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LAB 3 – PARAMETRIC & POLAR PLOTS Parametric Equations: If the coordinates (x, y) of a point P on a curve are given as functions x f (t ), y g (t ) of a third variable or parameter t, the equations x f (t ), y g (t ) are called parametric equations of the curve, and we say that the curve has been parametrized. The domain of the parameter, a t b , is the parameter interval. The point ( f (a ), g (a )) is the initial point of the curve, and ( f (b), g (b)) is the terminal point of the curve. EXAMPLES: 1. Circle Cartesian Equation: x 2 y 2 r 2 Parametric Equation: x r cos , y r sin , 0 2 2. Parabola Cartesian Equation: y ax 2 Parametric Equation: x t , y at 2 , t Cartesian Equation: x ay 2 2 Parametric Equation: x at , y t , t 3. Ellipse x2 y2 1 a2 b2 Parametric Equation: x a cos , y b sin , 0 2 Cartesian Equation: 4. Cycloid (A wheel of radius a rolls along a horizontal line without slipping. The path traced by a point on the wheel's edge is called a cycloid). Parametric Equation: x a( sin ), y a(1 cos ), EXERCISES: Sketch the following curves using Maple and indicate the direction in which the curve is traced as the parameter increases: 1. x 2 cos , y 2 sin , 0 2 2. x 1 2t , y t 2 4, 0 t 3 1 3. x 2 cos , y sin , 0 2 2 4. x 3 cos , y 1 sin , 0 2 5. x t sin 2t , y t sin 3t , 0 t 2 The graph of a Polar Equation r f ( ) consists of all points P that have at least one polar representation (r , ) whose coordinates satisfy the equation. The polar equation r f ( ) may also be written in parametric form x f ( ) cos , y f ( ) sin . Exercises: 8 1. Consider the equation r sin 5 a) How many rotations are required until the curve starts to repeat itself? b) Write the equations in parametric form. c) Use the polarplot command on Maple to plot the graph of the equation. 2. Repeat Execise 1. for the equation r cos 2 . Commands: with(plots): polarplot([f(t),g(t),t=a..b); OR plot([r(t),t,t = a..b],coords=polar);