Miljenko Lapaine

A New Direct Solution of the Transformation Problem of Cartesian into Ellipsoidal Coordinates

For the spatial determination of points with respect to the rotational ellipsoid we often use geodetic coordinates: geodetic latitude o, geodetic longitude e and height h above or under the surface of the reference ellipsoid. The relationship between the Cartesian coordinates x,y,z of a space point and the geodetic coordinates o,e,h are well known. The inverse problem of computing o,e and h when x,y,z are given, has been considered by many authors who suggested different methods for its solution.

Map Projection Reconstruction of a Map by Mercator

The paper describes the beginning of research on Mercator’s map Sclavonia, Croatia, Bosnia cum Dalmatiae parte. This map has many editions and shows a great part of present-day Croatia. Zagreb, the capital of Croatia, is shown in two different places on the map. Naturally, this raises the question of its accuracy. One of the first steps in the research process was finding the mathematical basis, i.e. the map projection used to create the map. The research results showed it was a trapezoidal projection, but there are no references as to where an appropriate derivation of the equations can be found. Therefore, on the basis of Mercator’s own description, cited in this paper, the derivation of the equations for his trapezoidal projection is given. He used this map projection to produce the map Sclavonia, Croatia, Bosnia cum Dalmatiae parte, as well as many other maps.

TESTING THE PRECISION OF THE ELECTRO-OPTICAL DISTANCE METER MEKOMETER ME 5000 ON THE CALIBRATION BASELINE ZAGREB

Abstract The paper examines the precision (accuracy) obtained with the Leica-Kem Mekometer ME 5000 distance meter on the calibration baseline Zagreb. A new method for the processing and analysing the measurements on the baseline is presented It provides an estimate of the precision, and, with thefrequency in the distance meter measured, the approximate accuracy as wel. For distances up to 100 m the standard deviation of distance measurement in one direction S = 0.058 mm is obtained For distances in the range of 100 to 3000 m, applying the proposed method of data processing, the standard deviation of a measured distance (measured forward and backward, and on the basis of adjusting all measurements on the baseline) srDII = 0.12mm + 0.21mm. D km was obtained in 1988, and in 1996 s′rDII = 0.07 mm − 0.13 mm.D km + 0.37mm.D2 km .

Map of the Gospić-Senj Diocese

Abstract Please click here to download the map associated with this article. The paper describes previous maps of the Gospić-Senj Diocese, Croatia, and the production of a new map. It explains the motivation in producing a new map of this diocese, and describes the first working version. Additionally, it also describes the factors taken into account during the maps production, and catalogues the difficulties involved in the organization and presentation of certain thematic content. The project on which the paper is based is unique because it researches sacral topics and objects that have not been hitherto acknowledged or depicted cartographically. It attempts to produce a map that is built around sacral themes and that emphasizes details relating to the above-mentioned diocese. The procedure, though complicated by a shortage of available data, entailed commonplace data collection and analysis, and afforded the authors exposure to the difficulties of making and designing maps.

Minimum Data Collection Dimensions in Topographic Information Systems

Abstract An overview is presented of the main subjects connected with topographic information systems, such as feature objects, object generalisation, acquisition criteria and minimum dimension. The ISO 19110 standard and INSPIRE initiative are explained in relation to minimum dimensions and data collection. The main section of the article refers to experiences in criteria acquisition within ATKIS and STOKIS. These are shown in mutual comparison and in comparison with solutions in classical cartography. The problem of minimum collecting dimensions of buildings is given special treatment. Adequate solutions to this problem are offered on the basis of research conducted. Experiences in treating linear and area features and how to select minimum collecting dimensions for the purpose of achieving optimal results are also presented. The research results are applicable in countries attempting to create their own topographic information system, or wanting to produce a new edition of their feature catalogue, as is the case in Croatia.

Map projection aspects

ABSTRACT A projection aspect is usually defined in references as the relation to the so-called auxiliary surface. However, such surfaces do not usually exist in map projection theory, which raises the issue of defining projection aspects without reference to auxiliary surfaces. This paper explains how projection aspects can be defined in two ways which are not mutually exclusive. According to the first definition, the aspect is the position of a projection axis in relation to the axis of geographic parameterization of a sphere. The projection axis is the axis of the pseudogeographic parameterization of a sphere, based on which the basic equations of map projection are defined. The basic equations of map projection are selected according to agreement and/or custom. According to this definition, aspects can be normal, transverse or oblique. According to the second definition, an aspect is the representation of the area in the central part of a map, and can be polar, equatorial or oblique. Therefore, it is possible for a map projection to have a normal and polar aspect, but it can also have a normal and equatorial aspect. The second definition is not recommended for use, due to its ambiguity.

