scholarly journals Polar and Equatorial Aspects of Map Projections?

2019 ◽  
Vol 2 ◽  
pp. 1-6
Author(s):  
Miljenko Lapaine ◽  
Nedjeljko Frančula
Keyword(s):  
Perspective Projection ◽  
Developable Surface ◽  
Map Projection ◽  
Map Projections ◽  
Developable Surfaces

<p><strong>Abstract.</strong> There is no standard or generally accepted terminology of aspect in the theory of map projections. The term is probably derived from the concept that a graticule is produced by perspective projection of the meridians and parallels on a sphere onto a developable surface. Developable surfaces are widely accepted, and it is almost impossible to find a publication that deals with map projections in general and without developable surfaces story. If found, it usually classifies projections as cylindrical, conical and azimuthal, and applies developable surfaces to define the projection aspect. This paper explains why applying developable surfaces in the interpretation of map projections is not recommended, nor defining the aspect of all projections by the position of a midpoint as polar, equatorial, or oblique. In fact, defining a projection aspect this way is invalid in general, and obscures the fact that the aspect depends on the class to which a particular map projection belongs.</p>

Secant Parallels in Azimuthal Projections

2019 ◽  
Vol 946 (4) ◽  
pp. 39-54
Author(s):  
М. Lapaine
Keyword(s):  
Lateral Surface ◽  
Developable Surface ◽  
Converse Statement ◽  
Map Projections ◽  
Developable Surfaces

Map projections are commonly approached as mapping onto developable surfaces; cylindrical projections onto the lateral surface of a cylinder, conic projections onto the lateral surface of a cone, and azimuthal projections onto a plane. If an intermediate developable surface intersects the Earth’s sphere or ellipsoid, the projection is referred to as a secant projection. The intersection of a developable surface and the Earth’s sphere or ellipsoid, e.g. secant parallel is considered a standard parallel. In this paper the definitions of secant and standard parallel in azimuthal projections are given. The first conclusion is that the secant and standard parallels are two distinct notions. The second one is that a standard parallel, if such a parallel exists in an azimuthal projection, is a secant parallel, while the converse statement is not true in general. Furthermore, it is shown that there are azimuthal projections with only one secant parallel that is not standard, with only one standard parallel which is also secant one, with two different secant parallels, and with one standard and one secant parallel.

Geometric Design and Fabrication of Developable Bezier Surfaces

1992 ◽  
Author(s):  
R. M. C. Bodduluri ◽  
B. Ravani
Keyword(s):  
Geometric Design ◽  
Developable Surface ◽  
Developable Surfaces ◽  
Surface Patches ◽  
Bézier Surfaces ◽  
Point Representation ◽  
Cad Cam ◽  
C2 Continuity ◽  
Type Construction ◽  
Bezier Surfaces

Abstract In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop direct representations of developable surfaces in terms of point as well as plane geometries. The point representation uses a Bezier curve, the tangents of which span the surface. The plane representation uses control planes instead of control points and determines a surface which is a Bezier interpolation of the control planes. In this case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In design of piecewise surface patches, a computational geometric algorithm similar to Farin-Boehm construction used in design of piecewise parametric curves is developed for designing developable surfaces with C2 continuity. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.

Design of a Two-Piece Brassiere Cup From its Data Points Toward its Automation

2020 ◽  
Author(s):  
Kotaro Yoshida ◽  
Hidefumi Wakamatsu ◽  
Eiji Morinaga ◽  
Takahiro Kubo
Keyword(s):  
Three Dimensional ◽  
Developable Surface ◽  
Two Dimensional ◽  
Trial And Error ◽  
Design Efficiency ◽  
Developable Surfaces ◽  
Data Points ◽  
Dimensional Shape ◽  
The Given ◽  
Three Dimensional Shape

