Identify the curve defined by the parametric equations:

1. \(x = 2t-1, y=-t+4, t\in \R\)

2. \(x=\cos(2t), y=\sin(2t), t\in [0,\pi]\)

Defined as the locus (set) of points equidistant from a line (the directrix) and a point (the focus).

If the vertex is at \((0,0)\text{,}\) the focus is at \((0,p)\text{,}\) and the directrix is \(y=-p\text{,}\) then the parabola has equation

An ellipse is the set of all points \(P\) such that the sum of the distances from \(P\) to two points \(F_1,F_2\text{,}\) called the foci is constant.

If the centre is at \((0,0)\text{,}\) and the foci are at \((\pm c, 0)\text{,}\) the ellipse has equation

Vertices are at \((\pm a, 0)\) and \((0, \pm b)\text{.}\)

Sketch the ellipse \(\dfrac{x^2}{9}+\dfrac{y^2}{4}=1\text{.}\)

A hyperbola also has two foci. This time, we look for the set of all points where the difference of the distances between each point and the foci is constant.

Result:

• Usually we describe curves “explicitly”, as graphs of functions: \(y=f(X)\text{.}\)

• This is rather restrictive: lots of intesting curves are not graphs! (Like circles, ellipses, hyperbolas...)

• One option is defining curves implicitly using equations of the form \(f(x,y)=c\text{.}\)

• Another is to define them parametrically: we define \(x\) and \(y\) as functions of a third variable \(t\text{.}\)

• An example you've seen: \(x=\cos\theta\text{,}\) \(y=\sin\theta\text{.}\)

• From linear algebra: \(\langle x,y\rangle = \langle 2,3\rangle + t\langle 5,1\rangle\)

1. \(x=t^2-t, y=1-t^2, t\in [-3,3]\)

2. \(x=t^3-t+3, y=t^2+1, t\in [-2,2]\)

3. \(x=\cos(t), y=\sin(2t), t\in [0,\pi]\)

Sometimes we can gain insight on a parametric curve by converting back to an equation relating \(x\) and \(y\text{.}\)

Examples:

1. \(x = 3\cos(t)-1, y=2\sin(t)+3, t\in [0,2\pi]\text{.}\)

2. \(x=\cos(t), y=\cos(2t), t\in [0,\pi]\)

3. \(x=\dfrac{1}{2+t}, y = \dfrac{3t+9}{3+t}\)

Note: plotting curves on a computer often relies on being able to parametrize them. But (except for graphs) this isn't always easy!

A curve \(x=f(t), y=g(t)\) is considered smooth if \(f\) and \(g\) are both differentiable, and \(f'(t)\) and \(g'(t)\) are never simultaneously zero.

Some people will also require that a smooth curve has no self-intersections. Finding points of self-intersection is an algebraic nightmare.

Our definition of smooth curve is natural in one sense: it means we can find tangent lines!

Slope of the tangent is given by \(\dfrac{dy}{dx}\text{.}\) Chain rule gives:

If either \(x'(t)\) or \(y'(t)\) is undefined, we can't compute the slope of the tangent. If both are zero, the slope is indeterminate.

(If \(x'(t)=0\) but \(y'(t)\neq 0\text{,}\) we have a vertical tangent!)

(Aside:) a vector in the direction of the tangent line at \((x(t_0),y(t_0))\) is \(\langle x'(t_0), y'(t_0)\rangle\text{.}\)

Find the equations of the tangent lines as indicated:

1. \(x=t^2-1, y=t^3-t\text{,}\) at \(t=0\) and \(t=1\)

2. \(x=\tan(t), y=\sec(t)\text{,}\) at \(t=\pi/4\text{.}\)

We can compute area as usual. For area under a curve: \(A = \int_a^b f(x)\,dx\text{.}\)

If \(x=g(t), y=f(g(t))\text{,}\) get \(A = \int_{t_0}^{t_1} f(g(t))g'(t)\,dt\text{.}\)

This still makes sense for a general parametric curve if we're careful about interpretation.

For a closed curve \(x=g(t), y=h(t)\) with counterclockwise orientation, the area enclosed is

1. Find the area encolosed by the ellipse \(\di\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\text{.}\)

2. Find the area encolosed by the “teardrop” in the curve \(x=t^3-t, y=t^2-1\text{.}\)