Research on the Reliability of Mercator’s Map Sclavonia, Croatia, Bosnia cum Dalmatiae parte

This chapter describes further research into Mercator’s map Sclavonia, Croatia, Bosnia cum Dalmatiae parte, which depicts not only present-day Croatia, but Bosnia and Herzegovina, with parts of Slovenia and Serbia. Previous research determined the map projection and scale of the map. A great degree of divergence, which is evident regarding a number of towns on the map in relation to their real, mathematically calculated positions, led us to undertake a more detailed comparison of the geographic contents of Mercator’s map and the actual situation. In this chapter, the positions of river courses and their tributaries are compared, along with the positions of towns along river courses and those on the coast of the Adriatic Sea in the area of Dalmatia and the islands. The results of the research show that Mercator’s map Sclavonia, Croatia, Bosnia cum Dalmatiae parte is a good example of a lovely old map which is, unfortunately, unreliable. It is a reflection of mid-seventeenth century knowledge of the areas depicted and the cartographic capabilities of that era.

The Problem of Double Longitudes on Glavač’s Map

ABSTRACT Stjepan Glavač (1627-1680) produced a map of Croatia dated 1673, on which two meridians appear twice, 40°/51° and 41°/52°. The only written source regarding Glavač’s map is the dedication, which forms an integral part of the map. It does not say which prime meridians Glavač relied on, so in our research we attempted to establish what those prime meridians might have been. We approached the problem of the double longitudes using linear regression, through which we wanted to examine the relation of Glavač’s longitudes to modern values. An essential part of the research was a comparison of Glavač’s map with maps by his predecessors and contemporaries which show approximately the same prime meridians. The maps analysed, with their written values for the longitudes of certain places, indicated that the selection of prime meridians at that time was not entirely reliable. A further problem seemed to be the knowledge of the position of islands chosen as points of departure for longitudes. We concluded that the points of departure for Glavač’s dual longitudes were Palma, one of the Canary Islands, and the islands of Corvo and Flores in the Azores.

Famous People and Map Projections

Map projections have been developed in parallel with the development of map production and cartography in general. Development of sciences, technical achievements and needs of everyday life have gradually increased demands for production of various topographic and thematic maps in various scales and for various purposes, which required continuous improvement of map projections and mathematical basis of maps. Beginnings of map projections date as far as two thousand years ago, when ancient Greek scientists applied mathematical principles to projecting Earth and the starry sky and started applying the graticule. Hundreds of map projections have been invented since the antique. Many people have been interested in the theory of map projections and have written about them. Since there are so many of them, we decided to make a narrower selection of abut 40 people. Those people are presented in chronological rather than alphabetical order in this chapter. Their most important contributions to map projections are described and illustrated.

Short History on Map Projections

Map projections appeared more than two millennia ago, when ancient Greek scientists started applying mathematical principles to studying and representing the celestial sphere. Hundreds of map projections have been invented since. There is no limit to the number of possible map projections and their usefulness motivates scientists to develop new ones. Map projections have recently been developed to represent planets or asteroids which are too irregular to be modelled with a sphere or a rotational ellipsoid. In general, map projections are a field of mathematics, just like differential or projective geometry. Few map projections are based on perspective, while all other are some sort of mapping a continuous curved surface into a plane. Famous mathematician Leonhard Euler provided the first proof that a sphere’s surface can not be mapped onto a plane without distortion in 1777. Considering each map projection includes certain distortion, map projection theory primarily deals with researching map projection distortions. Most map projections can not be interpreted in a simple geometrical or physical manner and they are defined by mathematical formulae. Each map projection provides an image distorted in a different way. Studying map projections yields those distortions’ characteristics. Therefore, a cartographer should apply a map projection according to desired and arbitrary properties or conditions. Map projections have developed concurrently with development of map production and cartography in general. This chapter provides a brief overview of map projection development from their beginnings to the present days, mentioning famous names such as Gerhard Mercator, Johann Heinrich Lambert, Carl Friedrich Gaus, Nicolas Auguste Tissot, John Parr Snyder and many others.