Step By Step Calculus » 12.1 - Introduction to Parametric Curves

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A parametric curve in the Cartesian plane is a curve consisting of those points (x, y)(x, y) satisfying a parametric equation :
\displaystyle x\displaystyle x
\displaystyle =\displaystyle =
\displaystyle x(t), \qquad y=y(t) \displaystyle x(t), \qquad y=y(t)
Each point (x, y)(x, y) on the curve is expressed in terms of another variable tt called a parameter. Each value of the parameter tt gives a point (x, y)(x, y) on the curve; so one can plot the curve by tracing out (x, y)(x, y) as tt changes.
In general, the parameter tt covers an interval, perhaps infinite. If t \in [a, b]t \in [a, b], then the parametric curve x=x(t), \; y=y(t), \; a\leq t\leq bx=x(t), \; y=y(t), \; a\leq t\leq b has an initial point(x(a),y(a))(x(a),y(a)) and a terminal point(x(b),y(b))(x(b),y(b)). When these two points coincide, we have a closed curve.
We use the following TIPS procedure to sketch a curve represented by parametric equations.
(Tabulate) We tabulate a few points.
(Implicit Equation?) We find an implicit equation if one is obvious. A known form of implicit equation can give us the idea of the shape of the curve.
(Plot) Using the information from the implicit equation and the tabulated points, we sketch the curve.
(Sign) We label increasing tt with arrows. We do this from the tabulated points obtained in Step Tabulate.
A parametric curve intersects itself if there exist t_1t_1 and t_2 t_2 such that x(t_2)=x(t_1)x(t_2)=x(t_1) and y(t_2)=y(t_1)y(t_2)=y(t_1), that is, you get the same coordinates for the two distinct values of tt. It then intersects itself at the point (x(t_1),y(t_1))= ( x_(t_2), y(t_2))(x(t_1),y(t_1))= ( x_(t_2), y(t_2)).
Parametric representation of a curve is not unique. That is, we may come up with different possible representations for the same curve.