The rectangular coordinate system (or the Cartesian coordinate system) can be easily imagined as a system of mutually perpendicular lines (x axis, y axis, etc.) that are used in order to determine the position of the specific point (or any other geometrical figure) in space. Usually, when we are talking about the term ‘space’ from the point of view of the traditional rectangular coordinate system, we mean the term ‘manifold’. In practice, the Cartesian coordinate system is mainly used as the system of numbers that uniquely determine the concrete position of the point in the three-dimensional Euclidian space or the two-dimensional plane. In fact, this system of axes can be used not only in the two-dimensional space. Obviously, the Cartesian coordinate system for the two-dimensional space maintains two perpendicular directed lines. The x axis is a horizontal line and the y axis, which is perpendicular to the x axis, is a vertical line. The rectangular coordinate system that can be sued for determination of the concrete position of some geometric figure in the three-dimensional space requires selecting three mutually perpendicular lines: x axis, y axis and z axis. In fact, the use of the Cartesian coordinate system is not limited to these rates of space dimensions. This system of coordinates can be successfully used in higher dimensions. However, the scheme of this system is too over sophisticated for beginners in the sphere of classical and/or modern geometry. Therefore, let us summarize the facts about the coordinate system. Each line that forms the coordinate system describes the position of the specific geometric figure in one dimension. The horizontal line is usually called the x axis, whereas the vertical line is called the y axis. The point that determines the location in which these two lines intersect each other is called the origin. The number, which is associated with the sought-for point, is usually called the coordinate of the point. In fact, we can describe positions of all existing or potential geometric figures using length indicators that mark the distance from the origin to the point where there can be found the resulting normal carried out from the initial point to the axis (x axis, y axis, etc.). Therefore, we can determine the specific position of any point in n-dimensional space using the n-dimensional system of coordinates in which the n-axis determines the position of the figure in the n-dimension. In other words, the x axis represents the ‘first’ dimension, the y axis – the ‘second’ dimension, etc. Usually, various mathematical illustrations of the classical two-dimensional Cartesian systems present the first coordinate, which is traditionally called the abscissa, as a measure of distance on the x axis, oriented from left to right. Hence, the x axis is usually located horizontally. Analogically, the second coordinate, which is called the ordinate, is measured along a vertical axis (y axis) that is usually oriented from bottom to top. Nevertheless, modern software that is designed in order to provide advanced computer graphics and image processing often maintain a rectangular coordinate system in which with the x axis is oriented horizontally on the display. This convention has been developed in the 1960s because of the methods of storage images in display buffers. However, this method of visualization is not widely used in the lion’s share of classical mathematical literary sources.
However, the idea of the rectangular coordinates generalizes to permit axes that are not perpendicular to each other, and/or various units of measurement along each specific axis. In other words, we can describe the standard Cartesian coordinate system for the two-dimensional space as the specific issue of the general coordinate system. From this point of view, each single coordinate can be determined by projecting the point onto one axis (for example x axis) along a direction that is parallel to the other axis (y axis).
Doubtlessly, nowadays the rectangular coordinate system is widely used in virtually all areas of scientific research no matter how highly specialized is their subject of study. For example, even a simple business paper just cannot be written, eschewing the use of the classical coordinate system. Nowadays, all types of presentations apart from highly specialized and extremely sophisticated financial and stochastic computations require the use of different graphs, diagrams, tables, schedules, delineations and schemes. Furthermore, it is a common place to mention that information is much more easily perceived in a form of demonstrative and comprehensive visual object than in a form of an audio presentation. Of course, one has to build her geometrical figures in space, using the classical coordinate system.
