Analytic geometry

The basic concept introduced by analytic geometry is that the position of a point in space can be expressed in terms of numerical quantities. From this starting point, all geometrical properties and relationships can be reduced to numerical terms; conversely, the properties of sets of numbers can be presented in the form of geometrical diagrams.

Overview

Analytic geometry brings the powerful tools of algebra to describe and analyze the properties of geometrical objects, points, lines, circles and more complicated curves. The crucial step is to define numerically the position of a point. It is helpful initially to restrict consideration to points that lie in a plane—that is, points that can be visualized as being on a flat sheet of paper; in this way, the techniques of analytic geometry can describe the results of plane geometry.

A single number is required to describe the position of a point on a line. A reference point, called the "origin," must first be selected. All other points on the line are then specified by their distance from the origin. An arbitrary choice must be made that one side of the origin corresponds to positive distances, the other to negative. This signed number is called the coordinate of the point. The sign convention allows a general rule for calculating the distance between two points. One coordinate is subtracted from the other, irrespective of whether the points are on the same or different sides of the origin. For example, if points A and B have coordinates 10 and -15, respectively, the distance between them is 10 - (-15). By the rules of arithmetic for negative numbers, subtracting a negative number gives the same result as adding the corresponding positive one. Thus, the distance between the points is 25 units, which, as can be seen from a simple diagram, is the correct result. The "world" lying entirely on a line is one-dimensional.

Two coordinates are required to locate a point on a plane. The "Cartesian coordinates" of a point are obtained as follows: Define the basic reference point, the origin. Draw a simple diagram; mark a point in the middle of a sheet of paper and label it "origin." Draw a straight horizontal line through the origin, extending all the way across the paper on both sides. Through the origin, draw a second line that is vertical (at a right angle) to the first. Each of these lines is called an "axis" (plural "axes"). Mark a point anywhere on the sheet of paper. From the point, draw a line parallel to the vertical axis until it crosses the horizontal axis. The horizontal distance from the origin to this point (points to the right of the origin are reckoned to be positive) is one coordinate. Draw a horizontal line from the point until it crosses the vertical axis. The distance along the vertical axis from the origin to the crossing point, where distances measured upward are considered positive, is the other coordinate. It is conventional to state them in the above order (horizontal, vertical). For historical reasons, the horizontal and vertical axes are often referred to as the x- and y-axes respectively, and x and y are often used to designate the coordinates of a general point the position of which is not specified explicitly with numerical values.

A line or curve is a collection of points that have a special relationship between the two coordinates. The points ,0), (1.5, 0), ,0), (-5, 0), and ,0) lie somewhere on the x-axis. In general, the point, (x, 0) is part of this line, whatever value x is given. For all points on the x-axis, the y-coordinate is equal to zero (y = 0). This statement of the common feature for all the points on the line is called the "equation of the line." Similarly, x = 0 is the equation of the y-axis. All the points on a line parallel to the x-axis and one unit of length above it have coordinates (x, 1); the equation of this line is y = 1. Similarly, x = -3 is the equation of a line parallel to the y-axis and three units to the left.

The set of points such as ,, ,, (-5, -5) and the like share the feature that the x-coordinate equals the y-coordinate. These and all similar points lie on a straight line passing through the origin and making an angle of 45 degrees with the x-axis. The equation of this line, which defines the relationship between the coordinates for all points on the line, is y = x. A set of points such as ,, ,.5), (-4, -2), ,.5) and the like have the y-coordinate one-half the x-coordinate; the corresponding equation is y = ½x. These points lie on a straight line passing through the origin and making an angle of about 26½ degrees with the x-axis. The inclination of the line also can be described by its "slope" or "gradient," which is defined as the number of units the y-coordinate increases when the x-coordinate is increased by 1. In this example, y increases by ½ when x is increased by 1, no matter what part of the line is examined: the segment between the points ,0.5) and , or between the points , and ,.5), for example. The slope of the line y = x is 1. The relationship between slope and the angle with the x-axis is more complicated than a simple proportionality, however; note, for example, that the angle for a slope of ½ is not one-half of 45 degrees.

