Polar Coordinate Systems - 3D Math Primer for Graphics and Game Development - page 195

are some “special” places, like the origin, and the axes, but those are just
like marks on the bottom of the pan—the Jell-O itself is the same there
as everywhere else. But when you imagined the fabric of polar coordinate
space, something was different. Polar coordinate space has some “seams”
in it, some discontinuities where things are a bit “patched together.” In
the infinitely large circular pan of Jell-O, there are multiple sheets of Jell-O
stacked on top of each other. When you put your spoon down a particular
place to get a bite, you often end up with multiple bites! There's a piece of
hair in the block of Jell-O, a singularity that requires special precautions.
Whether your mental image of polar space was of Jell-O, or some other
yummy dessert, you were probably pondering some of these questions:

1. Can the radial distance r ever be negative?

2. Can θ ever go outside the interval [−180 o ,+180 o ]?

3. The value of the angle θ directly “west” of the origin (i.e., for points
where x < 0 and y = 0 using Cartesian coordinates) is ambiguous.
You may have noticed that none of these points are labeled in Fig-
ure 7.3. Is θ equal to +180 o or −180 o for these points?

4. The polar coordinates for the origin itself are also ambiguous. Clearly
r = 0, but what value of θ should we use? Wouldn't any value work?

The answer to all of these questions is “yes.” 3 In fact, we must face a

rather harsh reality about polar space.

For any given point, there are infinitely many polar coordinate pairs that
can be used to describe that point.

This phenomenon is known as aliasing. Two coordinate pairs are said
to be aliases of each other if they have different numeric values but refer to
the same point in space. Notice that aliasing doesn't happen in Cartesian
space—each point in space is assigned exactly one (x,y) coordinate pair;
the mapping of points to coordinate pairs is one-to-one.
Before we discuss some of the di culties created by aliasing, let's be
clear about one task for which aliasing does not pose any problems: inter-
preting a particular polar coordinate pair (r,θ) and locating the point in
space referred to by those coordinates. No matter what the values of r and
θ, we can come up with a sensible interpretation.

3 Even question 3.