Nevertheless, it is a quite intricate assignment to find the sphere of study in which the coordinate system is used more often than in geography. In fact, the modern geographic coordinate system can be represented in the form of a three-dimensional reference system, which permits us to locate specific points on the Earth’s surface. The standard unit of measure, which is used in the geographic coordinate system, is decimal degree. Each point on the Earth’s surface can be determined through two fundamental coordinate values that measure angles. The first measure is longitude, which is also called meridian (or a line of longitude), can be found measuring the degree of the specific angle between a reference plane and the specific plane, which passes through the points of the North and South poles. The reference plane is also widely known as the prime meridian. Nowadays, it is commonly known as the Greenwich meridian. The second measure is latitude that can be defined as the angle, which is formed by the intersection of a line perpendicular to the surface of the Earth at a point and the Equator plane. Therefore, it is obvious that unlike other geographical measures latitude values range from - 90 degrees to + 90 degrees. In specialized scientific references as well as in various examples of fictional literature one can often found another term that describes the same mathematical conception - ‘parallel’. In truth, the term ‘parallel’ is ubiquitously used not only amid dilettantes in the sphere of geography and geographical topology, but also amid specialists in these fields of study, because it correctly describes the physical sense of this geographic term: a particular value of latitude forms a notional circle, which is parallel to the Equator.
In fact, the geographic coordinates are angular units. However, in order to simplify the computations and eschew unnecessary complications when transferring units from one form to another ArcSDE stores and treats them as if they are planar. One does not have to be confused meeting this type in various dissertation abstracts or scientific articles that are dedicated to the topic of study. Therefore, longitude values are represented as those that belongs to the x axis, whereas the latitude values are represented as those that appertains to the y axis.
A standard geographic coordinate system comprises four fundamental elements. Here is a list of these components:
A standard projected coordinate system can be represented as a two-dimensional planar surface. Nevertheless, the surface of the Earth is, obviously, three-dimensional. In order to transform a three-dimensional space into a two-dimensional surface, we have to perform a specific operation called ‘projection’. Regardless of the specific type of the projection formulas, all of them can be presented as mathematical expressions, which allow us to convert data from a certain geographical location (latitude and longitude) on a spheroid to a corresponding location (x axis and y axis) on a two–dimensional surface. Thereby, the standard projected coordinate system maintain two different axes: the x axis, which represents east-west, and the y axis that represents north-south. The whole scheme is analogical to those that is used in the Cartesian coordinate system for the two-dimensional space. The x axis intersects with the y axis in the certain point (origin). Obviously, the coordinates of the origin are 0,0. The x axis refers to the distance along the horizontal line, and the y axis represents the distance along the vertical line. The projected or planar coordinate system locates points relative to the point of origin (0,0) and the x axis and y axis. Points below the x axis or to the left of the y axis have negative values.
A standard projected coordinate system also comprises the following elements: parameters, projection and units. In this coordinate system, units represent the geographical coordinates on the plane (two-dimensional) surface. In fact, the projection is a standard mathematical operation. In this case, it is used with an eye to converting geographic coordinates to projected coordinates. Therefore, actual geographic coordinates suffer transformation in the abstract geometric coordinates that are represented as measures of distance between the origin and the resulting perpendiculars on the axes (x axis and y axis). At last, parameters are the concrete units of measurement that are specific for each single case of study. Of course, all but these specific techniques of converting geographic data are interesting and necessary only for specialists in these spheres of study. Nevertheless, even different common writing assignments, which study some specific geographical position, require a knowledge of fundamental principles of work with a standard geographic coordinate system.
The rectangular coordinate system was invented in the XVII century by Rene Descartes – a famous French philosopher, physicist, engineer, mathematician and physiologist. Nowadays, Descartes is known as one of the greatest minds in human history because of his revolutionary philosophical conceptions, great achievements in mathematics and linear algebra, and a great number of significant inventions in different scientific disciplines. In fact, he is also famous as a talented publicist because of his extraordinary creative writing techniques and a polished literary style. Rene Descartes is one of the founders of analytic geometry, which he developed together with Fermat. From the point of view of modern geometric science, he deserves attention as an inventor of the coordinate system, which bears his name. Firstly, this conception was presented to the public in 1637 in his famous Geometry. Additionally, he was the first scientist who expressed the law of conservation of momentum, proposed the conception of impulse and laid the foundations of classical mechanics. In truth, the rectangular coordinate system proposed by Descartes was so elegant and effective that it has not suffered any significant transformation, despite the time that has passed since its invention. The Cartesian coordinate system, which is based on two perpendicular lines (y axis and x axis), invented by Rene Descartes is the same as the modern system of coordinates that is used in diverse sample case study papers.