09 Parametric Equations - Handout.pdf

Parametric Curves

Institute of Mathematics University of the Philippines-Diliman

What are parametric equations?

What are parametric equations?

The curve cannot be expressed as the graph of an equation of the form y₌f(x).

Sometimes, it’simpossibleto find an equation relating thex₋and y₋coordinates of the particle.

But we canexpressthex₋and they₋component of the particle’s position as functions of timet, i.e.

x ₌ x(t)

y = y(t)

In this process, we say that we areparametrizingthe curve above.

The equations definingxandyas functions oftare calledparametric

equations, andtis called aparameter.

Parametric Curves

Example

Sketch the curve defined byC :x=t,y=t2, wheret∈R.

t _→ (x,y)

−2 _→ (₋2, 4) −1 _→ (₋1, 1) 0 _→ (0, 0) 1 _→ (1, 1) 2 _→ (2, 4) 3 → (3, 9)

Indeed, if we relatexandydirectly by eliminatingt, we havey₌x2_.

Parametric Curves

Example

Sketch the curve defined byx₌costandy₌sint, where 0_Ét_É2π.

t _→ (x,y)

0 → (1, 0)

π

4 → (

p 2 2,

p 2 2 ) π

2 → (0, 1)

π → (−1, 0)

3π

2 → (0,−1)

2π _→ (1, 0)

Indeed, if we relatexandydirectly, eliminatingt, we havex2+y2=1. The direction of trace is known as thedirection of increasing parameter. In this case, it’s counterclockwise.

The point corresponding tot₌0 is theinitial point. The point corresponding tot₌2_πis theterminal point.

Parametric Curves

Remark

The functiony=f(x) can be parametrized as

½

x ₌ t

y ₌ f(t)

Example

Parametrize the line connecting the points (0, 1) and (3, 7).

Using the two-point form of a line,

y₋1₌7−1

3−0(x−0) ⇔ y=2x+1

Hence, we can parametrize the line as

½

x ₌ t

Examples

Parametrization of circles...

x = sint

y ₌ cost

0Ét_É2π

’clockwise’ circle

x = 2 cost

y ₌ 2 sint

0ÉtÉ2π

’counterclockwise’ circle of radius 2 units

Examples

x = h+rcost

y ₌ k₊rsint

0ÉtÉ2π

circle of radiusrcentered at (h,k)

x = cos 2t

y ₌ sin 2t

0Ét_Éπ

Interesting Examples

x ₌ sin 2t

y = sin 3t

Lissajous curves

x ₌ cost₊0.5 cos 7t₊

µ₁

3

¶

sin 17t

y ₌ sint₊0.5 sin 7t₊

µ₁

3

¶

cos 17t

0_Ét_É2π

Examples

Consider the curve defined by

x₌sect

y₌tant

−π 2 <t<

π 2.

Note that

x2₌sec2t

y2₌tan2t Thus,

Exercises

1 Describe the curve given by the parametric equationsx₌2 costand

y₌3 sint.

2 GivenA(2, 0) andB(₋1, 1). Give a parametrization for 1 the line containgAandB.

2 the line segment connectingAandB.

3 _{What can be said about the parametric equationsx=cost,y= −sint}

andx=cos 2t,y= −sin 2t?

Slope of Parametric Curves

Smooth Curve

A parametric curveC:x=f(t),y=g(t) is said to besmoothiff andgare differentiable and no value oftsatisfiesf0(t)=g0(t)=0.

IfCis a smooth curve, then its slopedy_dxis given by

dy dx=

dy dt dx dt

Also, the concavityd2y

dx2 is given by

d2y dx2=

d³dy_dx´

dt dx dt

6=y

00_(t)

Tangent Lines to Parametric Curves

Example

Find the equation of the tangent line to the right branch of the hyperbola x=sect,y=tant, where−π₂<t<π₂, at the point (p2, 1).

Solution: Note that the point (p2, 1) corresponds tot=π4. Thus,

dy dx=

dy/dt dx/dt=

sec2t

secttant=csct

dy dx

¯ ¯ ¯

(p2,1)=csct ¯ ¯ ¯

t=π/4=

p 2

Hence, the equation is

y₋1₌p2³x₋p2´

y=p2x−1

Concavity

Example

Findd_dx2y2 ifx=t−t2,y=t−t3.

Solution: dy dx=

1−3t2 1₋2t Thus,

d2y dx2 =

d³1−3t2

1−2t ´

dt dx dt

=

(1−2t)(−6t)−(1−3t2₎₍₋₂₎ (1−2t)2

1₋2t

= 6t

2

−6t₊2 (1−2t)3

t<1₂ t>1₂

Caution

d2y dt2

d2x=

−6t −2 =3t6=

Length of a Parametric Curve

Length of Arc

LetC:x=x(t),y=y(t),aÉtÉb, be a smooth curve traced exactly once. Then the length ofCis given by

L₌ Zb a s µ_dx dt ¶2 + µ_dy dt ¶2 dt

Proof.The length of arc of the smooth curvey₌F(x), fromt₌atot₌b, is given by L= Z b a s 1+ µ_dy dx ¶2 dx.

Using differentials, we transform the integrand such that

s 1+ µ_dy dx ¶2 dx= v u u u

t1+

Ãdy dt dx dt !2 dx dt·dt=

s µ_dx dt ¶2 + µ_dy dt ¶2 dt.

Arc Length

Example

Find the length of theastroid

x=cos3t

y=sin3t 0ÉtÉ2π

L ₌ 8

Zπ/4 0

q

(₋3 cos2_tsint)2_{+(3 sin}2_tcost)2_dt

= 8

Zπ/4 0

3 costsint dt

= 12

Z π/4 0

sin 2t dt

= −6 cos 2t¯¯ ¯

π/4 0

Arc Length

Example

Set-up the necessary definite integral that will solve the following.

1 _{Perimeter of the circlex=4 cost,y=4 sint.} 2 Perimeter of the ellipsex₌2₊9 cost,y₌1₋4 sint. 3 _{Arclength of the curve}x₌₄et_,y₌₅₋t2_{, with}t_{∈[1, 3].}

Solution:

1

Z 2π 0

p

(−4 sint)2_{+(4 cost)}2_dt

2

Z 2π 0

p

(−9 sint)2_{+(−4 cost)}2_dt

3

Z 3 1

p

16e2t_+4t2_dt

Exercises

1 _{Find the slope of the curve with parametric equations}

x= 1 t2,y=t

3_{−5tat the point (1,−4).}

2 Find the concavity of the curvex₌t2₋4t,y₌5t3₊t₋3 whent₌2.

3 Set-up the necessary definite integral that will solve the following. 1 Perimeter of the hypocycloid

 



x = 2 cost−3 cos

³

2t 3

´

y = 2 sint+3 sin

³

2t 3

´

witht∈[0, 2π].