Definition: A plane curve is a set C of ordered pairs (f(t), g(t)), where f and g are continuous functions on an interval I. The graph of C consists of all points P(t) = (f(t), g(t)) in the xy-plane, for t in I. If I is a closed intervel [a, b] and P(a) ≠ P(b), then P(a) and P(b) are called endpoints. If P(a) = P(b), then C is a closed curve. If P(a) = P(b) and C does not intersect itself at any other point, then C is called a simple closed curve Section 9.1: Continued

Example 1 A point moves in a plane such that its position P(x, y) at time t is given by Describe the motion of the point graphically.

Parametric equations of the form where are constants, occur in electrical theory. The graph of the following curve is called a Lissajous figure.

Definition A curve is smooth if it has a parametrization x = f(t), y = g(t) on an interval I such that the derivatives are continuous and not simultaneously zero, except possibly at endpoints. A curve is piecewise smooth if the interval I can be partitioned into closed subintervals where C is smooth on each subinterval. Note: a smooth curve will have no corners or cusps.

Cycloids A curve traced by a fixed point P on the circumference of a circle as the circle rolls along a line in a plane is called a cycloid. Cycloid Applet The parametric equations for a cycloid formed by a circle of radius a and moving from left to right are: The cycloid is piecewise smooth since it has cusps at points corresponding to