104.48 and 120°. The associated rotation axes were obtained and are plotted in stereographic projection. These angles and axes are solely determined by the symmetries of the point groups under consideration, and the only input data needed are the symmetry operators of the lattices.

scholarly journals On Certain Mappings of Riemannian Manifolds

1965 ◽  
Vol 25 ◽  
pp. 121-142
Author(s):  
Minoru Kurita
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Conformal Mapping ◽  
Riemannian Manifolds ◽  
Stereographic Projection ◽  
Point Of View ◽  
The Other ◽  
Central Projection ◽  
Special Cases ◽  
Special Case

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

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