Curve

Informally, one can picture a curve as either a line, a line segment, or a figure obtained from a line or a line segment by having the line or line segment bent, stretched, or contracted in any way. A plane curve, such as a circle, is one that lies in a plane; a curve in three dimensional space, such as one on a sphere or cylinder, is called a skew curve.

Plane curves are frequently described by equations such as y = f(x) or F(x,y) = 0. For example, y = 3x + 2 is the equation of a line through (1,5) and (2,8) and x² + y² 9 = 0 is the equation of a circle with center at (0,0) and radius 3. Both of the curves described by the equations y = 3x + 2 and x² + y² - 9 = 0 are examples of algebraic curves. On the other hand, a curve described by an equation such as y = cos x is an example of a transcendental curve since it involves a transcendental function.

Another way of describing a curve is by means of parametric equations. For example, the parabola y = x² can be described by the parametric equations x = t, y = t² and the helix by x = a cos θ, y = a sin θ, I = b θ.

A closed curve is a curve with no endpoints such as a circle or a figure eight. A curve that does not cross itself is a simple curve. So a circle is a simple closed curve whereas a figure eight is closed but not simple. Figure 1 shows another simple closed curve, a curve that is simple but not closed, and a curve that is closed but not simple.

Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group.