The set of points ,, ,.5), (-4, -1), ,.5) and the like each lie one unit above the set of points lying on the line with equation y = ½x. This relationship, common to all points, is written y = ½x + 1. The set of points lies on a straight line parallel to the line y = ½x and one unit above it. The slope of the line is also ½, but instead of going through the origin, the line crosses the y-axis at the point (0, 1), one unit above the origin. This quantity is called the "y-intercept" of the line. In the equation of the line, the slope appears as the factor multiplying the value of x, and the y-intercept is the quantity added on to this. A line with equation y = 2x -3, therefore, has a slope of 2 and a y-intercept of -3. The equation of any line can be written as y = mx + b, where m stands for any numerical value and is the slope of the line and b is the value of the intercept. A complete line can thus be specified by only two numbers, its slope and its intercept. From these quantities, the coordinates of any point on the line can be calculated.

A geometrical result is that two lines intersect in one point; that is, one point is common to both collections of points. In the examples above, the point , appears on both the line y = x and the line y = ½x + 1. The point , is thus the point of intersection of these two lines. This result can be obtained directly from the equations of the lines; it corresponds to the following word problem: "What values must be given to x and y so that y is the same as x and, at the same time, y is equal to one-half of x plus one?" This problem does not refer to lines or points or coordinates but asks to find a pair of numbers satisfying the conditions of the problem. The result is obtained by algebra through an operation called "solving the equations."

This example illustrates what analytic geometry can do. A geometrical problem ("Where do two straight lines intersect?") has been transformed into an algebraic problem ("What is the solution to a pair of equations?"). The solution of the equations is something that can then be found, in turn yielding the solution of the geometrical problem without the necessity of drawing a diagram. The converse operation can be useful to visualize an algebraic problem in terms of a corresponding geometrical diagram. For example, it is known from geometry that parallel lines do not intersect; consequently, it can be stated that there are pairs of equations—ones that represent parallel lines—that have no solution. The geometrical property thus reveals something about the properties of the equations. As an example, consider the two equations y = x and y = x + 1, which represent lines that both have a slope of 1. It is impossible to find a pair of values, x, y that will make both equations true at the same time.

Geometry is more than points and lines; there are more complicated figures. If one plots the set of points (-3, 9), (-2, 4), (-1, 1), (0, 0), ,, ,, ,, they appear to be part of a U-shaped curve. The feature shared by all the points is that the y-coordinate is the square of the x-coordinate; the equation of the curve thus described would be y = x². Additional points can be calculated with the aid of this equation to fill in the gaps in the figure and show the shape more clearly. The curve is an example of a parabola, a variety of curve that has many useful and important properties. For example, the path taken by a projectile near the surface of the earth is a parabola (though upside-down compared with the one given as an example). The relationship between the two coordinates, y = x², is the equation of this particular parabola. Equations describing other parabolas can be more complicated, but any parabola can be represented by its own equation.

A straight line can intersect the parabola in two points. Suppose the line has the equation y = x + 1; the problem of determining the points of intersection is similar to the problem of determining the point of intersection of two straight lines. One must ask, "What values of x and y make both equations, y = x² and y = x + 1, true at the same time?" The solution can be calculated, giving the coordinates of the two points as (1.62, 2.62) and (-0.62, 0.38) approximately.

Another familiar shape, the circle, can also be described by an equation. A circle with radius 2 units with its center at the origin (0, 0) is composed of all the points that lie a distance of 2 units away from the origin, irrespective of the direction from the origin. The figure above shows a point P with coordinates (x, y); P is on the circle, which is shown with a dotted line. The point, P, the origin, O, and the point on the x-axis directly below P make up a right-angled triangle. A famous result from geometry, the Pythagorean theorem, shows that the square of the length from O to P (OP) is equal to the sum of squares of the lengths of the other two sides. These two sides have lengths equal to x and y, as will be clear from a review of how coordinates are defined. The length of OP is the distance of the point from the origin. In the example, this has a value of 2 units. Thus, if the point (x, y) is on the circle, it lies 2 units away from the origin, yielding the equation x² + y² = 2². All pairs of numbers (x, y) that make this equation true represent points on the circle. Therefore, x² + y² = 2² is the equation of a circle centered on the origin with a radius of 2 units. All other circles with other centers and other radii have similar equations, though more complicated.

