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 x-axis and 2 units up the y-axis. 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 x-axis. Therefore, the ray Θ = π/2 becomes the positive y-axis. 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 vice-versa.
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 x-axis, sine yields symmetry along the y-axis).
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 "snail-like".
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 non-overlapping 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 x-axis (y ≥ 0):
β ________________
S = ∫ 2πr sin θ √ r² + (dr / dθ)² dθ
α
Revolution about the y-axis (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. |
