**6. Conclusions**

Some typical mathematical problems in map projections are solved by means of computer algebra system which has powerful function of symbolical operation. The main contents and research results presented in this chapter are as follows:


Transverse Mercator Projection (or UTM) is usually implemented. Mathematically speaking, there is no essential difference between UTM and Gauss projections. The only difference is that the scale factor of UTM is 0.9996 rather than 1. With a slight modification, the non-iterative expressions of the forward and inverse Gauss projections can be extended to UTM projection accordingly.

#### **Author details**

Shao-Feng Bian *Department of Navigation, Naval University of Engineering, Wuhan, China* 

Hou-Pu Li

22 Cartography – A Tool for Spatial Analysis

2 0

(1 )

*w q il*

the conformal latitude

be explained as follows:

**6. Conclusions** 

the inverse Gauss projection indeed.

research results presented in this chapter are as follows:

latitude (40).

 

*x iy a eK*

13 5 7 9

 

arcsin(tanh ) *q* (80)

  (78)

 

arctanh(sin ) sin sin 3 sin 5 sin7 sin9

Therefore, the isometric latitude *q* and longitude *l* is known. Inserting *q* into (78) yields

Then one can compute the geodetic latitude through the inverse expansion of the conformal

(77) is the solution of the inverse Gauss projection by complex numbers. Its correctness can

The two equations in (78) are all elementary complex functions, so the mapping defined by (78) form *z x iy* to *w q il* meets the conformal mapping constraint. When *l* 0 , the imaginary part disappears and (78) restores to (77). Therefore, (78) meets the second and third constraints of Gauss projection when *l* 0 . Hence, it is clear that (78) is the solution of

Some typical mathematical problems in map projections are solved by means of computer algebra system which has powerful function of symbolical operation. The main contents and

1. Forward expansions of rectifying, conformal and authalic latitudes are derived, and some mistakes once made in the high orders of traditional forward formulas are pointed out and corrected. Inverse expansions of rectifying, conformal and authalic latitudes are derived using power series expansion, Hermite interpolation and Language's theorem methods respectively. These expansions are expressed in a series of the sines of the multiple arcs. Their coefficients are expressed in a power series of the first eccentricity of the reference ellipsoid and extended up to its tenth-order terms. The accuracies of these expansions are analyzed through numerical examples. The results show that the accuracies of these expansions derived by means of computer algebra system are

improved by 2~4 orders of magnitude compared to the formulas derived by hand. 2. Direct expansions of transformations between meridian arc, isometric latitude and authalic latitude function are derived. Their coefficients are expressed in a power series of the first eccentricity of the reference ellipsoid, and extended up to its tenth-order terms. Numerical examples show that the accuracies of these direct expansions are improved by 2~6 orders of magnitude compared to the traditional indirect formulas. 3. Gauss projection is discussed in terms of complex numbers theory. The non-iterative expressions of the forward and inverse Gauss projections by complex numbers are derived based on the direct expansions of transformations between meridian arc and isometric latitude, which enriches the theory of conformal projection. In USA, Universal

*Department of Navigation, Naval University of Engineering, Wuhan, China Key Laboratory of Surveying and Mapping Technology on Island and Reef, State Bureau of Surveying, Mapping and Geoinformation, Qingdao, China* 

#### **Acknowledgement**

This work was financially supported by 973 Program (2012CB719902), National Natural Science Foundation of China (No. 41071295 and 40904018), and Key Laboratory of Surveying and Mapping Technology on Island and Reef, State Bureau of Surveying, Mapping and Geoinformation, China (No.2010B04).

#### **7. References**

	- [9] Li Houpu, Bian Shaofeng (2008). Derivation of Inverse Expansions for Auxiliary Latitudes by Hermite Interpolation Method. Geomatics and Information Science of Wuhan University, 33(6): 623-626. (in Chinese)

**Chapter 0**

**Chapter 2**

**Web Map Tile Services for Spatial Data**

Ricardo García, Juan Pablo de Castro, Elena Verdú, María Jesús Verdú and Luisa María Regueras

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/46129

**1. Introduction**

Quality of Service (QoS).

similar initiatives.

cited.

**Infrastructures: Management and Optimization**

Web mapping has become a popular way of distributing online mapping through the Internet. Multiple services, like the popular Google Maps or Microsoft Bing Maps, allow users to visualize cartography by using a simple Web browser and an Internet connection. However, geographic information is an expensive resource, and for this reason standardization is needed to promote its availability and reuse. In order to standardize this kind of map services, the Open Geospatial Consortium (OGC) developed the Web Map Service (WMS) recommendation [1]. This standard provides a simple HTTP interface for requesting geo-referenced map images from one or more distributed geospatial databases. It was designed for custom maps rendering, enabling clients to request exactly the desired map image. This way, clients can request arbitrary sized map images to the server, superposing multiple layers, covering an arbitrary geographic bounding box, in any supported coordinate

However, this flexibility reduces the potential to cache map images, because the probability of receiving two exact map requests is very low. Therefore, it forces images to be dynamically generated on the fly each time a request is received. This involves a very time-consuming and computationally-expensive process that negatively affects service scalability and users'

A common approach to improve the cachability of requests is to divide the map into a discrete set of images, called tiles, and restrict user requests to that set [2]. Several specifications have been developed to address how cacheable image tiles are advertised from server-side and how a client requests cached image tiles. The Open Source Geospatial Foundation (OSGeo) developed the WMS Tile Caching (usually known as WMS-C) proposal [3]. Later, the OGC released the Web Map Tile Service Standard (WMTS) [4] inspired by the former and other

> ©2012 García et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

©2012 García et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

reference system or even applying specific styles and background colors.

