Geometric Polar Plane
Figure 5-12 shows a variant of the polar coordinate plane on which the radial scale is graduated geometrically, rather than in linear fashion. The point corresponding to 1 on the r axis is halfway between the origin and the outer periphery, which is labeled ∞ (the “infinity” symbol). Succeeding integer points are placed halfway between previous integer points and the outer periphery. In this way, the entire polar coordinate plane is, in effect, portrayed inside an open circle having a finite radius.
Fig. 5-12. A polar coordinate plane with a “geometrically compressed” radial axis.
The radial scale of this coordinate system can be expanded or compressed by multiplying all the values on the r axis by a constant. This allows various relations and functions to be plotted, minimizing distortion in particular regions of interest. Distortion relative to the conventional polar coordinate plane is greatest near the periphery, and is least near the origin.
This “geometric axis compression” scheme can also be used with the axes of rectangular coordinates in two or three dimensions. It is not often seen in the literature, but it can be interesting because it provides a “view to infinity” that other coordinate systems do not.
Mathematician’s Polar Vs Cartesian
Figure 5-13 shows a point P = ( x _{0} , y _{0} ) = ( θ _{0} , r _{0} ) graphed on superimposed Cartesian and polar coordinate systems. If we know the Cartesian coordinates, we can convert to polar coordinates using these formulas:
θ _{0} = arctan ( y _{0} / x _{0} ) if x _{0} > 0
θ _{0} = 180° + arctan ( y _{0} / x _{0} ) if x _{0} < 0 (for θ _{0} in degrees)
Fig. 5-13. Conversion between polar and Cartesian (rectangular) coordinates. Each radial division represents one unit. Each division on the x and y axes also represents one unit.
We can’t have x _{0} = 0 because that produces an undefined quotient in the conversion formula to θ _{0} . If a value of θ _{0} thus determined happens to be negative, you can add 360° or 2π rad to get the “legitimate” value.
Polar coordinates are converted to Cartesian coordinates by the following formulas:
x _{0} = r _{0} cos θ _{0}
y _{0} = r _{0} sin θ _{0}
These same formulas can be used, by means of substitution, to convert Cartesian-coordinate relations to polar-coordinate relations, and vice versa. The general Cartesian-to-polar conversion formulas look like this:
θ = arctan ( y / x ) if x > 0
θ = 180° + arctan ( y/x ) if x < 0 (for θ in degrees)
θ = π + arctan ( y/x ) if x < 0 (for θ in radians)
r = ( x ^{2} + y ^{2} ) ^{1/2}
The general polar-to-Cartesian conversion formulas are:
x = r cos θ
y = r sin θ
When making a conversion from polar to Cartesian coordinates or vice versa, a relation that is a function in one system is sometimes a function in the other system, but that is not always the case.
Compression and Conversion Practice Problems
Practice 1
Provide an example of a graphical object that can be represented as a function in polar coordinates, but not in Cartesian coordinates.
Solution 1
In polar coordinates, let θ represent the independent variable, and let r represent the dependent variable. Then when we talk about a function f , we can say that r = f(θ) . A simple function of θ in polar coordinates is a constant function such as this:
f (θ) = 3
Because f (θ) is just another way of denoting r , the radius, this function tells us that r = 3. This is a circle with a radius of 3 units.
In Cartesian xy -coordinates, the equation of the circle with radius of 3 units is more complicated:
x ^{2} + y ^{2} = 9
(Note that 9 = 3 ^{2} , the square of the radius.) If we let x be the independent variable and y be the dependent variable, we can rearrange the equation of the circle to get:
y = ±(9 – x ^{2} ) ^{1/2}
If we say that y = g ( x ) where g is a function of x in this case, we are mistaken. There are values of x (the independent variable) that produce two values of y (the dependent variable). For example, when x = 0, y = ±3. If we want to say that g is a relation, that’s fine, but we cannot call it a function.
Practice 2
Consider the point ( θ _{0} , r _{0} ) = (135°, 2) in polar coordinates. What is the ( x _{0} , y _{0} ) representation of this point in Cartesian coordinates?
Solution 2
Use the conversion formulas above:
x _{0} = r _{0} cos θ _{0}
y _{0} = r _{0} sin θ _{0}
Plugging in the numbers gives us these values, accurate to three decimal places:
x _{0} = 2 cos 135° = 2 × (–0.707) = –1.414
y _{0} = 2 sin 135° = 2 × 0.707 = 1.414
Thus, ( x _{0} , y _{0} ) = (–1.414, 1.414).
Practice problems for these concepts can be found at: Polar Coordinates Practice Test