III.TRACING OF PARAMETRIC III.TRACING OF PARAMETRIC CURVESCURVES

To trace a cartesian curve defined by To trace a cartesian curve defined by the parametric equations x = f(the parametric equations x = f(), y ), y = g(= g(), we use the following ), we use the following properties.properties.

If either x or y is a periodic function of If either x or y is a periodic function of with period T then trace the curve with period T then trace the curve in one period, say for in one period, say for [0, T]. [0, T].

Symmetry:Symmetry:

(i)(i) About X-axis if f(-About X-axis if f(-) = f() = f(), g(-), g(-) = - g() = - g())

(ii)(ii) About Y-axis if f(-About Y-axis if f(-) = - f() = - f(), g(-), g(-) = g() = g())

(iii)(iii) About Opposite Quadrants ifAbout Opposite Quadrants if

f(-f(-) = f() = f(), g(-), g(-) = g() = g())

Region : Find the region for Region : Find the region for for which x for which x and y are well defined.and y are well defined.

Tabulation: Tabulate the values of x and y Tabulation: Tabulate the values of x and y for selected values of for selected values of ..

Derivatives : Find dy/dx defined byDerivatives : Find dy/dx defined by

( / ) '( )

( / ) '( )

dy dy d g

dx dx d f

The tangents to the curve at different points The tangents to the curve at different points can be determined by evaluating dy/dx at can be determined by evaluating dy/dx at that point.that point.

If dy/dx >0, in an interval then the curve If dy/dx >0, in an interval then the curve increases at that interval.increases at that interval.

If dy/dx < 0, then the curve decreases.If dy/dx < 0, then the curve decreases.

If dy/dx = 0, the curve has a stationary point.If dy/dx = 0, the curve has a stationary point.

Example : Trace the curve Example : Trace the curve x = a(x = a( - sin - sin ), y = a(1-cos ), y = a(1-cos ); 0 ); 0 ≤ ≤ ≤ ≤ 22;a ;a

>0.>0.

Note that f(-Note that f(-) = -a() = -a( - sin - sin )= - f( )= - f(); ); g(-g(-) = a(1-cos ) = a(1-cos ) = g( ) = g().). Therefore the curve is symmetric about the Therefore the curve is symmetric about the

y-axis. y-axis.

Also y is a periodic function of Also y is a periodic function of with period 2 with period 2. . It is sufficient to trace the curve for It is sufficient to trace the curve for [0, [0, 22]. ].

For For [0, 2 [0, 2], x and y are well defined. ], x and y are well defined. Note that y Note that y ≥ 0. Entire curve lies above the Y-≥ 0. Entire curve lies above the Y-

axis.axis.

/ sin

/ (1 cos )

dy dy d a

dx dx d a

2

2sin( / 2)cos( / 2)cot( / 2)

2sin ( / 2)

00 22x =a(x =a( - sin - sin ) )

y = a(1-cos y = a(1-cos ) )

dy/dxdy/dx 0

0

0 0

a

2a

2a

At At = 0, dy/dx = = 0, dy/dx = . Tangent to the curve . Tangent to the curve at at = 0 is perpendicular to x-axis. = 0 is perpendicular to x-axis.

At At = =, dy/dx = 0. Tangent to the curve is , dy/dx = 0. Tangent to the curve is parallel to x-axis at parallel to x-axis at = = ..

At At =2 =2, dy/dx = , dy/dx = . Tangent to the curve is . Tangent to the curve is again perpendicular to x-axis at again perpendicular to x-axis at = 2 = 2..

For 0<For 0<<<<< , , dy/dx > 0.dy/dx > 0.

Therefore the function is increasing in this Therefore the function is increasing in this interval.interval.

For For <<<2<2, , dy/dx <0. Therefore the function dy/dx <0. Therefore the function is decreasing in this interval.is decreasing in this interval.

cot( / 2)dy

dx

X

Y

(a,0) (2a,0)(0,0)

(a, 2a)

The curve is called a cycloid

2. Trace the following curve with explanation.2. Trace the following curve with explanation.

x = a(x = a( + sin + sin ) , y = a(1 - cos ) , y = a(1 - cos ); -); - ≤ ≤ ≤ ≤ ; a >0.; a >0.

Note that f(-Note that f(-) = ) = -a(-a( + sin + sin )= - f()= - f(), ),

g(-g(-) = a(1 - cos ) = a(1 - cos )) = = g(g().).Therefore the curve is symmetric about the Therefore the curve is symmetric about the y-axisy-axis..

Also y is a periodic function of Also y is a periodic function of with period 2 with period 2. It is . It is sufficient to trace the curve for sufficient to trace the curve for [- [- , , ]. ].

For For [- [- , , ], x and y are well defined. Note that y ], x and y are well defined. Note that y ≥ 0. ≥ 0. Entire curve lies above the Y-axis.Entire curve lies above the Y-axis.

/ sin

/ (1 cos )

dy dy d a

dx dx d a

2

2sin( / 2)cos( / 2)tan( / 2)

2cos ( / 2)

- - 00 x =a(x =a( + sin + sin ) )

y = a(1-cos y = a(1-cos ) )

dy/dxdy/dx 0

a

2a

a

2a

0

0

For 0<For 0<<<<< , , dy/dx > 0 .dy/dx > 0 .

Therefore the function is increasing in this Therefore the function is increasing in this interval.interval.

For -For -<<<0<0, , dy/dx <0 . Therefore the function dy/dx <0 . Therefore the function is decreasing in this interval.is decreasing in this interval.

tan( / 2)dy

dx

X

Y

(a,0)(-a,0) (0,0)

(a, 2a)(-a, 2a)