Analytic Geometry

Analytic geometry is a branch of mathematics which uses algebraic equations to describe the size and position of geometric figures on a coordinate system. Developed during the seventeenth century, it is also known as Cartesian geometry or coordinate geometry. The use of a coordinate system to relate geometric points to real numbers is the central idea of analytic geometry. By defining each point with a unique set of real numbers, geometric figures such as lines, circles, and conics can be described with algebraic equations. Analytic geometry has found important applications in science and industry alike. During the seventeenth century, finding the solution to problems involving curves became important to industry and science. In astronomy, the slow acceptance of the heliocentric theory of planetary motion required mathematical formulas which would predict elliptical orbits. Other areas such as optics, navigation and the military required formulas for things such as determining the curvature of a lens, the shortest route to a destination, and the trajectory of a cannon ball. Although the Greeks had developed hundreds of theorems related to curves, these did not provide quantitative values so they were not useful for practical applications. Consequently, many seventeenth-century mathematicians devoted their attention to the quantitative evaluation of curves. Two French mathematicians, Rene Descartes (1596-1650) and Pierre de Fermat (1601-1665) independently developed the foundations for analytic geometry. Descartes was first to publish his methods in an appendix titled La geometrie of his book Discours de la methode (1637). The link between algebra and geometry was made possible by the development of a coordinate system which allowed geometric ideas, such as point and line, to be described in algebraic terms like real numbers and equations. In the system developed by Descartes, called the rectangular Cartesian coordinate system, points on a geometric plane are associated with an ordered pair of real numbers known as coordinates. Each coordinate describes the location of a single point relative to a fixed point, the origin, which is created by the intersection of a horizontal and a vertical line known as the x-axis and y-axis respectively. The relationship between a point and its coordinates is called one-to-one since each point corresponds to only one set of coordinates. The x and y axes divide the plane into four quadrants. The sign of the coordinates is either positive or negative depending in which quadrant the point is located. Starting in the upper right quadrant and working clockwise, a point in the first quadrant would have a positive value for the abscissa and the ordinate. A point in the fourth quadrant (lower right hand corner) would have negative values for each coordinate. The notation P (x,y) describes a point P which has coordinates x and y. The x value, called the abscissa, represents the horizontal distance of a point away from the origin. The y value, known as the ordinate, represents the vertical distance of a point away from the origin.

1.2 Cartesian Coordinates in the Plane

In cartesian coordinates (or rectangular coordinates), the ``address'' of a point P is given by two real numbers indicating the positions of the perpendicular projections from the point to two fixed, perpendicular, graduated lines, called the axes. If one coordinate is denoted x and the other y, the axes are called the x-axis and the y-axis, and we write P=(x,y). Usually the x-axis is drawn horizontal, with x increasing to the right, and the y-axis is drawn vertical, with y increasing going up. The point x = 0, y = 0 is the origin, where the axes intersect. See Figure 1.

Figure 1:

In cartesian coordinates, P=(4,3), Q=(-1.3,2.5), R=(-1.5,-1.5), S=(3.5,-1), and T=(4.5,0). The axes divide the plane into four quadrants: P is in the first quadrant, Q in the second, R in the third, and S in the fourth. T is on the positive x-axis.