All other curves are described in the same way, by defining the relationship that exists between the x- and y-coordinates of all points that lie on the curve. The positions of the intersection points of curves are obtained by the methods of algebra—that is, by solving the equations of the curves. All other geometrical properties, such as the tangents to the curve, angles between lines, and distances between lines and the curve, can also be expressed in terms of algebra.

The discussion thus far has considered plane geometry, in which the lines and points all lie on a single surface. The concepts of analytic geometry, however, are readily extended to describe the location of points in space. Three numbers or coordinates are required. The Cartesian coordinates are extended so that (x, y, z) gives the location of a point z units directly above the point previously described by the coordinates (x, y). In the imagined diagram drawn previously, the point would be located above (or below) the surface of the paper. A point on the paper would now have the coordinates, (x, y, 0). Since three coordinates are required to specify the location of a point in space, we are now dealing with a three-dimensional situation. Equations that specify relations between the coordinates describe not only lines and curves in space but also planes and curved surfaces. For example, all the points in space that are 2 units from the origin lie on a sphere of radius 2 units with its center at the origin. A sphere in three dimensions is the counterpart of a circle in two dimensions. In fact, its equation is very similar: x² + y² + z²= 2² for a sphere with a radius of 2, for example. The concept of coordinates can be extended even further, so that (x, y, z, u) are the coordinates of a point in a four-dimensional world. Analytic geometry allows results to be obtained for the geometry of such a world, even though it is beyond human power to visualize it.

Applications

The most prevalent application of analytic geometry is in the presentation of numerical data in the form of graphs. Examples abound of such things as the progress of average wages over the past few years or weeks. A set of pairs of numbers, for example (year, average wage for that year), are interpreted as coordinates of points, and a diagram is drawn plotting the points. Two columns of numbers are thus translated into a picture. The combination of human eye and brain is an effective pattern analyzer, much more readily digesting information that is presented in a visual format than information presented as a set of numbers. It is said that "a picture is worth a thousand words"; this aphorism is even more true of numbers, where the exchange rate between pictures and numerical values can be much higher.

Previously, it was shown that the position of the intersection of two lines is found by solving the equations of the lines. Conversely, the solution to the equations is found by plotting the lines they represent and locating the intersection point. More complicated equations, such as can occur in the study of electrical engineering or navigation, can be solved approximately by plotting the curves that they represent and determining the coordinates of the points of intersection. This technique is employed in "nomograms," which were once widely used to obtain the solution of difficult mathematical problems graphically. The programmable hand calculator has largely superseded this geometrical method of solving equations.

Computers operate with numbers, whereas humans are more comfortable dealing with images. Analytic geometry provides the bridge between these two worlds. An image, picture, or diagram is made up of geometrical objects: points, lines, curves. The geometry must be translated into numbers for a computer to store a diagram and make modifications to it. Whatever properties of the diagram are of interest—areas, intersection points, or lengths, for example—can then be calculated.

Architects use computer programs that consider all aspects of a building, such as wall locations, doors, windows, wiring, and piping, in three dimensions. Before computers, these data were stored in geometrical form in the plans of the building. Analytical geometry allows the translation of the spatial description of the building into numbers. In addition to calculating simple things such as areas and lengths, a computer program can show a perspective view of the building, calculated from the plans. The final step in this procedure is the translation from numbers back to geometrical objects. Similarly, a spacecraft follows a trajectory in three dimensions. The trajectory is computed based on physical laws that govern the motion of all objects; the computations provide numbers in the form of the coordinates of the spacecraft, which are interpreted according to the concepts of analytic geometry.