Abstract A method to design the two-dimensional shapes of patterns of two piece brassiere cup is proposed when its target three-dimensional shape is given as a cloud of its data points. A brassiere cup consists of several patterns and their shapes are designed by repeatedly making a paper cup model and checking its three-dimensional shape. For improvement of design efficiency of brassieres, such trial and error must be reduced. As a cup model for check is made of paper not cloth, it is assumed that the surface of the model is composed of several developable surfaces. When two lines that consist in the developable surface are given, the surface can be determined. Then, the two-piece brassiere cup can be designed by minimizing the error between the surface and given data points. It was mathematically verified that the developable surface calculated by our propose method can reproduce the given data points which is developable surface.

scholarly journals Developable mechanisms on developable surfaces

2019 ◽  
Vol 4 (27) ◽  
pp. eaau5171 ◽  
Author(s):  
Todd G. Nelson ◽  
Trent K. Zimmerman ◽  
Spencer P. Magleby ◽  
Robert J. Lang ◽  
Larry L. Howell
Keyword(s):  
Mechanical Systems ◽  
Relative Orientation ◽  
Developable Surface ◽  
Curved Surfaces ◽  
Surface Type ◽  
Complex Tasks ◽  
Rigid Link ◽  
Developable Surfaces ◽  
Flat Sheet ◽  
Wide Range

The trend toward smaller mechanism footprints and volumes, while maintaining the ability to perform complex tasks, presents the opportunity for exploration of hypercompact mechanical systems integrated with curved surfaces. Developable surfaces are shapes that a flat sheet can take without tearing or stretching, and they represent a wide range of manufactured surfaces. This work introduces “developable mechanisms” as devices that emerge from or conform to developable surfaces. They are made possible by aligning hinge axes with developable surface ruling lines to enable mobility. Because rigid-link motion depends on the relative orientation of hinge axes and not link geometry, links can take the shape of the corresponding developable surface. Mechanisms are classified by their associated surface type, and these relationships are defined and demonstrated by example. Developable mechanisms show promise for meeting unfilled needs using systems not previously envisioned.

scholarly journals Parameterisation of developable surfaces by asymptotic lines

1996 ◽  
Vol 54 (3) ◽  
pp. 411-421 ◽  
Author(s):  
Vitaly Ushakov
Keyword(s):  
Explicit Form ◽  
Gaussian Curvature ◽  
Developable Surface ◽  
Developable Surfaces ◽  
Asymptotic Lines

An example of a “non-developable” surface of vanishing Gaussian curvature from W. Klingenberg's textbook is considered and its place in the theory of 2-dimensional developable surfaces is pointed out. The surface is found in explicit form. Other examples of smooth developable surfaces not allowing smooth asymptotic parametrisation are analysed. In particular, Hartman and Nirenberg's example (1959) is incorrect.

V. On certain developable surfaces

1863 ◽  
Vol 12 ◽  
pp. 279-280 ◽  
Keyword(s):  
Developable Surface ◽  
Developable Surfaces

If U = 0 be the equation of a developable surface, or say a developable, then the hessian HU vanishes, not identically, but only by virtue of the equation U = 0 of the surface; that is, HU contains U as a factor, or we may write HU = U. PU. The function PU, which for the developable replaces, as it were, the hessian HU, is termed the prohessian; and since, if r be the order of U, the order of HU is 4 r —8, we have 3 r —8 for the order of the prohessian. If r =4, the order of the prohessian is also 4; and in fact, as is known, the prohessian is in this case = U.

scholarly journals A Semantic Representation of Map Projections Knowledge

2019 ◽  
Vol 1 ◽  
pp. 1-1
Author(s):  
E. Lynn Usery
Keyword(s):  
Knowledge Base ◽  
Semantic Representation ◽  
Appropriate Use ◽  
Easy Method ◽  
Map Projection ◽  
Map Projections ◽  
Semantic Organization ◽  
Body Of Knowledge ◽  
Fundamental Part

<p><strong>Abstract.</strong> A body of knowledge for cartography requires representing knowledge of the specific sub topics in the field. Map projections is a fundamental part of the knowledge base for cartography and a wealth of material exists on knowledge of map projections. Semantic organization of such knowledge is of primary importance to the access and use of map projections knowledge. This project builds a semantic representation for the fundamental parts of map projection knowledge. The semantics capture the concepts and relations between these concepts providing the user an easy method to access the knowledge and apply it to specific problems. The semantics represent classes of projections and the properties associated with those classes as well as the appropriate use. Such a representation can be accessed by humans or machines to arrive at appropriate selection and use of map projection theory.</p>

scholarly journals Reference frame and map projection for irregular shaped celestial bodies

