Parametric Equations

Overview of Chapter 11

• new ways to define curves in the plane (on the $y$-axis)

• polar coordinates using $r$ and $\theta$ $(r\cos \theta, r\sin \theta)$

• geometric definitions and standard equations of parabolas, ellipses, and hyerperbolas

Parametric Equations: expressing both $x$ and $y$ as functions of $t$

If $x$ and $y$ are given as functions $$x=f(t),\qquad y=g(t)$$ over an interval $I$ of $t-$values, then the set of points $(x,y)=(f(t),g(t))$ defined by these equations is a parametric curve. The equations are parametric equations.

• $t$: parameter for the curve

• $I$: parameter interval

• If $I=[a,b]$: $(f(a),g(a))$ and $(f(b),g(b))$ are the initial point and the terminal point of the curve.

Example:

$x=t^2,~y=t+1$ for $t\in R$

A given curve can be represented by different parametrizations. ($x=t^2-2t+1,~y=t$ for $t\in R$)

Cycloids

A wheel of radius $a$ rolls along a horizontal straight line. Find parametric equations for the path traced by a point $P$ on the wheel's circumference. The path is call a cycloid.

The problem with a pendulum clock whose bob swings in a circular arc is that the frequency of the swing depends on the amplitude of the swing. The wider the swing, the longer it takes the bob to return to center (its lowest position).

• This does not happen if the bob can be made to swing in a cycloid. In 1673, Christian Huygens designed a pendulum clock whose bob would swing in a cycloid.

There is a toy called the Spirograph that lets you draw interesting curves using a collection of wheels. We can produce these pictures using Sage.

Experiment with different values of $a$ and $b$. If the curve looks incomplete, then increase tmax.

For example, try $a=21,~1/2,~\sqrt{2}$

Tangents and Areas

A parameterized curve $(x,y)=(f(t),g(t))$ is differentiable at $t$ if $f$ and $g$ are differentiable at $t$. If at a point on a differentiable parameterized curve, $y$ is also a differentiable function of $x$, then we have the Chain Rule: $${dy\over dt}={dy\over dx}\cdot{dx\over dt}\Rightarrow {dy\over dx}={dy/dt \over dx/dt}$$

Example

$(x,y)=(\sec t, \tan t)$ at the point $(\sqrt{2},1)$

Find the area enclosed by the asteroid with parameter $t$. $$A=4\int_0^1 y(x)dx$$

Since we have $dx = x'(t)dt$, we need to find the integral region for $t$.

Length of a Parametrically Defined Curve}

If a curve $C$ is defined parametrically by $x=f(t)$ and $y=g(t)$, $a\leq t\leq b$, where $f'$ and $g'$ are continuous and not simultaneously zero on $[a,b]$, and $C$ is traversed exactly once as $t$ increases from $t=a$ to $t=b$.

The length of the line segment from $t$ to $t+dt$ is $$\begin{align*}&\sqrt{(x(t+dt)-x(t))^2 +(y(t+dt)-y(t))^2}\approx & \sqrt{(x'(t))^2 +(y'(t))^2}dt\end{align*}.$$

Then the length of $C$ is the definite integral

Example

Find the length of $(x,y)=(r\cos t, r\sin t)$ for $0\leq t\leq 2\pi$

Example

Find the perimeter of the ellipse ${x^2\over a^2}+{y^2\over b^2}=1$.

We can have the parameterization $x(t)=a\cos(t)$ and $y(t)=b\sin (t)$ for $0\leq t\leq 2\pi$.

Find the centroid of the first quadrant arc

We have $$ds= \sqrt{x'(t)^2 + y'(t)^2} dt=3|\sin t\cos t|dt=3\sin t\cos tdt$$ The $x$ coordinate of the centroid is $${\int_0^{\pi/2} x(t) ds\over\int_0^{\pi/2} 1ds}={\int_0^{\pi/2} \cos^3 t \times 3\sin t\cos t dt\over\int_0^{\pi/2} 1dt}$$

Areas of Surfaces of Revolution

We can think about concentrating all the weight on to the graph in 2D. Note that the weight depends on the length of the circle.

Calculate the area of the surface of revolution swept out by this parametrized curve $(\cos(t),~\sin(t)+1)$ for $t\in(0,2\pi)$.