Context

The concepts of geometry—points, lines, triangles, circles, and the like—and their various relationships are visualized as diagrams on paper, although it should be noted that physical diagrams are approximations to the abstract geometrical concepts. A point has no size, only position, while a line has no width, only length, unlike the penciled dots and lines of a diagram. Nevertheless, it was probably the reverse thought process that, many centuries ago, transformed the ropes and posts of surveyors into the abstract concepts of lines and points. The practical relationships observed while surveying thus probably started the development of geometry as an abstract study. The word "geometry" is derived from Greek words meaning, literally, "earth measurement."

The definition of a point does not provide a quantifiable description of its position. Geometrical arguments require only that the position exists. Lines have quantifiable properties: length and direction. A surveyor might describe the boundaries of a piece of land thus: "From the start, 100 yards northwest to a point, 200 yards northeast to the large oak, 150 yards southeast to a point, 150 yards southwest to the stream, 50 yards northwest to a point, and finally 50 yards southwest to the starting point." The line segments are easily described, but the location of points are referred to by landmarks. It is simple from this description to calculate the perimeter of the property; what, however, is the area? The reader will probably find it necessary to translate the description into a geometrical diagram to find that the area is 27,500 square yards.

Analytic geometry allows the description of geometrical figures more complex than the example above, entirely in terms of numerical quantities. All the properties of the figure can then be obtained directly by calculation. The crucial step is in developing a method to describe the position of a point. The powerful tools of algebra are then available to develop relationships between numerical quantities and the geometrical objects they describe.

The invention of analytic geometry is attributed primarily to the seventeenth century philosopher and mathematician René Descartes. As with all inventions, the idea did not spring fully-formed from his mind; many of the ideas and threads of argument had existed through the centuries. An excellent review of the historical development of analytic geometry is given by Carl B. Boyer in the article referenced below.

Principal terms

algebra: the study of properties and relationships between numerical quantities

axis: a fixed reference line used in defining the coordinates of a point

coordinate: one of a set of numbers all of which together describe the location of a unique point

equation: the statement of a relationship between numerical quantities, usually presented in symbolic form

geometry: the study of the properties and relationships of objects in space

origin (of coordinates): a fixed reference point from which all coordinates are defined

plane: an abstract geometrical concept that can be visualized as a "flat sheet"

point: an abstract geometrical concept with only a single property, position

square (of a number): the result of multiplying a number by itself, usually shown symbolically by a small superscript "two"; for example, the square of four is sixteen (4 × 4 = 16, or 4² = 16)

Bibliography

Abbott, Edwin A. Flatland: A Romance of Many Dimensions. New York: HarperPerennial, 1983. This small work of mathematical fiction, first published in 1884, describes life in a one-dimensional and a two-dimensional world, and further, how a three-dimensional visitor to the world would appear. It provides a useful and enjoyable mental exercise for coming to terms with the concept of dimensions.

Boyer, Carl B. "The Invention of Analytic Geometry." Scientific American, January, 1949, 40-45. This is a historical review tracing the development of what is now known as analytic geometry, from the Greeks and earlier up to Descartes and Pierre Fermat.

Cleveland, William S. The Elements of Graphing Data. Monterey, Calif.: Wadsworth Advanced Books and Software, 1985. This readable book concerns itself with the many considerations involved in presenting numerical data in the form of graphs—one of the applications of analytic geometry.

Descartes, René. "The Geometry." In The World of Mathematics 1. New York: Simon & Schuster, 1956. This is a facsimile, with translation on the facing page, of the first eight pages of the first edition of Descartes' seventeenth century work La Géométrie. Preceded by a commentary on Descartes and analytic geometry.

Kline, Morris. Mathematics in Western Culture. London: Oxford University Press, 1976. Chapter 12 of this fine book introduces the reader to René Descartes and gives a solid introduction to analytic geometry. The rest of the book is also highly recommended.

Paulos, John Allen. Beyond Numeracy. New York: Alfred A. Knopf, 1991. The fourth in this set of thought-provoking essays is entitled "Analytic Geometry." It provides another introductory view of the subject. The other essays are also worth the reader's time.