2020 ◽  
Vol 2 ◽  
pp. 1-2
Author(s):  
Krisztián Kerkovits ◽  
Tünde Takáts
Keyword(s):  
Celestial Body ◽  
Reference Frame ◽  
Reference Frames ◽  
Surfaces Of Revolution ◽  
Map Projection ◽  
Map Projections ◽  
Ellipsoid Of Revolution ◽  
Celestial Bodies ◽  
Best Fitting ◽  
Comet 67P

Abstract. Recent advancements of technology resulted in greater knowledge of the Solar System and the need for mapping small celestial bodies significantly increased. However, creating a good map of such small objects is a big challenge for the cartographer: they are usually irregular shaped, the usual reference frames like the ellipsoid of revolution is inappropriate for their approximation.A method is presented to develop best-fitting irregular surfaces of revolution that can approximate any irregular celestial body. (Fig. 1.) Then a simple equal-area map projection is calculated to map this reference frame onto a plane. The shape of the resulting map in this projection resembles the shape of the original celestial body.The usefulness of the method is demonstrated on the example of the comet 67P/Churyumov-Gerasimenko. This comet has a highly irregular shape, which is hard to map. Previously used map projections for this comet include the simple cylindrical, which greatly distorts the surface and cannot depict the depressions of the object. Other maps used the combination of two triaxial ellipsoids as the reference frame, and the gained mapping had low distortion but at the expense of showing the tiny surface divided into 11 maps in different complicated map projections (Nyrtsov et. al., 2018). On the other hand, our mapping displays the comet in one single map with moderate distortion and the shape of the map frame suggests the original shape of the celestial body (Fig. 2. and 3.).

scholarly journals Map Projection Article on Wikipedia

2019 ◽  
Vol 1 ◽  
pp. 1-8
Author(s):  
Miljenko Lapaine
Keyword(s):  
Mathematical Approach ◽  
Map Projection ◽  
Map Projections ◽  
Mathematical Formulas ◽  
Near Future

<p><strong>Abstract.</strong> People often look up information on Wikipedia and generally consider that information credible. The present paper investigates the article Map projection in the English Wikipedia. In essence, map projections are based on mathematical formulas, which is why the author proposes a mathematical approach to them. Weaknesses in the Wikipedia article Map projection are indicated, hoping it is going to be improved in the near future.</p>

Newer Theory and More Robust Algorithms for Computer-Aided Design of Developable Surfaces

2005 ◽  
Vol 42 (03) ◽  
pp. 71-79
Author(s):  
B. Konesky
Keyword(s):  
Computer Aided Design ◽  
Developable Surface ◽  
Robust Algorithms ◽  
Developable Surfaces ◽  
Space Curves ◽  
Surface Areas ◽  
C Programming ◽  
Computer Aided ◽  
Reliable Solution ◽  
Aided Design

The use of developable surfaces in design is of engineering importance because of the relative ease with which they can be manufactured. The problem of how to make surfaces developable is not new. The usual technique is by using two space curves, defining the edges of the surface. These are first created, and then a set of rulings are constructed between the space curves under the constraint of being developable. A problem with existing algorithms for designing developable surfaces is the tendency to include nondevelopable portions of the surface: areas of regression. A more reliable solution to the problem of creating a developable surface is presented. The key to the method is to define the developable surface in terms of a normal directrix. The shape of the normal directrix defines the resulting developable surface. Algorithms are defined to compute the shape of a normal directrix from a pair of space curves. Intersecting adjacent developable surfaces and generating the flat plate layouts were also accomplished. This paper presents research and development that started around 1987. The algorithms were implemented using ANSI C++ programming language and commercial computer-aided design and manufacturing (CAD and CAM) software programs.

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