History: The Cartesian coordinate system is named after Rene Descartes(1596-1650), the noted French mathematician and philosopher, who was among the first to describe its properties. However, historical evidence shows that Pierre de Fermat (1601-1665), also a French mathematician and scholar, did more to develop the Cartesian system than did Descartes. The development of the Cartesian coordinate system enabled the development of perspective and projective geometry. It would later play an intrinsic role in the development of calculus by Isaac Newton andGottfried Wilhelm Leibniz. 3] Nicole Oresme, a French philosopher of the 14th Century, used constructions similar to Cartesian coordinates well before the time of Descartes. Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space. Cartesian coordinate system: A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).
In general, one can specify a point in a space of any dimension n by use of n Cartesian coordinates, the signed distances from n mutually perpendicular hyperplanes. Terms to remember: Coordinate Axes Three mutually perpendicular coordinate lines X-axis, Y-axis, Z-axis (intersecting at origin). • Coordinate Planes Three planes determined by coordinate axes XY-plane, XZ-plane, YZ-plane • Coordinates •Any point is determined through an ordered triple (a, b, c) •P has coordinates (a, b, c)means To locate P , we start from the origin, move a -units along X-axis, hen b-units parallel to Y-axis and then c -units parallel to Z-axis. Number line A line with a chosen Cartesian system is called a number line. Every real number, whether integer, rational, or irrational, has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum which includes the real numbers. Quadrants and octants The four quadrants of a Cartesian coordinate system. The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes.
These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are I (+,+), II (? ,+), III (? ,? ), and IV (+,? ). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right (“northeast”) quadrant. Similarly, a three-dimensional Cartesian system defines a division of space into eight regions or octants, according to the signs of the coordinates of the points. The octant where all three coordinates are positive is sometimes called the first octant; however, there is no established nomenclature for the other octants.
The n-dimensional generalization of the quadrant and octant is the orthant. [edit]Cartesian space A Euclidean plane with a chosen Cartesian system is called a Cartesian plane. Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with all possible pairs of real numbers; that is with the Cartesian product , where is the set of all reals. In the same way one defines a Cartesian space of any dimension n, whose points can be identified with the tuples (lists) of n real numbers, that is, with . 3–Dimensional Rectangular Coordinate System
In the 2–dimensional rectangular coordinate system we have two coordinate axes that meetat right angles at the origin (Fig. 1), and it takes two numbers, an ordered pair (x, y), tospecify the rectangular coordinate location of a point in the plane (2 dimensions). Each ordered pair (x, y) specifies the location of exactly one point, and thelocation of each point is given by exactly one ordered pair (x, y). The x and y values are the coordinates of the point (x, y) . The situation in three dimensions is very similar. In the3–dimensional rectangular coordinate system we have three coordinateaxes that meet at right angles (Fig. ), and three numbers, an orderedtriple (x, y, z), are needed to specify the location of a point. Eachordered triple (x, y, z) specifies the location of exactly one point,and the location of each point is given by exactly one ordered triple(x, y, z). The x, y and z values are the coordinates of thepoint (x, y, z). Fig. 3 shows the location of thepoint (4, 2, 3) . Right–hand orientation of the coordinate axes (Fig. 4):Imagine your right arm along the positive x–axis withyour hand at the origin and your index finger curlingtoward the positive y–axis.
Then, in a right–handcoordinate system, your extended thumb points alongthe positive z–axis. Other orientations of the axes arepossible and valid (with appropriate labeling), but theright–hand system is the most common orientation and is the one we will generally use. Relations between Cartesian, Cylindrical, and Spherical Coordinates Consider a cartesian, a cylindrical, and a spherical coordinate system, related as shown in Figure 1. Figure 1: Standard relations between cartesian, cylindrical, and spherical coordinate systems. The origin is the same for all three.
The positive z-axes of the cartesian and cylindrical systems coincide with the positive polar axis of the spherical system. The initial rays of the cylindrical and spherical systems coincide with the positive x-axis of the cartesian system, and the rays =90° coincide with the positive y-axis. Then the cartesian coordinates (x,y,z), the cylindrical coordinates (r, ,z), and the spherical coordinates ( , , ) of a point are related as follows: Applications Each axis may have different units of measurement associated with it (such as kilograms, seconds, pounds, etc. ).
Although four- and higher-dimensional spaces are difficult to visualize, the algebra of Cartesian coordinates can be extended relatively easily to four or more variables, so that certain calculations involving many variables can be done. (This sort of algebraic extension is what is used to define the geometry of higher-dimensional spaces. ) Conversely, it is often helpful to use the geometry of Cartesian coordinates in two or three dimensions to visualize algebraic relationships between two or three of many non-spatial variables. The graph of a function or relation is the set of all points satisfying that function or relation.
For a function of one variable, f, the set of all points (x,y) where y = f(x) is the graph of the function f. For a function of two variables, g, the set of all points (x,y,z) where z = g(x,y) is the graph of the function g. A sketch of the graph of such a function or relation would consist of all the salient parts of the function or relation which would include its relative extrema, its concavity and points of inflection, any points of discontinuity and its end behavior. All of these terms are more fully defined in calculus. Such graphs are useful in calculus to understand the nature and behavior of a function or relation.
