# Analytic Geometry

## Three-dimensional Coordinate Systems And Beyond

Geometric figures such as points, lines, and conics are two-dimensional because they are confined to a single plane. The term two-dimensional is used because each point in this plane is represented by two real numbers. Other geometric shapes like spheres and cubes do not exist in a single plane. These shapes, called surfaces, require a third dimension to describe their location in space. To create this needed dimension, a third axis (traditionally called the z-axis) is added to the coordinate system. Consequently, the location of each point is defined by three real numbers instead of two. For example, a point defined by the coordinates (2,3,4) would be located 2 units away from the x axis, 3 units from the y axis, and 4 units from the z axis.

The algebraic equations for three-dimensional figures are determined in a way similar to their two-dimensional counterparts. For example, the equation for a sphere is x2 + y2 + z2 = r2. As can be seen, this is slightly more complicated than the equation for its two-dimensional cousin, the circle, because of the additional variable z2.

It is interesting to note that just as the creation of a third dimension was possible, more dimensions can be added to our coordinate system. Mathematically, these dimensions can exist, and valid equations have been developed to describe figures in these dimensions. However, it should be noted that this does not mean that these figures physically exist and in fact, at present they only exist in the minds of people who study this type of multidimensional analytic geometry.

## Resources

### Books

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Paulos, John Allen. Beyond Numeracy. New York: Alfred A. Knopf Inc., 1991.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.

Perry Romanowski

## KEY TERMS

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Abscissa

—The x-coordinate of a point representing its horizontal distance away from the origin.

Conic

—A geometric figure created by a plane passing through a right circular cone.

Coordinate system

—A system that relates geometric point s to real numbers based on their location in space relative to a fixed point called the origin.

Directrix

—A line which, together with a focus, determines the shape of a conic section.

Ellipse

—An eccentric or elongated circle, or oval.

Focus

—A point, or one of a pair of points, whose position determines the shape of a conic section.

Hyperbola

—A conic section created by a plane passing through the base of two cones.

Intercept

—The point at which a curve meets the x or y axes.

Linear equations

—A mathematical equation which represents a line.

Locus

—The set of all points that make an equation true.

Ordinate

—The y-coordinate of a point representing its vertical distance away from the origin.

Pythagorean theorem

—An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is used to find the distance between two points.

Slope

—Slope is the ratio of the vertical distance separating any two points on a line, to the horizontal distance separating the same two points.