Section 11.1

Curves Defined by Parametric Equations

PARAMETRIC CURVES

Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations

x = f (t) y = g(t)

(called parametric equations). As t varies, the point (x, y) = ( f (t), g(t)) varies and traces out a curve C, which is called a parametric curve.

INITIAL AND TERMINAL POINTS OF A PARAMETRIC CURVE

If the parametric curve is given by

x = f (t) y = g(t) a ≤ t ≤ b,

where a and b are finite numbers, the point ( f (a), g(a)) is called the initial point and the point ( f (b), g(b)) is called the terminal point.

ORIENTATION OF A PARAMETRIC CURVE

Let

x = f (t) y = g(t)

be the parametric equations for a parametric curve. The direction in which the curve is traced for increasing values of the parameter t is called the orientation of the curve. When sketching a parametric curve, we draw arrows to indicate the orientation.

UNIQUENESS OF PARAMETRIC EQUATIONS

There can be more than one set of parametric equations for the same parametric curve. In this sense, parametric equations are not unique.

THE CYCLOID

The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid. See: http://www.ies.co.jp/math/java/calc/cycloid/cycloid.html

If the circle has radius r and rolls along the x-axis and if on position of P is the origin, then the parametric equations for this cycloid are

)cos1()sin( ryrx