Intro to polar coordinates
To understand polar coordinates, we must first think of the most fundamental and well known
coordinate system  the
Cartesian plane. In this plane, points are represented by a unique pair of coordinates along a x (
horizontal) and y (
vertical) axis. For example,
y

 .(3,2)

x
O


The drawing above represents a typical point in the
Cartesian Plane. The
coordinate pair (3, 2) means the point is 3 units to the right on the xaxis and 2 units up the yaxis. The
O represents the point (0, 0) which is referred to as the
origin.
To define
coordinates in the
polar plane though, we first fix an origin
O (like in the Cartesian Plane) called the
pole and an initial
ray from that origin
O.



> Initial ray (θ = 0)
O


Thus, each point
P can be located by assigning it a polar coordinate pair (
r ,
Θ). In this pair,
r gives the directed distance from
O to
P and
Θ gives the directed angle from the initial ray to ray
OP.
To complicate matters further, each point
P can be represented in infinitely many ways, unlike in the Cartesian plane. Since
Θ is positive when measured
counterclockwise and negative when measured
clockwise, the
angle associated with a given point is not
unique.
Thus, (2, π/6) can be represented as (2, 11π/6) or generally as (2, π/6 + 2kπ) or (2, 11π/6  2kπ )
Converting Polar Coordinates to Cartestian coordinates
When we use both polar and Cartesian coordinates in a plane, we place the two origins
O together and take the polar initial ray as the positive
xaxis. Therefore, the ray Θ = π/2 becomes the positive
yaxis. The two coordinate systems are then related through the following equations.
x = r cos Θ
y = r sin Θ
x ² + y ² = r ²
Thus, Polar (r, Θ) = Cartesian (r cos Θ, r sin Θ)
We can use these equations and some
trigonometry/
algebra to rewrite polar coordinates and polar equations into Cartesian coordinates and equations and viceversa.
Common Polar Graphs
1.) Θ = k
This equation represents a
line through the
origin that follows the
angle from the initial ray of k.
2.) r = k
This equation represents a
circle with center
O and of
radius k. Its Cartesian equivalent is x ² + y ² = k ²
3.) A ± B (sin or cos) Θ
This formula has four
subsets, depending on the
absolute value of the
ratio of A to B ( A / B ). The
sine and
cosine functions effect the
axis of symmetry (cosine yields
symmetry with the xaxis, sine yields symmetry along the yaxis).
3.1) If  A / B  < 1
This polar graph is called a
limaçon with a loop or a looped limaçon. It has an inner loop when graphed on the polar plane. The word limaçon comes from the
French word meaning "
snaillike".
3.2) If  A / B  = 1
This polar
curve is named a
cardioid because of its
heart shape. If B is positive, the
dimple of the heart is in the negative half of the plane, whereas if B is negative the dimple lies in the positive half of the plane. The inward point of the heart is at the origin.
3.3) If 1 <  A / B  < 2
The equation above represents a dimpled limaçon. The dimple comes from the cardioid shape but because A is slightly larger, the dimple does not reach the origin like in the case of a cardioid.
3.4) If  A / B  ≥ 2
Because of the even larger A, the dimple disappears completely and the graph becomes a convex limaçon. Where the dimple would be in the case of a cardioid and the dimpled limaçon, there is now a horizontal or vertical line.
Calculus of Polar Curves
Slope
The slope of the
tangent line to a polar curve r = ƒ(Θ) is given by
dy /
dx, not by r' =
dƒ /
dΘ. To see why, think of the graph of ƒ as the graph of the
parametric equations:
x = r cos Θ = ƒ(Θ) cos Θ
and
y = r sin Θ = ƒ(Θ) sin Θ
(Hereafter referred to as "the above equations")
If ƒ is a
differentiable function of Θ, then so are x and y and when
dx /
dΘ ≠ 0, we can calculate
dy /
dx from the
parametric formula for slope:
dy dy / dΘ
 = 
dx dx / dΘ
d/dΘ (ƒ(Θ) sin Θ)
= 
d/dΘ (ƒ(Θ) cos Θ)
dƒ/dΘ sin Θ + ƒ(Θ) cos Θ
=  (Product Rule)
dƒ/dΘ cos Θ  ƒ(Θ) sin Θ
Therefore,
dy ƒ'(Θ) sin Θ + ƒ(Θ) cos Θ
 = 
dx ƒ'(Θ) cos Θ  ƒ(Θ) sin Θ
Area in Polar Coordinates
In a
region bounded by the rays θ = α and θ = β and the curve r = ƒ(θ). We can then
approximate the region with
n nonoverlapping circular
sectors based on a
partition P of the angle (β  α). The typical sector has radius r = ƒ(θ) and a central angle of
radian measure ΔΘ.
Its area is:
1 1
A =  r²ΔΘ =  (ƒ(Θ))²ΔΘ
2 2
Thus, the area of our region is approximately
n
 1
\ _ ƒ(Θ ))²ΔΘ
A = / k
 2
k=1
If ƒ is
continuous, we expect the approximations to improve as n > ∞, and we are led to the following formula for the region's area:
n
 1
A = lim \ _ ƒ(Θ ))²ΔΘ
n→∞/ k
 2
k=1
Using the
Riemann Sum Definition, this turns into:
α 1
A = ∫  r² dθ
β 2
Length of a Polar Curve
We can obtain a polar coordinate formula for the length of a curve r = ƒ(θ), where α ≤ θ ≤ β, by
parametrizing the curve using the same method as above for the slope as well as the area. Then, by
substituting these
formulae into the parametric length formula:
β _________________________
L = ∫ √ (dx / dθ)² + (dy / dθ)² dθ
α
we get:
β _________________
L = ∫ √ r² + (dr / dθ)² dθ
α
When the parameterized equations are substituted for x and y.
Nota Bene: This only is
valid if ƒ(θ) has a
continuous first derivative for α ≤ θ ≤ β and if the point
P(r, θ)
traces the curve r = ƒ(θ) exactly
once as θ runs from α to β.
Area of a Surface of Revolution
Once again, we parametrize the curve ƒ(θ) with the above equations and substitute them into the parametric surface area equation, yielding:
Revolution about the xaxis (y ≥ 0):
β ________________
S = ∫ 2πr sin θ √ r² + (dr / dθ)² dθ
α
Revolution about the yaxis (x ≥ 0):
β ________________
S = ∫ 2πr cos θ √ r² + (dr / dθ)² dθ
α
Nota Bene: This also only is valid if ƒ(θ) has a
continuous first derivative for α ≤ θ ≤ β and if the point
P(r, θ) traces the curve r = ƒ(θ) exactly
once as θ runs from α to β.