Polar coordinates.

In class Monday, we introduced the idea of polar coordinates.  In rectilinear coordinates (x, y), the first number tells us how far right (positive) or left (negative) we move on the horizontal axis and the second number tells us how far up (positive) or down (negative) we move on the vertical axis.

In polar coordinates we have (r, theta).  There is a central point, not unlike (0, 0) in the xy-axis system, and we consider that we are pointing in a direction we call angle 0 in radians.  The standard is to make that direction to the right, the same as the positive x-axis in rectilinear.  r is the distance from the origin and theta is the angle given in radians.

It is acceptable to have negative values for distance and for angles.

Unlike rectilinear coordinates, polar coordinates are not unique.  The easiest example of this are the coordinate pairs (0, 0) and (0, 1).  The first zero means we are a distance of 0 from the origin.  That means we are on the origin.  The angle we turn doesn't change where we are.

Even when points aren't the origin, the polar coordinates are not unique. The simplest example is

(1, 0) = (1, 2pi)

What this says is if we are 1 away from the origin and pointing to the right, this is the same as being 1 away from the origin and turning a full circle. In fact, adding or subtracting any multiple of 2pi to an angle brings us back to pointing in the exact same direction.

We will be working more in polar coordinates for the rest of the week, possibly longer.

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