Week 2

Polar Coordinates

• To avoid confusion, for polar coordinates, we use square brackets: [,] while for Cartesian coordinates we use round brackets.

• A point P in the plane having Cartesian coordinates (x,y) is represented by the pair [r,θ] in polar coordinates if and only if x = rcos θ and y = rsin θ. For example, the pair [2,π/3] in polar coordinates and (1,√3) in Cartesian coordinates represent the same point in the plane. θ is the angle between the OP line and the x-axis.
• There is an ambiguity caused by the fact that the real number r is allowed to be negative as well (however, its absolute value is always the length of OP: |r|=|OP| (r²=x²+y²). For example, [2,π/3] and [-2,4π/3] determine the same point, which has Cartesian coordinates (1, √3). In general, [r, θ] ≡ [-r, θ+ π], i.e. they determine the same point, since: cos (θ+ π)= - cos θ and sin (θ+ π)= - sin θ.
• If one has to determine *all* possible representations of a given point P in Cartesian coordinates, we proceed as follows: i) first we determine the “natural” representation [r, θ], where r = |OP| (r is positive here !) and θ = arctan (y/x), and from here take all possible [r, θ+2nπ] for all integers n; ii) we then allow the ambiguity (take “r” to be negative), since we know P is also represented by [-r, θ + π], so that all “other” possible representations in polar coordinates are [-r, θ + π + 2nπ].

Example: for P (1,-1), the natural representation (involving r positive) has r = |OP|=√2, while θ = arctan (-1)

= -π/4. This gives the following set o possible representations of P in polar coordinates are [√2, - π/4+2nπ];

the “other” not-so-natural (when we allow “r” to be negative) representations are [-√2, - π/4+ π +2nπ],

i.e. [-√2,3π/4+2nπ ].

• Polar coordinates help us represent (i.e. to give a mathematical equation for) certain curves (or shapes) that are otherwise hard to describe using plain old Cartesian coordinates. Some of these shapes are very good mathematical models of real-life objects.

Examples: Archimede’s spiral, the cardioid (heart shape), the limaçon, etc..