Polar Coordinates

I.  Quiz

II.  Homework

III. Definition of Polar Coordinates

Recall that we define a point (x,y) on the plane as x units to the right of the origin and y units to the left of the origin.  This works great for lines and parabolas, but circles have somewhat messy equations.  As an alternative we define a new coordinate system where the first coordinate r is the distance from the origin to the point and the second coordinate theta is the angle that the ray from the origin to the point makes with the positive x axis.  

From trigonometry, we have

x = rcos theta

y = rsin theta

Your calculator has a special mode for polar coordinates.  We will use the calculator to graph

r = 5cos theta and r = cos(sqrt2 theta) and others.

IV.  Circles

A circle centered at the origin has equation

x² + y² = R²

In polar form we have

r = R

For example the circle of radius 3 centered at (0,0) has polar equation

r = 3

V.  Lines

if y = mx + b we can write

rsin(t) = mrcos(t) + b or

r = b/(sin(t) - mcos(t))

VI.  Conic Sections

Recall that a conic section is defined as follows:

Let f ( called the focus) be a fixed point in the plane, m (called the directrix) be a fixed line, and e (called the eccentricity) a positive constant.  Then the set of points P in the plane with

|PF|/|Pm| = e

isa conic section.

If e < 1 then the section is an ellipse, if e = 1, then its a parabola,and if e = 0 then it is a hyperbola.

Note that if F is the origin m is x = d then

|PF| = r, |Pm| = d - rcos(t) so that the equation becomes

r = e(d - rcos(t)) = ed - recos(t)

or

r = ed/(1 + ecos(t))